2004 2005 - nepero.net · AeN[4gL[ SbhV %$ ] USB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

�� ! "�!�

#$&%('*),+-)�./%('0'�1/2*3434$!5�6(7/898;:�<(=9=?>A@�6B=DCFEF<(G�=H6JILKH7(=D:MINC�GO>BPH>JI�>;=/Q

RTSBSBUWV*XYXYV*Z;[Y\M]^X_U2004/2005` UaXY[YS0bN[cZB[Yd�XYUfehgLU ` UfbLb�i gjgNVkRlinma[�gLbN]oSB]

prq 2*s�s�tu%n$v)-w(x,xDy

��

� �� ! i !#" U bhV%$�] U*SB[ i i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i i &! i � ' Uf\)(A] SAV*$Y]oUfSB[+(AVfeN]^X_[�S*bjeN]^XYV i i i i iJi i i i i i i i i i i iJi i i i i &! i , -�ehVfg/. Ufej\kV%$Y]oUfSA](V%0MSA]-i i i i i i i iJi i i i i i i i i i i iJi i i i i 1! i 2 3HeNU;Z;UfbLbjU4. ehV65f[YbLbNU*eN] i i i i i i i iJi i i i i i i i i i i iJi i i i i !87! i 9;: S*bjeNU;Z=<>$Y]oUfSB[ V dodo[ X�<Be?5f[i i i i i i iJi i i i i i i i i i i iJi i i i i !*!! i @ A�]CBnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[!Z;]�<ASAVMXE<Be?5�VriJi i i i i i i i i i i iJi i i i i ! �

! iGF;H�<BSJI�KB[�$8$�V Z;[YdodML VfejXYU i i i i i i i i iJi i i i i i i i i i i iJi i i i i ! 2! i & ' <BeN5�V bN<BehV i i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i i ! @! i 1 -�U*ejgN] U*SB[ i i i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i i ! F! i !O7QP gL[�\BB]Di i i i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i i ! F

! i !87 i ! m�BB]oejVfd [ doU*I*VfeN] bN\M]^XYV i i i i iJi i i i i i i i i i i iJi i i i i ! &! i !87 i �R' ]oXYd U*]oZ;[ i i i i i i i i i i iJi i i i i i i i i i i iJi i i i i ! &! i !87 i , P d ]^XYV�i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i i �=!

S;T4��VUW�X��ZY4[��*�\�]� S�^� i ! 3/U*d ]oSBU*\ ]9ZB]`_O[YejSAg bj[Y]oS i i i i i i iJi i i i i i i i i i i iJi i i i i � 9� i � ' <BeN5*[JZB]`_Za�$Y]o[Ye�i i i i i i i i i i iJi i i i i i i i i i i iJi i i i i � 1� i , ` [8Ifej[Y[ [Ydo[�5�V�bj] U*S i i i i i i i i i iJi i i i i i i i i i i iJi i i i ib,*,� i 2 R dDIfUfej] bN\MU ZB]�Z;[ ' VfgLbN[�d cLV*((Z;]D5a]ogN]oUfS�i i i i i iJi i i i i i i i i i i iJi i i i ib,]F� i @ ' <BeN5*[JZB]`_Za�$Y]o[Ye�ehV%$�] U*SAV do]�i i i i i iJi i i i i i i i i i i iJi i i i ib,*1

� i @Bi ! ' U0U*ejZB] SAV bN[ Uf\MU�If[YSA[Y[ i i iJi i i i i i i i i i i iJi i i i ib,*1� i @Bi � ' <BeN5*[ ejV*$Y]oUfSAVfd ] i i i i i i iJi i i i i i i i i i i iJi i i i ib,*1

� iGF;H9[ X_U*SB]oXdKA[i i iJi i i i i i i i i i iJi i i i i i i i i i i iJi i i i ie2�F� ifFai ! ' U0U*ejZB] SAV bN[g(AVfeN]^X_[�S*bjeN]^XdKB[ i iJi i i i i i i i i i i iJi i i i ie2�F� ifFai � A V%BJBAeN[4gL[�S*bhV%$�] U*SB[JZB]�XE<Be?5f[ X_U*SB]^XdKB[ i i i i i i i iJi i i i ie2�1

h T4��VUW�Xi8j�k�� ^�^,Ai !l' <BeN5*[+_Za8$Y]o[YeNm&gNBBdo] SB[ i i i i i i i i iJi i i i i i i i i i i iJi i i i in9*9,Ai � ' UfS0bN]oS�<B] bdo6If[YU*\M[_bNej]^XYV i i i i i i iJi i i i i i i i i i i iJi i i i ib@ �

,Bi � i ! ' UfS0bN]oS�<B] bdo_pa8$Y]o[YeNm&gNBBdo] SA[ i iJi i i i i i i i i i i iJi i i i ib@*,,Ai , qJ<BSJ$�] U*SB]�gNBBdo] SB[ i i i i i i i i i i iJi i i i i i i i i i i iJi i i i ib@�9

,Bi ,Bi ! _OV*gL[ Z;[�d do[rB(U bN[�SJ$Y[JbjeNU*SAXYV bN[�i i i i i i i i i i i iJi i i i ib@]F,Bi ,Bi � _OV*gL[ Z;[�d do[s_pm&gNBBdo] SA[ i i i i iJi i i i i i i i i i i iJi i i i ib@*&,Bi ,Bi , _tm gNBBd ]oSB[rI*[YSB[�ejVfd ]D$�$4V�bN[�i i iJi i i i i i i i i i i iJi i i i iuF �

,Bi ,Bi 2 ' <BeN5*[r_tm gNBBd ]oSB[ i i i i i i iJi i i i i i i i i i i iJi i i i iuFV2,Bi ,BiG9 ' <BeN5*[r_tm gNBBd ]oSB[ XdKB]D<AgN[ i i iJi i i i i i i i i i i iJi i i i iuF�@,Bi ,Bi @ R dDIfUfej] bN\MUkZ;]�Z;[r_OUaUfe i i iJi i i i i i i i i i i iJi i i i ib& 7,Bi ,BifF R dDIfUfej] bN\MUkZ;]`v SBUfb/m&] SAgN[YeNbN]oUfS�i i i i i i i i i i i iJi i i i ib& !

,

2 ��

l�E��O��Vjt��k��>�*�� *�\�W�� ^2Ai ! : S0bN[�eNB(Ufd^V%$�] U*SB[+BAV ehV \M[_bjeN]^XYV iJi i i i i i i i i i iJi i i i i i i i i &]92Ai � -F[�XYSB]oXdKA[cZ;] BAV ehV \M[YbNej]C$8$�V%$�] U*SB[�i i i i i i i i i i iJi i i i i i i i i &�@2Ai ,;: S0bN[�eNB(Ufd^V%$�] U*SB[+BnU*d ]oSBUf\M]^V do[ri iJi i i i i i i i i i iJi i i i i i i i i &�@

2ni ,Ai ! : S0bj[Ye?BnU*doV*$Y]oUfSB[ Z;]`HFV%IfehV S>If[ i i i i i i i iJi i i i i i i i i &�@2ni ,Ai � : S0bj[Ye?BnU*doV*$Y]oUfSB[ Z;] " [�� bNU*S i i i i i i i i iJi i i i i i i i i &�F

2Ai 2 : S0bN[�eNB(Ufd^V%$�] U*SB[ Z;]�� [�eN\M] bN[ i iJi i i i i i i i i i iJi i i i i i i i i &�F2AiG9l: S0bN[�eNB(Ufd^V%$�] U*SB[ X_UfS gNBBd ]oSB[

C1iJi i i i i i i i i i iJi i i i i i i i i 1 7

2ni 9Bi ! -F[�XYSB]^XdKB[gBn[�e d\L V%B>BBeNU0gNgN]o\MV*$Y]oUfSB[JZ;[Ydodo[ ZB[Yej]C5�V�bj[ i i i i i i i 1 �

2Ai @;: S0bN[�eNB(Ufd^V%$�] U*SB[ X_UfS gNBBd ]oSB[cX�<J(B]^XdKB[C2

i i i i i i iJi i i i i i i i i 1�@2AifF m0bjejVf] S [�SB[Ye?I�� i i i i i i i i iJi i i i i i i i i i iJi i i i i i i i i 1�12Ai &

ν− gNBBdo] SB[ i i i i i i i i iJi i i i i i i i i i i iJi i i i i i i i i i i !87�72Ai 1;: S0bN[�eNB(Ufd^V%$�] U*SB[ X_UfS _pm&gNBBdo] SA[ iJi i i i i i i i i i iJi i i i i i i i i !O7 ,^�� j�j��V��iOi8��J�*�\�W��J�� J��=�8� ��

9;i ! ' V*gLU6. <BSJ$�] U*SAV do[�i i i i i i i iJi i i i i i i i i i iJi i i i i i i i i !�!*!9;i � ' V*gLUBAVfejVf\ [YbNej]oXYU�i i i i i i iJi i i i i i i i i i iJi i i i i i i i i !�! 2

9Bi � i ! 3�VfejVf\ [YbN[�e XYUfejeN[4X bj] U*S�iJi i i i i i i i i i iJi i i i i i i i i !�! 2�;T4�`�VU��X� �ti8�]� ��J

@Bi ! : S0bNejU;Z=<J$�] U*SB[Ti i i i i i i i i iJi i i i i i i i i i iJi i i i i i i i i !�! F@Bi � ' <Be?5f[ U�!,gN[_b�[rBJ<BS0bj]�g bjejVfUfehZ;]oSAV ej] i i i i i i i i iJi i i i i i i i i !�! 1@Bi , "�!DgL[YbTZ;]�<BSnV4BnV ehV%((Ufd^V i i i iJi i i i i i i i i i iJi i i i i i i i i ! �%7

HFV ' Uf\BJ<BbN[Ye��JejV*BJKB]^XYgcXYU*gLbN] bN<B]^gNXY[kU*I�If] <BSBU Z;[Y] BA]��]o\BnU*eLbhV S0bN]OgL[YbLbNU*eN]�Z;[Ydod\L ]oS=m. U*eN\kV bN]^XYVBi�� <BSAV Z;]^gNXY]CBBdo]oSAV"XdKB[��(AVfgjV SAZBU*gN]�gN<�\M[_bNU;Z;]O[�Z V dDIfUfej] bN\M]�Z;] R SnV do]ogN]" <B\M[Yej]oX�V��H]oSAgN] [�\M[ V BAV eNbN]^X_U*doVfeN]?bN[4X_SBU*d U�If]o[�[ g bjeN<B\M[�S*bj] KAV ehZ � V ej[��X_U*SAgL[�S0bN[ Z;]5a]^g/<nV do]C$8$�V ej[JU*I*I*[_bNbN]D\M[�Z;]^V S0bj[J]od�X�V d^X_Ufd^V�bjUfej[fi

H�[JV*BJBBdo]oX�V%$�] U*SB](ZB[YdodoV ' Uf\BJ<;bj[Ye�JehV%BJKB]^XYgOgNBAV%$�]oVfSBUl]oS 5�V ej]DgL[YbLbjUfej]��Bn[�e [�gN[Y\6mBA] Urd^V _Z<AgN] SA[�gjg��JehV%BJKA]oX�g �H]od XE<A]OU�(J(B]o[_bNbN]D5fU��]<B[�d doUrZ;] BBeN[4gL[�S0bjV ej[ ZB[_bN[�eN\M]oSAV�bj]ej]^g/<Bd bjV bN](X_UfSkdML V%<AgN] do]oUcZ;]WI*ejV��nX_]�� ]oSW\MU;Z;UlZBV ej[YSAZB[YejSB[ BB]��k] \M\M[�ZB]oV bjV d^VcX_U*\BBeN[�S=mgN]oUfSB[*i

� S Vfd bjeNUTgN[_bNbNU*eN[OZ;]0V*BJBBdo]oX�V%$�] U*SB[��<A[Ydod UJZ;[Ydod\L V SA] \kV%$�] U*SB[�9g/. e?<;bLbhV SAZ;U!d\L [Ydo[�5�V bjV5*[YdoU;X_] b?oTZB[Y];XYVfdoXYUfd^V�bNU*eN]f\ U;Z;[�eNSA]�� gNUfSBUJg bhV�bj]0XYeN[4V�bN]]BBejU*I*ejVf\ \M]*] SsI*ejV*Z;UTZ;];VfSB] \kVfeN[ZB]ogN[�I*SB]�X_UfS <BSAV eN]^gNUfdD<J$Y]oUfSB[ X_U0g��T[Ydo[�5�V�bhV ZAV ZBVfeN[ doV gN[YSAgjV%$�] U*SB[ Z;[�d \ U�5a]o\M[YS0bNUXYUfS0bN]oS�<BUAi-/V do[MgN[_bNbNU*eN[��<;bN]od ]D$�$4V�bjU ]oSr\MUfd bN]HXYVf\BB]HV%BJBAd ]^XYV bN]D50]��JdoV BBej[�BAVfejV*$Y]oUfSB[ZB[Y]�5a]^Z;[YU�If]oUaXdKA]��;Z;]��nd \��Bdo[cV SAVfd ]^gN]9X_]oSB[Y\kV�bj]oXdKA[ ZB] . [YSBU*\M[YSB]vi

� SBU�Z;[�] BBej] \M]TX�V \BB]TV*BJBBdo]oX�V�bN]D5a] Z;[Ydod^V ' Uf\BJ<;bj[Ye��JehV%BJKA]oX�g�� g bhV�bNUr]oSBUfd bNej[�]<B[�d doU"Z;[�d d^V BAeNU�If[_bNbjV*$Y]oUfSB[��sBn[�e��]<AVfS0bNUej]CI�<AV ehZBV ] d gN[_bLbjUfej[W[Ydo[_bNbNejUfSB]^X_U��Fdo[ BBej]oS=mXY]CBnV do] V%BJBBdo]^XYV%$�] U*SB] 5�V SBSBU�ZBVfd d^V BBeNU�If[YbLbjV*$Y]oUfSB[ Z;[Y] XY] ehXE<A] bj]�] S0bN[8IfehV�bj] V��]<B[�d d^VrZ;]]oS0bN[�eN[MgNXdKA[�Z;[��,XYUfS BAeNU;Z=<J$�] U*SB[ V*<;bNU*\kV�bN]^XYV�Z;[Y]�Z;]^gL[8IfSB]�[4gL[4XE<;bj]C5a]/[ Z;[�d d^VWdo]^g bhV Z;[�]XYUf\B(UfSB[�S*bj]nSA[�X_[4gNgjV ej]nVfd d^VrBBejU;Z=<J$�] U*SB[fi�R SnXdKB[TSB[�d d^VsBAeNU�If[_bNbjV*$Y]oUfSB[ VfejXdKB] bN[YbLbjUfSB]^XYVd^V ' Uf\BJ<BbN[Ye��JehV%B>KB]oX�g gL] �Mej]C5*[Yd^V�bhV�[4g bjeN[�\MVf\M[YS0bN[6<;bN]odo[fi H9[ BAeN]oSAX_]DBAVfd ]HV*BJBBdo]oX�V�m$�] U*SB]Bej]CI�<AV ehZBVfSBUTdML V*(B] bjV%(A] do] bdo Z;[�I*d ]BVf\)(B]o[YS0bj]��*d\L V ejeN[4ZBV \M[�S*bjU!�udML VfSAV do]ogN]BgLbNe?<;bLb?<BejVfd [*iH�[ BnU0gNgN]D(B] do] b?o IfehV"�nXdKA[ Z;[�d?X_U*\6B><;bN[�ecgLU*SBU U�I*I*]�Vf\BB]oVf\M[YS0bN[<;bj] do]C$8$�V bN[WV SnXdKB[M] S�]<B[�d dML ] \B(UfeNbjV S0bj[ gN[_bLbjUfej[ bN[4X_SB]^X_U*m&VfeLbj]ogLbN]^X_UkXdKB[45�VWgLUfbLbjUW] d/SBU*\ [ Z;]�BBeNU�If[YbLbjV*$Y]oUfSB[]oSAZ=<ng bjeN]^V do[$# : SAZ=<ng bjeN]^V d ` [�gN]CI*S�% iQ: d XYVf\6B(UrV%BJBBdo]^XYV�bj]C5*U ZB[Ydod\L : SAZ=<AgLbNej]^V d ` [4gL]DIfS&�5�V*g bj]ogjgL]o\MU!�aX_U�BBeN[ bN<;bNbN]D],gL[YbLbNU*eN](SB[�]'�]<AVfd ]��0VfdWBAeNU�If[_bNbNU bj[�XYSB]oXYUl[4gL[4XE<;bj]C5*UlZ;]�<BS�Z;[�mbj[Yej\M] SAV bNU BBejU;Z;U bNbNU��(� SB[�XY[�gjgNVfeN]oU V*gNgNU;X_]^V ej[ <ASBUkg b?<AZ;]oUk[�gLbN[_bj]oXYUWV%BJBAeNU*. UfSAZ;] bNUkZ;[�dej]^g/<Bd bjV bNUcXdKB[ gN]AZ;[4gL]^Z;[�ejVJU bNbN[�SB[Yej[fi�: dnXYUf\BJ<;bj[YeHU�!(ej[ d\L U*BJB(UfeNbN<ASB] bdo Z;]ngN[Y\BBdo]��nXYVfeN[[�SBUfej\M[Y\M[YS0bj[ ]od�d^VV5fUfejU�Z;]HgN]oS*bj[�gN]HZ;] <BS�L ]^Z;[�V V eNbN]^gLbN]^XYV��DXYU*g)� V SAXdKB[6Ifdo]�g bj] do]^g bj]H[�Z ]XYeN[4V�bjUfej]�Z;] \MU;ZBV gNUfSAU V�bNbN<AVfd \M[�S*bj[ <;bj[YS0bN]�Z;] BAV*X+*�V%If[4gcU*eN]o[YS0bjV bN] V dodoV gLU*dC<>$Y]oUfSB[ehV%BA]oZBV Z;] BBejU*(Ad [�\ ] I*ejV��nX_],VfgjgNUaXY]oV bN],V��<A[�gLbNU bN]DBnUkZ;]9V bLbN]D5a] bdoMV d bjV \M[�S*bj[JXYeN[4V�bj]C5*[fiHFV6BBeNU�If[YbLbjV*$Y]oUfSB[J\M[�X�XYV SA]oX�V,� dML V*g/B([_bNbNUBB]�� SBU bjUMZB[Yd�Z;]^gN[�IfSAU V*gNgN]ogLbN] bNUkZBVfd9XYVfdoXYU%md^V�bjUfej[fi P gjgNV-� g/B([�gjgLUk]^Z;[�S*bj].�(XYV�bhVkX_U*S ' i Rli ` i��,X_]oU/� doV BBeNU�If[YbLbjV*$Y]oUfSB[ V*<;bNU*\MV bN]^XYVSA[Yd�gN<BUkX_U*\6BAd [4gNgNU [4Z0��B]oS eN[4V d b?o��a]od�XYV*BnU0g bj]CBA] bj[ ZB]DbN<;bNbN[cdo[ V d bNej[ V%BJBBdo]^XYV%$�] U*SB]vi

: S1��<A[�gLbNUW[�doV*(nU*ejV bNUk]oS*bjeNU;Z=<AeNej[Y\MU�Z;]D5f[�ejgN[ ehV%BJBBej[�gN[YS0bhV%$Y]oUfSA]�Bn[�eTdo[ X�<BeN5*[c[ldo[ej[Yd^V�bj]C5*[ U�Bn[�ejV*$Y]oUfSB]�If[�Uf\M[_bjeN]^XdKB[*i ' ]HgLU�!([�eN\M[�eN[�\ U V dC<BS>IfU g/<BdHZB]ogN[�I*SBU ZB]HXE<Be?5f[<ngNVfSAZ;UZ;[Ydodo[WV%BJBBejU*gjgN] \kV%$�] U*SB]2# XE<AeN5*[ Z;]p_�[8$Y]o[Ye3��XE<AeN5*[6_pm g/BBdo]oSB[�/XE<Be?5f[ " � A _�m�%[4Z [�5a]oZB[YSJ$�] [�eN[�\ UI*d ]9V*g/B([_bNbN]�XYUf\BJ<;bhV%$Y]oUfSnV do](]oSB[�eN[�S*bj]D]�BBejU*(Bdo[Y\M] I*[YUf\M[YbNej]oXY] i

9

@ ��

� �� !"��#��%$��'&(��)�*��$��'��*,+

-/.�0�1#24351#6#7�8�9;:

1��=<?>A@ �CBED �F@ �

� S bN]DBnUZ;]HehV%B>BBeN[4gL[�S0bjV%$�] U*SB[lZ;]HX�<Be?5f[ [MgN<JB([Ye �nX_]HXdKA[ ej]^g/B(UfSAZB[ Vfd do[M[�gN]CI*[YSJ$�[ Z;[�d' R� ` ��]<B[Ydod^V BAVfejVf\M[_bNej]^XYVHG�i ` Vfd�B><BS0bNU Z;] 5a]ogLbjV�\MV bN[�\MV bN]^X_U��,bjV do[ ejV*BJBBej[�gN[YS=mbhV%$�] U*SB[ U !,eN[��;eN]^g/B([_bNbNUWVfd d^V4. Ufej\MV V SnV do] bj]oX�V��;\MUfd bN]�5�VfS*bhV%I�If]�. ejV��]<AVfd ]��

• �r<BSAV ehV%BJBAeN[4gL[�S*bhV%$�] U*SB[!Z;]9bN]DBnUk[4g/BBdo]^X_] bNUJI

• B([Yej\ [YbLbj[ Z;]Z<AgjV ej[Wd^V . Ufej\kV 5f[YbLbjUfej]oVfd [��/XdKB[ KAV ]odZBBeN[8If]oU"Z;]O[�gjgL[�eN[ X_UfSnX_]^gNV[ XYUfSAgN[YS0bj[lZ;]/[�gNBBeN]o\M[Yej[cd^V�XE<AeN5�VWXYUf\M[ XYUf\)(A] SAV*$Y]oUfSB[ do]oSB[�VfeN[lV X_Ua[E0WXY] [�S*bj]5f[YbLbjUfej]oVfd ]DZ;] . <ASJ$Y]oUfSB]FgNX�V d^V ej]%I

• � ]oSAZ;]DBn[�SAZ;[YS0bj[cZBV d9gN]ogLbN[�\kV ZB]9eN]C. [Yej]o\ [�S0bNUJI

• B([Yej\ [YbLbj[ Z;]TXYVfdoXYUfd^V ej[ gN[�BAVfejV bjVf\ [�S0bN[ d [X_U*\BnU*SB[YS0bN]�[ ZB] <AgNVfeN[ d^Vrg bj[�gjgNVejU*<;bj] SB[ZB([Ye�] d;X�V d^X_UfdoU!ZB]aXY]oV*gNX�<BSAV!X_Uf\B(UfSB[�S0bN[OeN]^Z=<AXY[YSAZ;U X_U0g��Ad^V!XYUf\BBdo[�gjgL] b?oZ;[�]�BBejU*IfehV \M\M]�I

• B([Yej\ [YbLbj[�Z;]*[�gNBBeN]o\M[Yej[HX_Ua[E0WXY] [�S*bj]*VfSJIfU*doVfeN]�]oS(�ASB] bN]0]od*XdKB[ �t<BSAV X�V ehV�bLbj[Yej]ogLbN]^XYVZ;[4gL]^Z;[YehV%(A] do[JZAV dD\MUf\M[YS0bjU XdKB[ gNBn[4gNgNU ] BBeNU�Ado],ZB[�Ifdo]DU*I�If[YbLbN]�KAVfSBSBU bjVfSJIf[�S*bj]5f[�eLbj]oX�V do] i

: S$�]<B[4g bjU X�V%BB] bNU*d U 5fU�Ifdo]oVf\ U . UfejSB] ej[6<BSAV (Bej[�5f[ ]oS*bjeNU;Z=<>$Y]oUfSB[MV ]�X_U*SAX_[YbLbj]HZ;] (AVfgN[ZB[YdodoV bNehV�bNbjV*$Y]oUfSB[��9Z;[ �ASA[YSAZ;U d^V SBU bhV%$Y]oUfSA[ XdKB[6<AgN[Yej[Y\MU [6BAeN[4gL[�S*bhV SAZBU ]HXYUfSAXY[_bLbj][4gNgN[YS>$Y]^V do]9Z;]�If[�Uf\M[_bjeN]^V Z;] !,[Yej[YS>$Y]^V do[fi

K�L .M:%8�N�9%1O.P2�8�6#6#8RQ�S,9T0U8V8XW%S�Y�8�9TZ�Q�1[Y�.\9%.\]A8�:%9T1#Q�^�8_8�9%.a`�1ObM7)8�cdc�N\:�.ac�8�`�6#1[.\c�c�1eW�8�W%W�.\cU:�.a8Q�N�WT:�1 :%S�1 0\.fS,c�.AY�.\9T:%8[7)8�c?.\W%W�8�W%:�.�:�.f2,8�6#6O.'`�8�N�]A8�:%9%1O.,g�h S�9;0U8i8jW%S�Y�8�9;Z�Q�1"Y�.\9%.\]A8�:%9T1#Q�^�8�k)Y)8�9%.\6 :�9TN�kc�N�cM8�9�.\c,N?W%:�.�:�8AWT:�S�2,1O.�:�8f2�.lS,c_Y�S�cU:%N?2�1(0�1#W%:�.lQ�N�]AY�S,:�.\m�1#N�c�.\6#8�k"Y)8�9;:�.\cU:%NX7�8�c_Y�NnQ�N?8�9%.4c�N\:%NW�S�6#6O.e6#N�9TN[Y�N\:%8�c�m�1O.\6#1 :�b[2�1�S,:�1#6#1#m�m�Nj2�1�S�cAW�1#WT:�8�]f.jh o/pRq�h o�-igUr".\cn:�s t�Q�^�8�6#NjW%:%S�2�1#Nj2�1�:�.\6#8/.\]AY�1O.Y�9%N�7,6#8�]f.�:%1#Q�.u^�.uQ�N�W%:%1 :%S�1 :�N�kFY�9%.�:�1#Q�.\]A8�cU:�8'2�.fW%N�6#N�k)1#6vc�S�Q�6#8�Nu2�1"7�.\W�8i2�1vS�c�.fc�S,Nw0�.f2�1#W%Q�1#Y,6#1#c�.A.W�xeW%:�.\cU:�8�k�1#6vh�g o[g yig -ignQ�1#Nntih N�]AY�S,:%8�9zo�1#2�8�24y�8�N�]A8�:�9T1#Q[-�8�W�1#`�cHg

F

& ��

(x,y)y

x

(x,y)y

xV % (�%q/]CI�<BehV ! i ! ��[YbLbjUfej]

�� ! �"#$��%'&

' U*SR

2]oSAZ;]^XdKB]^V \MU d\L ] SngL]o[Y\M[rX_U*gLbN] bN<A] bjU-ZBVfd do[ XYU*BJBA] [ UfehZ;] SnV�bN[ Z;] S�<B\M[Yej]Jej[�V%m

do]��cXY] U/�R

2 = {(x, y) : x ∈ R, y ∈ R} i �Jdo]c[Ydo[Y\M[�S*bj]�� U BJ<AS*bj]��cZB]R

2gNVfejVfSBSBU

ehV%BJBAeN[4gL[�S*bhV�bj]TSB[�d dML <ng/<AVfd [ ej] . [�eN]o\M[YS0bNU XYV eNbN[4gL]^V SAU # �>I�<BejV ! i ! V %kU-bjV dD5fU*d bhV�]^Z;[YS=mbj].�nXdKA[Yej[Y\MU

(x, y)XYUf\M[M] d 5*[_bNbNUfej[

v = (x, y)V%B>BBd ]^XYV bNU V dod\L Ufej]DIf]oSB[ ZB[�Ifdo]HVfgjgN]2# �Jm

I�<BejV ! i ! (�%_i R SnV doU*I*Vf\M[YS0bN[ ] SAZB]oXdKB[�eN[�\MU XYUfSR

3dML ]oSAgN] [�\M[ Z;[�d do[ bN[�eNSB[ UfehZ;]oSAV�bj[

R3 = {(x, y, z) : x ∈ R, y ∈ R, z ∈ R} I�<;bN]od ]D$�$�[Yej[Y\MU V SnXdKB[k] S �]<B[4g bjU XYV*gLUgN]oV d^VehV%BJBAeN[4gL[�S*bhV%$�] U*SB[HXYVfeLbj[�gN]oVfSAV Z;]�BJ<BS0bjUJXdKB[�doV!SBUfbjV%$�] U*SB[t5*[_bNbNUfej]^V do[fi �Jdo];g/BAV*$Y]ag/< XE<B]

d^VV5fU*eN[�\ UcgjV ehV SASBU �]<B] SnZ;]R

d�;X_U*S

d = 2, 3i�3H]��]oS I*[YSB[�ejVfd [��ngjgjV�bjVs<BS�L Ufej]CI*] SB[

O�

d^VV5fU*eN[�eN[�\ U SB[�d doU gNBAV%$�] UWV%0kSB[E

d ={P = O + v : v ∈ R

d} i!m;]oX�X_Uf\M[cd^VWgNUf\M\kV

Z;]`5f[_bNbNU*eN] �cV SnX_UfehV4<BS 5f[_bNbNU*eN[��]B(U*gjgNU [4g/BAeN]o\M[Yej[ <BS 5*[_bLbjUfej[b ∈ R

dX_Uf\M[

b =

n∑

j=0

ajbj

XYUfSbj ∈ R

d[

aj ∈ R�j = 0, . . . , n

i

��)( *+�-,/.0#$%' �"#$��%'&/.' �1�#$2&�%��31�#�24

5 �76�� 98��;:=<?>�@BA?CD<FE�G)@�G)HJILK�<MC9N�O�<BHPC9@QK)NJRS<BG)@$C=TURS<BG)N

b =n∑

i=0

ajbj

E�@a0 + . . . + an = 1 V

W <B]oSAZ;]��BSB[�d BB]^V SBUkV�0kSB[E

d��<A[�gLbjVMZ;[ �nSB]C$�] U*SB[cZ;]D5f[�S*bhV

b = b0 +

n∑

i=0

ajbj −n∑

i=0

ajb0 = b0 +

n∑

i=0

aj (bj − b0)

5 �76�� 98 SYX!CZNLG[HPIQK�<MC9N\O�<BHJC9@QK)NJRS<BG)@�CDT]RS<BG)N^ES<_>P<BG)@-G)HJC=`a@$E)E$NLE�@aj ≥ 0 b @�R

j = 0, . . . , n V5 �76�� 98Dh�:=<c>�@BA?CD<FE�G)@0G[HPCD`J@edgfihijMj?jlk <MC=Tm@�R)E�@[O�<BHPC9@n>J<oTUhiT]TU<�<pG[HPCD`J@SE)ES<cG)fq@G)HJC=Tm@�C7r7HJCZHL< b hiCDT]< b0, . . . ,bn

�� q ;��4��g�p� �� 1

x

y

x’

y’

O

z=z’

θ

θ

qF]DI*<AejV ! i � � A�UfbjV%$�] U*SB[

�-� � 1� ��J��1�, -"4#$�-%�# � %�#

" [Ydod\L V \4(B] bNU ZB[YdodoV ' U*\BJ<;bN[�e �JehV%BJKA]oX�g �nd [lbNehVfg/. Ufej\MV*$Y]oUfSB] BA]�� <BbN]od ]D$�$4V�bN[4Bn[�e!gNBnU*mgLbjVfeN[ <BS U*I*I*[_bNbNUcU \MUaZB].�nX�V ejSB[T]oS \ U;Z;U4IfehVfZ=<nV�bNUld [ Z;]o\ [�SAgN] U*SB]��BgNUfSAUldo[TbjejV*g . U*e/m\kV*$Y]oUfSB]9V%0MSA] i5 �76�� 98 :9<BN

Φ : Rd → R

d V ΦES<o>J<BG)@�hiC9NnT]ReNJE��$HPRSIQN\O�<BHJC9@0N� CZ@LE�@��

j <MC9@)NJRe@�RS<FE b @$T]TmH�NJjMj N x � G�<BH��E�@��^> @�j T]< b H��

Φ(x) = Ax + c

G)HPCA ∈ R

d×d @c ∈ R

d V� . VfXY] do[t5*[Yej].�(XYV ej[HXdKB[�d [�bNehVfg/. Ufej\kV%$Y]oUfSA[HV�0kSB]]KAV SBSAUTd^V BBejU*BBej]o[_b?oTZ;]aSBUfSc\MU;Z;]��nXYV%mej[?do[�XYUf\4(B] SnV%$Y]oUfSA]*(AVfeN]^X_[�S*bjeN]^XdKB[��]oS=. V�bNbN]�# ZAV dodoVTd ]oSB[�VfeN] b?o!Z;]

Φ[OZ;[�d d^V gNUf\M\kV�bjUfej]oV % �

Φ

(n∑

i=0

aibi

)

=

n∑

i=0

aiAbi +

n∑

i=0

aic =

n∑

i=0

ai (Abi + c) =

n∑

i=0

aiΦ(bi)

H�[rBB]�� XYUf\4<BSB],bNehVfg/. Ufej\kV%$�] U*SB](V%0kSB]�gNUfSBU�� JiOk��J�� ]oSXE<B]

A = I[c� ] d 5*[_bNbNUfej[ Z;]DbjejV*gLd^V%$�] U*SB[ 5*[YejU [rBBejU*BAeN]oUAi

� �>� �J�*�\�W�� ]oS X�<B]c = 0

[�AgN[ eN<BUfbN]^V \MUM] S0bjUfejSBU Vfd dML VfgjgL[z�

A =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

: S BAV eNbN]^X_U*doVfeN[��<ASAVMeNUfbNUfbNehVfgNdoV*$Y]oUfSB[�r<ASAV eNUfbjV%$�] U*SB[ X_UfSc 6= 0

i�� >k��! ]oS X�<B]

c = 0[

A =

d1 0 00 d2 00 0 d3

X_U*Sd1, d2, d3 > 0

� S gLUfbLbjUaXYdoV*gNgN[ ] \B(UfeNbjVfS*bj[ Z;[Ydodo[ bNehVfg/. Ufej\MV*$Y]oUfSB]�V%0MSA]TgLU*SBUr] ILHJ`\<MIQ@�CDT]<�RS< r J<U <FE�HJIL@$T]RS<B@ i -FVfd [kbj]CB(U ZB]?bjejV*g . U*eN\kV%$�] U*SB] KAVfSBSBU d^V\kV�bjeN]^X_[

AUfeNbNU*I*UfSAVfd [��FXY] U/�

AAT = AT A = I� [

c = 0i�H L ] \B(UfeNbjVfSJ$�VJZ;]!�]<B[4g bj[�bjejV*g . U*eN\kV*$Y]oUfSB] ��XdKB[ doV*gNXY]oVfSBU

!O7 ��

θv

w

q/]CI�<BejV ! i ,��3HejU;Z;U bNbNU 5*[_bLbjUfej]oVfd [

]oSAV d bN[�ejV bN[�do[?Z;]^g bhV SJ$�[�. ejV ]�BJ<BS0bN]*[ Ifdo]fVfSJIfU*d ]��4XY] U/��SBU*ScZ;[E. U*eN\kVfSBU doV2�WI*<BehVBi�: SJ. V�bLbj]��gN[ X_UfSngL]^Z;[Yej]^V \MU

x1,x2,x3 ∈ Ed�B[+. VfX�X_]^V \MUM] d�BAeNU;Z;UfbLbNUMgjXYV d^V ej[

(x2 − x1)T (x3 − x1)

V*BJBBdo]oX�V SAZ;U ]od9\MU�50]o\M[YS0bNUMej]CI*]oZBU!�0bjVfd [ �]<AVfS*bj] bdo eN]o\kV SB[ ]oS]5�VfeN]^V�bhV

(Φ(x2) − Φ(x1))T

(Φ(x3) − Φ(x1)) = (A(x2 − x1))T

A(x3 − x1) =

= (x2 − x1)T AT A︸︷︷︸

I

(x3 − x1) =

= (x2 − x1)T (x3 − x1)

maU*SBUM\MU�50]o\M[YS0bN]Dej]DIf]^Z;]Dd [ ejU bhV%$Y]oUfSA]D[Jdo[cgN] \M\M[_bjeN]o[ eN]^g/B([_bNbNUkV<BS VfgjgL[*i

�� 1��3�!� �a1� ��&�3�!��1�#

A�]^X_U*ejZB]oVf\ UMV d^XE<ASB[ ] \B(UfeNbjV S0bj] BBeNU�BBej] [Yb?oMZ;[Yd BAeNU;Z;UfbLbNU6. ehV65f[_bNbNU*eN]vi5 �76�� 98 ^ :=<�> @UA C=<FE�G[@ b ReH\> HPTUT H E�G)NJj NPRe@��)ReN >JhD@gà@�T]TmHPRS< u,w ∈ E

d @0ES<<MCZ>J<BG)N�G)HPC

u · w � j N��hDNJC=TU<MT

u ·w = |u| · |w| · cos θ

5 �76�� 98 ��:=<4>�@BA?CD<FE�G)@ b ReH\> HPTUT H�à@�TUT HJRS<BNPj @ �)ReN u,w ∈ Ed � @-ES< <MC9>P<BG)N'G)HPC

u ∧ w � <Mj�à@�T]TmHPRe@ G)fq@-fDNg>J<MRe@)O�<BHJC9@�> NPTmNg>�NJjMj NgRe@mr�HPj Nn> @�j b NJReNPjMj @$j Her ReNJIÎLN � `J@$R)E�H>�NJT H > NPjMj N�Re@mr�HPj N'> @�jMj N ILNJCZHQES<MCD<FEST]ReN @-ILH�>Phij H |u||w|| sin θ| V5 �76�� 98 :=<�> @UA CD<FE�G)@ b R[H�>�HJT]TmH I^<FESTmH �)ReN u,v,w ∈ E

d @�E)<c<MC9>P<BG[N G)HPCu · (v ∧ w)

j N��hDNJCDT]<MT > NPTmN'> Nu · v + u · w V

: SAZB]oXdKB[�eN[�\MU ] S-gL[8I*<B] bNU"XYUfSv × w

] dZBBejUaZBU bLbjU 5f[YbLbjUfej]oVfd [ . ehVv,w

X_U*S�\MUaZJ<Bd U|v||w| sin θ

i : SBU*d bjeN[ ej]ogN<Bd bjV��

v1

v2

v3

×

w1

w2

w3

=

v2w3 − v3w2

v3w2 − v1w3

v1w2 − v2w1

�� ^�� o�!� � �p�!�� !*!

x

y

v

po

r

qF]DI*<AejV ! i 2!� A�[YbLbhV ]oS . Ufej\kV4BnV ehV \M[_bjeN]^XYV

�-�� Z%��31��'"4#��%�& �$&/2��1 ��&

: SA]C$�]oVf\ U X_UfSngL]^Z;[YehV SnZ;U�doV eN[YbLbjV SB[Yd4BB]^V SBUni 3�U*gjgL]^V \MU�ehV%BJBBej[�gN[YS0bhV ej[ d^V�ej[_bNbjVbjejVf\M] bj[ d\L [ �]<AV*$Y]oUfSB[J] \BBdo]^X_] bjV

ax + by + c = 0

XYUfSa, b

SBUfS [YS0bNehV \4(B]DS�<Bdod ]��BU*B>BJ<Bej[TbjejVf\ ] bN[ dML [ �]<AV%$�] U*SB[J[�gNBBd ]^X_] bjV

y = mx + q

: SkVfSAV do]ogN];\kV�bj[Y\kV�bj]oX�VJgN]=<;bj] do]C$8$�V gNUfdo] bhV \M[YS0bj[2�]<B[4g bOL <Ad bj] \kV . Ufej\kV!]oS-�]<AV S0bNU+. U*eNSB]CmgjX_[ <ASk\ U;Z;U Z;]oej[_bLbjUrBn[�eHehV%BJBBej[�gN[YS0bhV ejdoV IfehV"�nX�V \M[YS0bN[�� V SAXdKA[ gL[�]oS bjV do[ . Ufej\kVJSBUfS�B(U*gjgL]D(B]od [ ehV%BJBBej[�gN[YS0bhV ej[cd [ ej[_bLbj[kZ;[Yd/bN]DB(U

x = ki4-�<;bLbhVV50]^V��MgN[Y\BBdo]oXY[6BnVfgjgNVfeN[

ZAV dodoV ehV%BJBBej[�gN[YS0bhV%$Y]oUfSA[T[�gNBBdo]oXY] bhVM[ ��<A[YdodoV ]o\6BAd ]^X_] bjV!�B] SJ. V�bLbj]��

ax + by + c = 0 → y = −a

bx − c

bcon b 6= 0

" [YdodoV"g/BAV*$Y]oU bNej]oZB] \M[YSngL]oUfSAVfd [kd^V ej[_bLbhV��ZBV bjVZBVfd dML ]oS0bN[�ejgN[�$�] U*SB[kZ;]�Z=<A[ BB]^V SB]�SBUfSBnV ehV dod [�d ]��

{ax + by + cz + d = 0

a′x + b′y + c′z + d′ = 0[ZBV bjVfd [ . Ufej\kV�� Z;]C0kXY] do[ eN]DBnU*eLbhV ejdoV"] S . Ufej\kV [�gNBBdo]oXY] bhVBi-m;Ufe?If[�Vfd doUfehV dML [4gL]DIf[�S=m$4V ZB]?bjeNU�5�V ej[ <BSnV . U*eN\kV Vfd bj[YejSAV�bj]C5�V"ZB]OehV%B>BBeN[4gL[�S0bjV%$�] U*SB[WXdKB[ . VfX_]odo] bj]O]od�Z;]^gL[8IfSBUZB] <BSAV��]<AV dD<BS!�]<B[X�<BeN5�VAi W <B[4g bjU"bj]CB(U�Z;] ehV%B>BBeN[4gL[�S0bjV%$�] U*SB[ BBej[YSnZ;[ ] d SAUf\M[Z;]R[N b b Re@$E$@�CDT N�O�<BHPC9@ b NJReNPIQ@�TURS<BG)N i i 3�U0gNgN]oVf\MU ]oS=. V�bNbN]9ehV%B>BBeN[4gL[�S0bjV ej[Td^V eN[YbLbhV XYUf\M[

p = p0 + tv

XYUfSp,p0,v ∈ E

d[�Z ]od�BAVfejVf\ [YbNejU

t ∈ Rit: S0bN<A] bj]C5�V \M[�S*bj[

pehV%BJBAeN[4gL[�S*bhVl]od�BJ<AS*bjU

UfbLbj[YS�<;bNU�ZAVp0

g/B(U*gLbjVfSAZ;U0gL]�Z;]tSB[Yd 5f[YehgNU Z;]

vi 3/[�ekejV*BJBBej[�gN[YS0bjVfeN[ XE<Be?5f[gN<Bd

X�V d^X_U*doV bNUfej[� <;bN]od ]D$�$�[Yej[Y\MU doV ejV*BJBBej[�gN[YS0bjV*$Y]oUfSB[ BnV ehV \M[_bjeN]^XYVAiRHtL [3�]<AV%$�] U*SB[ Z;[�d d^VXY] ehX_U*S=. [Yej[YS>$�V ] S . Ufej\kV6BAV ehV \M[_bjeN]^XYV gL]�U bLbj] [�SB[+. VfXY] do\M[YS0bN[��B[�gjgNV ej]ogN<Bd bjV

X(t) =

(cos tsin t

)

XYUfS0 ≤ t ≤ 2π

# X_]oU �tejV*BJBBej[�gN[YS0bjV�d\L V SJI*UfdoU %_i " U*SBU*gLbjVfS0bN[ ]+5�V S0bjV*I*If]JZ;[Ydod^V

ehV%B>BBeN[4gL[�S0bjV%$�] U*SB[ BAV ehV \M[_bjeN]^XYV!��Vrdo]C5*[Ydod U�XYUf\BJ<;bhV%$�] U*SB[ eN]o\kV SB[]od BBejU*(Bdo[Y\kV�Z;[�dX�V d^X_U*d U ZB[Ydod [X. <BSJ$Y]oUfSA]!bjejV*gNXY[YSAZB[YS0bN]

cos t[

sin ti 3�U0gNgN]oVf\MU�V dod U*ejV�XY[YehXYVfeN[ <BSnV

! � ��

ehV%BJBAeN[4gL[�S*bhV%$�] U*SB[�BAV ehV \M[_bjeN]^XYV Vfd bj[YejSAV�bj]C5�VBi/ma[�XYUfSAgN]oZB[YejU gLU*d U�d^V gN[Y\M]oXY] ehX_U*S=. [Yej[YS>$�VB(U*gN] bj]C5�V!�BV dod U*ejV

X(t) =

(1−t2

1+t22t

1+t2

)

− 1 ≤ t ≤ 1

-/V do[ \ U;Z;UkZ;]`BBeNU;XY[�Z;[�eN[ gL]FXdKB]oVf\kVlej]CBnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[!Z;]�<ASAVMXE<Be?5�VBi

�� #��' �1� -, &�!1�#$"" �"4#��%�& ��# �'%� 2 ��1 �p

R (J(B]^V \MUIf] o ]oS0bNejU;Z;U bNbNUMdoV ehV%BJBAeN[4gL[�S*bhV%$�] U*SB[ BAVfejVf\M[_bNej]^XYV Z;] <BSAVkX�<BeN5�VMX_U*S

X : [a, b] → Ed

� ] \B(UfeNbjV S0bj[�DV �]<B[4g bjU BJ<BS0bNU ]oS*bjeNU;Z=<AeNej[c]od�XYUfSAXY[_bNbNU�Z;] >P< �o@$R[@$CiO$<BN�K$<Mj <MT [ Re@mr7H��j NPRS<MT i5 �76�� 98 � X!CZN�G�hiRS`aN

X : [a, b] → Ed ES<!>P<BG)@�>J< �o@�Re@�C O�<BN K�<Mj @�<MC (x(t), y(t))E�@ ∃X(t)

G)HJCDT]<MCDhDH V5 �76�� 98� X!CZN G$hiRS`JN

X(t)ES<?>P<BG[@�Re@mr7HJj NPRe@�E$@ �->P< �o@�Re@[O�<BN�K$<Mj @ @

X(t) 6= 0b @$R-H[r CD< t ∈ [a, b] V3�U*gjgN]oVf\ U V dodoUfehV BAVfeNd^V ej[rZ;]cej]CBAVfejVf\M[_bNej]D$�$�V*$Y]oUfSB[ Z;]s<BSAV�X�<Be?5uV!� Z;[ �ASA[YSAZ;U <BSnV. <BS>$Y]oUfSB[

τ : [a, b] → [c, d][g<;bj] do]C$8$�VfSAZ;UX(τ(t))

V d�B(U*gLbNUkZ;]X(t)

ima[

τ�r<BSnV6. <BSJ$Y]oUfSA[cZ;[Yej]C5�V*(B] do[ [cV \M\M]^gNgN]C(A] do[�AXY] U/�

τ ′ > 0 ∀t ∈ [a, b]Vfd doUfehV d [

BBejU*BAeN]o[_bdokZ;]XgLU*SBUWX_U*SAgL[�eN5�V bN[cV SnXdKB[cSB[�d d^Vkej]CBAVfejVf\M[_bNej]D$�$�V*$Y]oUfSB[*i ' U*SAgL]^Z;[�eN]^V \MU

Vfd doUfehVl]od 5f[�ejgNUfej[e1(t) =

X(t)

|X(t)|B([Ye XYUf\M[JV*(J(B]^V \MUMZ;[ �ASA] bjU d^V eN]DBAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[��*gN[ �]<B[4g bOL <Bd bN]o\kV��!ej[�I*Ufd^V ej[�0] d5*[YehgLU*eN[

e1ej]o\MVfSB[ ] S�5�V ej]^V�bNUniF]oS=. V bLbN]

dX

dτ=

dX

dt· dt

dτB([Ye X�<B]

e1(τ) =X(τ)

|X(τ)|=

X(t)

>0︷︸︸︷

dτ

dt|X(t)||dτ

dt |= e1(t)

i8� � j�� =8\��' U*SAgL]^Z;[�eN]^V \MU doV ej[_bLbhV ] S . Ufej\kV6BAV ehV \M[_bjeN]^XYV

X(t) =

(tt

)

XYUfSt ∈ [−1, 1]

i�: S��]<B[4g bjUkXYVfgNU!�

X(t) =

(11

)

�� c�� p�� e� �Z�� o� �p�� i� ! ,

ZJ<BS!�]<B[

e1(t) =X(t)

|X(t)| =

(11

)

√2

=

(1√2

1√2

)

' U*SAgN]oZ;[�eNU6<BS�L V d bNehV6BAV ehV \M[YbNej]C$8$�V%$�] U*SB[!ZB[YdodoV ej[_bLbhV��

X(t) =

(t3

t3

)

XYUfSt ∈ [−1, 1]

i�: S��<A[�gLbNUkXYV*gLUB([Ye��!�

X(t) = 3

(t2

t2

)

X(0) =

(00

)

-�<;bNbjVV5a]oV!�BgL[cX�V d^X_U*d ]^V \MU ] d 5*[YehgLU*eN[e1V*(J(B]^V \MU

e1(t) =X(t)

|X(t)| =

3

(t2

t2

)

3√

2t4=

(1√2

1√2

)

� U*gLbNej]oVf\ U!UfehV <BS [�gN[Y\BB]oU!Z;]BX�<Be?5uVT] SMXE<A] �ZBBej[�gN[YS0bN[O]oS)<BS6BJ<BS0bNU+<BSAVJgN] SJI*Ufd^V�mej] b?o��0X_]oU/��SBUfS�� B(U*gjgL]D(B]od [ Z;[ �ASB]oeN[�]oS bjVfd [ BJ<AS*bjUc]odJ5f[YbLbNU*eN[ObhV SJI*[YS0bN[ <BSA]C5*UaX�V \M[YS0bj[fi iO� � j��\� �98��' U*SAgN]oZ;[�eN]^V \MU doVMX�<BeN5�V

X(t) =

(t3

t2

)

XYUfSt ∈ [−1, 1]

i : S . Ufej\MV [�gNBBd ]^X_] bjV1�]<B[�gLbjV XE<Be?5�V � ZAV�bjV ZBVy =

3√

x2i ' VfdoXYU%m

d^V SnZ;UX(t)

SBUfbN]^V \MU�XdKB[ ]oSt = 0

��<A[�gLbNU 5uVfd [ (00

) ��]<B] SnZ;] KBU <BSAVrgN] SJI*Ufd^V ej] bdoBi��[4Z;]^V \MUkX_Uf\M[ gL]DXYUf\BnU*eLbhVl]od 5f[�ejgNUfej[ ]oS \MU;Z;UMZBV gNV*Bn[�eN[ gL[ �g<BS BBeNU�(Bdo[Y\kVlZ;[�d d^VBnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[TUWZ;[�d d^VMXE<Be?5�VBi A�]^g/<Bd bjV!�

e1 =X(t)

|X(t)| =

(3t2

2t

)

√9t4 + 4t2

=

(3t2√

9t4+4t22t√

9t4+4t2

)

W <A] SAZB]��=. V*X_[�SAZ;UM] d�d ]o\M] bj[cZ;[�gLbNejUM[ gN]oSB]ogLbNejUkZ;]e1(t)

] St = 0

�

limt→0−

e1(t) = limt→0−

(3t2

t2√

9+4/t2

2t−t

√9t2+4

)

= limt→0−

(3√

9+4/t2

2−√

9t2+4

)

=

(0

−1

)

[

limt→0+

e1(t) = limt→0+

(3t2

t2√

9+4/t2

2tt√

9t2+4

)

= limt→0+

(3√

9+4/t2

2√9t2+4

)

=

(0

1

)

W <A[�gLbNU Z;] \MU0g bjejV XdKB[ doV gL]oSJI*Ufd^V ej] bdo-�BBeNU�BBej]oVWZB[YdodoV XE<AeN5�V��,SBU*S ZB[YdodoV BAV eNbN]^X_U*doVfeN[BnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[ <;bj] do]C$8$�V bjVBi

! 2 ��

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−1

−0.5

0

0.5

1

t3 / t2

Derivata

q/]DI*<BehV ! iG9(� P gN[Y\BB]oU ! i !�!

�� %��'&�"4" ��'& �� 1�2�

� S Vfd bjeNU6BAVfejVf\ [YbNejU ]o\BnU*eLbhV S0bN[ � d^V j hiC7rafq@)O)O�N�> @�jMj N'G$hiRS`JN �(Z;[ �ASA] bhV ZAV

s(t) =

∫ t

a

∣∣∣∣

dX

dt

∣∣∣∣dt

HFVr. <BSJ$�] U*SB[!gjXYVfdoVfeN[s� <BSAVs. <BSJ$Y]oUfSA[JXYeN[4gNXY[YS0bN[��f]oS*bjeN]oSAgN[�X�VlZ;[�d d^VlX�<BeN5�V

X�0]oS=. V bLbN]��

gN[ XYV \4(B]^V \MU4BnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[�

s(τ) =

∫ τ

c

∣∣∣∣

dX

dτ

∣∣∣∣dτ =

∫ t

a

∣∣∣∣

dX

dt· dt

dτ

∣∣∣∣

dτ

dtdt =

∫ t

a

∣∣∣∣

dX

dt

∣∣∣∣dt

` V�bjU XdKB[s(t)

� <BSAV . <J$Y]oUfSB[ X_ej[�gjX_[�S*bj[ # [ Z;[Yej]C5�V*(B] do[3% BnU0gNgNU <;bj] do]C$8$�VfeNd^V X_U*\M[ej]CBnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[�*XY] U/�

X(s) con 0 ≤ s ≤ L

RTd doUfehV,�]<B[�d doUWXdKB[ U bNbN[�SJIfU��

X′(t) =dX

ds=

dX

dt

1dsdt

=dX

dt

1∣∣dsdt

∣∣

=X(t)∣∣∣X(t)

∣∣∣

= e1(t)

m;V*BJBB]^V \MUkXdKB[X′(s)X′(s) ≡ 1

` [Yej]C5�VfSAZ;U-�]<B[�gLb8L <Bd bN]o\MVM[4g/BAeN[4gNgN] U*SB[�0U bNbN[�SB]oVf\MU

2X′(s)X′′(s) ≡ 0

�� a �o� �� Z� ��?�!�� o� �� ! 9

e

e

e1

2

3

q/]DI*<BehV ! i @!� qBej[YSB[Yb . ejVf\ [

XY] U/� ]�5f[_bNbNU*eN]X′ [ X′′ gLU*SBU�U*eLbjU*IfU*SAV do]��nX_U*SAZ;]D$Y]oUfSB[ XdKB[lSAUfSX5�Vfd [4Bn[�e!b?<;bLbj]�]Fbj]CBA]ZB]�BAV ehV \M[YbNej]C$8$�V%$�] U*SB[�0]oS=. V�bNbN]��

dX

dt=

dX

ds

ds

dt

[

d2X

d2t=

d2X

d2s

(ds

dt

)2

+dX

ds

d2s

d2t6= 0

m;]oX�X_Uf\M[X′ [ X′′ gNUfSBU UfeNbNU*I*UfSAVfd ]��(] SAZB]C5a]^Z=<AV SAU <BS BB]^V SBU XdKB[�� XdKB]^V \kV bNU b <BNJCZHHJE�G$hij NPTmHPRe@ i : d BAVfejVf\ [YbNejU

s�ZBn[�e �]<AV S0bjU 5a]^g bjU��ASBUfehV1�U bLbj] \MU��\kV�� BnU0gNgN]D(B] do[

ZB] \MU*gLbNehV ej[ XdKB[ d [g. <ASJ$Y]oUfSB]FX_Uf]oS�5fU*d bj[JgNUfSBUMZ;]Dbj]CB(Uk]oeNehV%$�] U*SAV do[�(�]<B]oSAZ;]9dML <BbN]od ]D$�$�UWZ;]bhV do[ BAVfejVf\ [YbNejU BJ< � B(UfeNbjV ej[kV BBejU*(Bdo[Y\M]?ZAV d BJ<AS*bjU"Z;] 5a]^g bhV X_U*\BJ<;bjV*$Y]oUfSAVfd [*i �B(U*gjgN]C(B]odo[cZ;[ �ASA] ej[g<BSgL]^g bj[Y\kVMZ;]9ej] . [�eN]o\M[YS0bNUMdoU;XYV do[�;ZBV bNUkZBV

e1(s) = X′(s)

e2(s) =X′′(s)

|X′′(s)|e3(s) = e1(s) ∧ e2(s)

ZBU�5f[M] 5f[�ejgNUfej]e1, e2, e3

BAeN[�SAZ;UfSAU ej]^g/B([_bNbN]D5uVf\M[YS0bN[M]odHSBUf\M[kZ;] `J@�R)E$HPRe@ TmNPC7r7@$C=Tm@ �`J@�R)E$HPRe@�C9HPRSIQNPj @ b RS<MC9G�I*KB[8$�$�V gN]�<;bN]odo]C$8$�V6<BS BAV ehV \M[YbNejU4I*[YSB[�eN]^X_U

e1(t) =X(t)∣∣∣X(t)

∣∣∣

e2(t) =X(t) −

(

e1(t) · X(t))

e1(t)∣∣∣X(t) −

(

e1(t) · X(t))

e1(t)∣∣∣

e3(t) = e1(t) ∧ e2(t)

-/V do[cgN]ogLbN[�\MVMZ;]Dej]C. [Yej] \M[YS0bjU6BAeN[�SAZ;[ ] d�SBUf\M[cZ;] �)Re@�C9@�T �)ReNJIL@ i

! @ ��

�� * ��1 �p �� 1�

5 �76�� 98��JS��!N��hDNJC=TU<MT

κ(t) =

∣∣∣X(t) ∧ X(t)

∣∣∣

∣∣∣X(t)

∣∣∣

3

RS<FE b @$T]TmH�N > hiC�r�@�C9@�RS<BG)H b NJR[NJIL@$T]ReH t �Zb Re@�C9> @�<Mj CZHJIL@�>P<4G�hiRS`aNPTUhiReN Vma[ [4g/BAeN]o\M]oVf\ U d^V X�<Be?5uV bN<BehVlbNehV \M] bj[ ] d`BAV ehV \M[_bjeNU

s�]<B[4g bhV ZB]C5*[YS0bjV

κ(s) = |X′′(s)|HFVWXE<Be?5�V�b?<BejV �)<BSAV��]<AV S0bN] bjVW] S0bNej]oSAgL[4XYVWZB[YdodoV XE<Be?5�V��(\ ]^gN<BejVWdoV Z;[�5a]^V%$Y]oUfSA[c] SX<BSBJ<AS*bjUkZBV dodoV bhV SJI*[YS0bN[*i ' U*SAgN]oZ;[�eN]^V \MU ] d�d [8I*V \M[+. ehV X�<Be?5uV bN<BehV [cgL]oSJIfU*doVfeN] b?o!�FgN[

κ(s) ≡ 0 ⇐⇒ X′′(s) ≡ 0 ⇐⇒ X(s) = αv + w

ma[ ]oS�5f[4X_[κ(s) = 0

B([Ye ��<nV d^XdKB[s�BVfd doUfehV

X′′(s) = 0��]<B] SnZ;]��

• gL[X′′′(s) 6= 0

Vfd doUfehV��a]oS <AS ]oS0bNUfejSBUkZ;]seN]^g/<Ad bhV��

X′′(s) ≈ X′′′(s)(s − s)

XdKB[� B(U*gN] bj]C5*UrBn[�es > s

�*SB[8I*V bN]D5fUcV d bNej]o\ [�S0bN]��*X_]oU/�� <BS B><BS0bNUlZ;]n] S��A[�gjgL]oUfSA[# XYVf\)(B]^V doVkXYUfS�5f[4gNgN] b?olZB[YdodoVMXE<AeN5�V�%_i

• gL[ ]oS�5f[�XY[X′′′(s) = 0

[X′′′′(s) 6= 0

V dod U*ejV

X′′(s) ≈ 1

2X′′′′(s)(s − s)2

XdKB[ SBUfS XYVf\)(B]^VMgL[8IfSBU SB[�d dML ]oS0bN[�eNSBUkZ;]s[ �]<B] SnZ;] �r<BS BJ<BS0bNU BA]oV bLbNUni

3�[Ye�U*I*SB]JBJ<BS0bjUlZ;[Ydod^VcXE<Be?5�V��BnU0gNgNUcZ;[ �nSB] ej[ ]od G)@�ReG[fi<BHLHaE�G�hij NJT HJRe@ �aXdKB[TgLb?ocgN<Bd>BA]oVfSBUU0gNX�<BdoV bNU*eN[J[�Z�� bjVfd [ XdKB[��] % BAVfgjgNV6B([Ye

X(t)

] ] % � bjV S>If[YS0bj[J]oSX(t)

VX(t)

]o] ] % KAVM] dDehV%I�If]oU ]oS�5f[�ejgjV \M[YS0bj[ BBejU*B(UfehgL]oUfSnV do[!V dod^V X�<Be?5uV bN<BehVr = 1

κ(t)

� U0g bjeN]^V \MU!UfehV XdKB[OdoVJXE<Be?5�V�b?<BejVTSBUfS Z;]DBn[�SAZ;[�ZBV dod^V BAVfeLbj]oXYUfd^V ej[ BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[Z;[�d d^VMXE<Be?5�VMgNXY[Yd bjV��

dX

dτ=

dX

dt

dt

dτ

d2X

d2τ=

d2X

d2t

(dt

dτ

)2

+dX

dτ

d2t

d2τ

W <B]oSAZ;]dX

dτ∧ d2X

d2τ=

(dt

dτ

)3(dX

dt∧ d2X

d2t

)

�� ;�� ! F

"TbNbN[�SB]oVf\MU6B([YeNbjVfS0bNU

κ(τ) =

(dtdτ

)3(

dXdt ∧ d2

X

d2t

)

(dtdτ

)3 ∣∣dX

dt

∣∣3 = κ(t)

" [Yd�X�VfgNUl]oS X�<B]9gN] 5fU*I*d ]^V SAU VV5f[�eN[ BB]�� ]oS=. Ufej\kV%$�] U*SB]Dg/<Bdod^V XYUfS�5f[4gNgN] bdolZ;[�d d^V XE<Be?5�V��(�B(U*gjgN]C(B]odo[cX_UfSngL]^Z;[YehV ej[Td^VMXE<Be?5�V�b?<BejVMXYUfS gL[8IfSBU��aZB[ �ASB] bjVkZAV��

κ(t) =X(t) × X(t)∣∣∣X(t)

∣∣∣

3

�-�� 1!�!#$��%'&

" [YdJXYVfgNU�gL] <;bN]od ]D$�$�] doU�g/BAV*$Y]oU bNej]oZ;]o\M[�gN] U*SAV do[� X_]!gL[�eN5*[ <AS�L ]oS=. U*eN\kV%$�] U*SB[V%I�If]D<BS=mbj]C5�V"eN]^g/B([_bNbNU V dodoV X�<Be?5uV bN<BehV��B(Uf]^XdK��Wd^V"XE<AeN5�V[4gNXY[�gL[�\BBeN[ ZBV dZBB]^V SBU U0gNX�<Bd^V�bNU*eN[# V d bNej] \M[�S*bj]�BJ< �M[4gNgN[Yej[ ejV*BJBBej[�gN[YS0bjV bjVlgN<Z;] <BS BB]^V SBU/% i5 �76�� 98��=h�:=<?> @UA C=<FE�G[@-T HJR)ES<BHJCZ@-j N��hDNJCDT]<MT �

τ(t) =det( .

X(t)..

X(t)...

X(t))

∣∣∣

.

X(t) ∧..

X(t)∣∣∣

2

>�HJ`J@ .

X(t)..

X(t)...

X(t)> @�C9HPTmN j N�IQNPTURS<BG)@

3 × 3G)fq@�fqN b @$R�G)HPj HJC=C9@�RS<FE b @�TUT]<M`JNJIL@$C=Tm@<?`a@�TUT HJRS<_<MCZ>J<BG)NJT]< V

HFV bNU*ejgN] U*SB[��Z;[ �ASA] bhV�gLU*d UrgN[doV�X�<BeN5�V bN<BehV��SAUfS S�<Bd d^V��O\M]ogN<BehV doV�Z;[85a]oV*$Y]oUfSB[ej]^g/B([_bNbNUWV d BA]oVfSBUMU*gjXE<Bd^V�bjUfej[JZ;[�d d^VMXE<Be?5�V [�Z��r<BSAV \M]ogN<BehV ]oS0bNej] SngL[4XYVMZ;[Ydod^V gLbN[4gNgjVBi

� bN]od ]D$�$4V SAZBU XYUf\M[ BAV ehV \M[YbNejU-d^V dC<BS>I*KB[8$�$�V Z;[�d d^V XE<Be?5�V��Jd\L [�gNBBej[�gjgL]oUfSA[ Z;[�d d^VX�<Be?5uV bN<BehVMZ;]D5f[YS0bhV��

τ(s) =det (X′(s)X′′(s)X′′′(s))

|X′(s) ∧ X′′(s)|2=

=(X′(s) ∧ X′′(s)) ·X′′′(s)

|X′′(s)|2=

=(e1 ∧ |X′′(s)|)X′′′(s)

κ(s)=

=|X′′′(s)| e3

κ(s): S (AV*gL[ V��]<B[�gLb8L <Bd bN]o\kVM[�gNBBeN[4gNgN]oUfSB[ Z;[Ydod^VlbjUfehgL]oUfSA[ B(U*gjgL]^V \MUMSBUfbjV ej[ XdKB[�

• gN[ld^VWbjUfehgL]oUfSB[�� S�<BdodoV�] S <BSXB><BS0bNUsVfd doUfehVk]oSX<BS ] S0bNU*eNSAU Z;] �]<B[Yd�BJ<AS*bjU�d^V

XE<AeN5�V bN[�SAZ;[cV eN]o\kV SB[�eN[ gN<Bd�BB]^V SBUMU0gNX�<BdoV bNU*eN[

• gN[Jd^V bNU*ejgN]oUfSB[ � ]oZ;[�S0bN]^XYV \M[�S*bj[ S�<Bd d^VMV dod U*ejV d^V X�<Be?5uVMgLb?oMgN< <BS BB]^V SAUAi

�-�� 3&�, �0#

��[4Z;]^V \MUrUfehV VfdoX�<BSB]�[�gN[Y\BB] ZB] XE<Be?5f[ g/< XE<A] . V ej[ XYUfSAgN]oZB[YehV%$Y]oUfSA]OgN<Bdod [Z;[�eN]D5�V�bN[��gN<BdodoVkX�<Be?5uV bN<BehV [ gN<BdodoV bjUfehgL]oUfSA[fi

! & ��

��

W <B[4g bjU bN]DBnUWZ;]9X�<Be?5uV6KAVM[3�]<AV%$�] U*SB[ BnV ehV \M[_bjeN]^XYV

X(θ) =

(eθ cos θeθ sin θ

)

0 ≤ θ ≤ 2kπ

' VfdoXYUfdo]oVf\ U d^V ZB[Yej]C5�V�bhV6BBeN]o\kV��

X(θ) =

(eθ(cos θ − sin θ)eθ(sin θ + cos θ)

)

: SBU*d bjeN[ �]<B[4g bhV X�<Be?5uV�� ej[�IfU*doVfeN[ BnU*]oXdK!� |X(θ)| SBUfSgN]9V SBS�<BdodoV!�∣∣∣X(θ)

∣∣∣ = eθ

∣∣∣∣

(cos θ − sin θsin θ + cos θ

)∣∣∣∣= eθ

√

(cos θ − sin θ)2 + (sin θ + cos θ)2 = eθ√

2

HFV ZB[Yej]C5�V�bhVMgL[4X_UfSnZBVlej]^g/<Bd bjV

X(θ) =

(eθ(cos θ − sin θ − sin θ − cos θ)eθ(cos θ + sin θ − sin θ + cos θ)

)

= 2eθ

(− sin θcos θ

)

' VfdoXYUfdo]oVf\ U6B([YeNbjV S0bjU d^VMXE<Be?5�V�b?<BejV!�

κ(θ) =

∣∣∣X× X

∣∣∣

∣∣∣X

∣∣∣

=

=

∣∣∣∣eθ

(cos θ − sin θcos θ + sin θ

)

× 2eθ

(− sin θcos θ

)∣∣∣∣

2 |eθ|3

= 2eθ ((cos θ − sin θ) cos θ + (cos θ + sin θ) sin θ)

2√

2e3θ=

=1√2e−θ

��! "#��$��%&�

3Hej]o\MVlZ;](ZAV ej[�doVg. U*eN\kVrBnV ehV \M[_bjeN]^XYV Z;[�d d^VlX�<BeN5�V XY]oXYd U*]oZ;[��0ZB[�gjX_ej]C5a]^V \MUcX_U*\ [ � If[�mSB[�ejV bjV��]<B[�gLbjVkX�<BeN5�V!��gN<JBJB(UfSA]oVf\ UkZ;]FVV5f[Yej[+<BSX_[YehXdKB]oU XdKA[Je?<BUfbjVMgL[�SJ$�VMgLbNej]ogjX_]^V ej[gNU*BBehV <BSnVWej[_bLbhV-�ngjgjV # K)NaE�@ %!] S \MUaZBU XdKB[ ] S U*IfSA]�]^g bhV S0bN[ d\L V ehX_U�ZB]�XY] ehX_U*S=. [Yej[YS>$�VXdKB[ gL]2� g/5a]odC<>BJBAV�bjU gN<BdodoVX(AVfgN[ gjV edo <JI�<AV do[ Vfd gN[�I*\M[YS0bNU ej[_bNbN]od ]oSB[�U B([YehX_U*ejgNU ZBV dXY[YS0bNejUAi q/]ogjgjV�bNU�Vfd doUfehV <BS �]<AVfdC<BS��<A[ BJ<BS0bNU

PZ;[Yd BB]^V SAU!� XdKB[gL]^V eN]DIf]^ZBV \M[�S*bj[

XYUfSBSB[4gNgNUcX_U*dnXY[YehXdKB] U��]<B[4g bjU \ U�5a]o\M[YS0bNUlZB[�gjX_ej]C5*[�doVcX�<Be?5uVlXdKB[TgN]nXdKB]^V \kV G�<BG$j HJ @ i3H]�� BBej[�X_]^gjV \M[YS0bN[MgN] BAV ejd^V Z;] G�<BG�j HJ�@nHJRe>P<MC9NJRS<BN gN[ ]od BJ<AS*bjUV%BJBAVfeLbj] [�SB[MV dodoV X_]oeNmXYUfS=. [�eN[�SJ$�V!�A\M[YS0bNej[lgN]`BAVfeNdo[YedoMZ;] G$<BG�j HJ�@ NPjMj hiC7r7NJT N U G$<BG�j HP @ N�G)G[HPReG�<BNJT N gN[c[4gNgNUI*]oV*X_[lVfd dML ]oS0bN[�eNSBU U Vfd dML [4g bj[YejSBU�Z;[�d�X_[YehXdKB]oUAi+H L [ �]<AV%$�] U*SB[sBAVfejVf\M[_bNej]^XYVWZ;[�d�X_]^X_doUf]^Z;[��

X(t) =

(at − h sin ta − h cos t

)

Z;U�5*[a, h > 0

[0 ≤ t ≤ 2π

i : d`BAV ehV \M[_bjeNUaehV%B>BBeN[4gL[�S0bjV ] d9ehV%I�If]oU Z;[�d�X_[�ejXdKA] U��;[�Z

h�MdoV Z;]^g bhV SJ$4V ZB[Yd BJ<BS0bjU

PZAV dHX_[�S0bNejU Z;[�dHX_[YehXdKB]oUAi ' U*SAgL]^Z;[�eN]^V \MU Vfd doUfehVWdo[ bNej[

�� 7 � � � �n� � ! 1

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

0 5 10 150

50

100

150

200

250

300

350

400

450Spirale logaritmica Curvatura

qF]DI*<AejV ! ifF/�Hm�BB]oehV do[JdoU*I0V ej] bj\M]oX�Vl[ ej[Yd^V�bN]D5�VMXE<Be?5�V�bN<AejV

B(U*gjgN]C(B]odo[ 5uVfeN]^V%$�] U*SB]�Z;[Y]tBAV ehV \M[_bjeN]a, h

i ' Uf\M[BBej] \MU X�VfgNU gN<JBJB(UfSB]^V \MUXdKB[MbhV do]BnV ehV \M[_bjeN]BgN]oVfSBUJ]oZ;[�S0bN]^X_]B[ B([Ye?gN[Y\BBdo]^X_] b?o+BnU*SB]^V \MU V SAXdKB[

a = h = 1i�HtL [3�]<AV%$�] U*SB[

ZB[Yd�XY]oXYd U*]oZ;[ Z;]D5f[�S0bjVMV dod U*ejV!�

X(t) =

(t − sin t1 − cos t

)

' VfdoXYUfdo]oVf\MU d^VkZ;[Yej]D5uV bjV4BBej] \kV

X(t) =

(1 − cos t

sin t

)

' U*S0bNejUfdod ]^V \MU6BnU*]9doV ej[�I*Ufd^V ej] bdolZ;[�d d^VMXE<Be?5�V∣∣∣X(t)

∣∣∣ =

√

(1 − cos t)2 + sin2 t =√

2√

1 − cos t

W <A] SAZB] Bn[�et = 2kπ

d^V XE<Be?5�V)BBej[�gN[YS0bhV)<BS`L ]oeNej[�I*Ufd^V ej] bdoBi!��[Yej].�nXdKA]oVf\ UMXdKB[ gN]nbjejV bLbhVZB]r<BS�L ] ejeN[8IfU*doVfeN] b?o�Z;[�d d^V XE<Be?5�V [ SBUfS Z;[Ydod^V BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[�� X_UfSngL]^Z;[YehV SnZ;U ]od5*[YehgLU*eN[

e1(t) =X(t)∣∣∣X(t)

∣∣∣

=1√2

( √1 − cos t

sin t√1−cos t

)

W <A] SAZB]��=. V*X_[�SAZ;UM] d�d ]o\M] bj[cgL]oSB]^g bjeNUM[ Z;[�gLbNejUBn[�et → 2π

�;U bNbN[�SB]oVf\MU

limt→2π−

e1(t) =1√2

(0−1

)

limt→2π+

e1(t) =1√2

(01

)

W <A[�gLbNU gL]DIfSB]��nX�V�XdKB[ ] S2kπ

KBU <BSAV XE<AgNBB]^Z;[fi P XYX_[YbLbjU XdKA[l]oS��]<B[4g bj]�BJ<AS*bj] BnU0gNgNUXYUf\4<BS!�]<B[ XYVfdoXYUfd^V ej[!d^VMXE<Be?5�V�bN<AejVAi

X(t) =

(sin tcos t

)

�*7 ��

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16 18 200

1

2

3x 10

13

Curvatura

Cicloide ordinaria

q/]CI�<BejV ! i &�� ' ]^X_doUf]^Z;[ UfehZ;]oSAV ej]oV

W <B]oSAZ;]κ(t) = − cos t − 1

2√

2(1 − cos t)√

1 − cos t= − 1

2√

2√

1 − cos t' U*SAgL]^Z;[�eN]^V \MU VfZ;[4gNgNUc]od(X�VfgNU ] S XE<B]

a > hi�3�[Ye gL[�\6BAd ]^X_] b?orB(UfSA]oVf\ U

a = 2, h = 1i

RTd doUfehV

X(t) =

(2t − sin t2 − cos t

)

X(t) =

(2 − cos t

sin t

)

��[�eN]��nXdKB]^V \MU doVMej[�I*Ufd^V ej] bdo��∣∣∣X(t)

∣∣∣ =

√

(2 − cos t)2 + sin2 t =√

5 − 4 cos t

W <B[4g bOL <Ad bj] \kV�]<AVfS*bj] bdo2�?gN[Y\BBej[ BnU0gL] bN]D5�V�� ]<B] SnZ;]*d^V�XE<AeN5�V �Hej[�I*Ufd^V ej[fiZ��[Yej]��nXdKB]^V \MUU*ejV doVMX�<BeN5�V bN<BehV��

X(t) =

(sin tcos t

)

W <B]oSAZ;]κ(t) =

2 cos t − 1

(5 − 4 cos t)√

5 − 4 cos t' U*SAgL]^Z;[�eN]^V \MUc]oS(�ASB[J]od(X�VfgNU

a < hB(UfSB[�SAZ;U4Bn[�e gN[Y\BBdo]^X_] b?o

a = 1, h = 2i�A ]ogN<Bd bjV

�� 7 � � � �n� � �=!

0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

Curvatura

Cicloide allungata

q/]DI*<BehV ! i 1!� ' ]oXYd U*]oZ;[cVfd dD<BSJI0V�bhV

X(t) =

(t − 2 sin t1 − 2 cos t

)

X(t) =

(1 − 2 cos t

2 sin t

)

��[�eN]��nXdKB]^V \MUMd^V ej[�I*Ufd^V ej] bdo��∣∣∣X(t)

∣∣∣ =

√

(1 − 2 cos t)2 + 4 sin2 t =√

5 − 4 cos t

W <A[�gLb8L <Bd bj] \kV-�]<AVfS*bj] bdo�# � ]^Z;[YS0bN]^XYVMVfd�XYV*gLU6BBej[�X_[4Z;[YS0bj[3% �cgL[�\BBeN[gB(U*gN] bj]C5�V!��<A] SAZB]d^VkXE<AeN5�V�� eN[8IfUfd^V ej[fi!��[Yej].�(XdKB]oVf\MU U*ejV d^V X�<Be?5uV bN<BehV��

X(t) = 2

(sin tcos t

)

W <A] SAZB]κ(t) =

2 cos t − 4

(5 − 4 cos t)√

5 − 4 cos t

��

' U*SAgN]oZ;[�eN]^V \MU UfehV <BS [�gN[Y\BB]oU ZB]HXE<Be?5�V�SB[�d doUg/BnV%$Y]oU!�Td\L [Ydo]oX�VBi6H L [ �]<AV%$�] U*SB[)BnV ehV�m\M[YbNej]oX�V ZB] �]<B[�gLbjVMXE<AeN5�V��

X(t) =

cos tsin tat

� � ��

−5 0 5 10 15 20−1

0

1

2

3

−5 0 5 10 15 200

200

400

600

Curvatura

Cicloide accorciata

q/]CI�<BehV ! i !O7 � ' ]oXYd U*]oZB[ V*XYXYUfehX_]^V�bjV

XYUfS0 ≤ t ≤ 2kπ

[aBAVfejVf\M[_bNejUcej[�Vfd [TZB]C5*[YehgLUlZAVs$Y[�eNUni�HFVlZ;[�eN]D5�V�bhVsBBej] \kVleN]^g/<Ad bhV��

X(t) =

− sin tcos t

a

�]<B]oSAZ;]��B]oS \MU;Z=<BdoUMeN]^gN<Bd bhV∣∣∣X(t)

∣∣∣ =

√

sin2 t + cos2 t + a2 =√

1 + a2

ma]^XYXYUf\M[cd^V�Z;[�eN]D50bjV BAeN]o\kV-� XYU*gLbjV S0bj[ �s(AVfSAV do[cej]oX�VV5uVfeN[ d^VWdD<BSJI�KB[�$8$�VkZ;[�d dML V ehX_U�Z;]X�<BeN5�V!�

s(t) =

∫ t

0

∣∣∣X(t)

∣∣∣ dt =

√

1 + a2t

' VfdoXYUfdo]oVf\ U6B(Uf]Dd^VkZ;[Yej]C5�V bjV gL[4X_UfSnZBV��

X(t) =

− cos t− sin t

0

A�]^XYVV5a]^V \MUMV dod U*ejV doVMX�<BeN5�V bN<BehV��

κ(t) =

∣∣∣∣∣∣

a sin t−a cos t

1

∣∣∣∣∣∣

(√1 + a2

)3 =1

1 + a2

�� 7 � � � �n� � � ,

−10

1

−1

0

10

10

20

−1

0

1

−1

0

10

1

2

−1

0

1

−1

0

1−1

0

1

−1

0

1

−1

0

1−1

0

1

X’’’X’’

X’X

q/]CI�<BejV ! i !�! � P d ]^XYVM[ ej[Yd^V�bj]C5*[ ZB[Yej]C5�V�bj[

W <A] SAZB]TV SAXdKA[ d^VrXE<Be?5�V�bN<AejV�� XYU*gLbjVfS*bj[fi ' V d^X_U*d ]^V \MU"U*ejV d^V bjUfehgL]oUfSB[��?ej]oX�VV5�V SAZ;UBAeN]o\kV d^VkZ;[�eN]D5�V�bjVlbN[�eN$4V��

...

X(t) =

sin t− cos t

0

HFV bNU*ejgN] U*SB[JeN]^g/<Ad bhV4B([YeNbjVfS0bNU

τ(t) =

det

− sin t − cos t sin tcos t − sin t − cos t

a 1 0

1 + a2=

a

1 + a2

` VMSBUfbjV ej[ XdKB[cV SAXdKA[Jd^V bNU*ejgN] U*SB[ �cX_U*gLbjVfS0bN[fi

� 2 ��

0 5 10 15 205.94

5.942

5.944

5.946

5.948

5.95

5.952

5.954

5.956

5.958

5.96x 10

−3

0 5 10 15 200.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

CurvaturaTorsione

q/]CI�<BehV ! i ! � � ' <Be?5uV bN<BehV [ bNUfehgN] U*SB[JZB[Ydod\L [Ydo]oX�V

� �v�� u�"��*� �� u��*��u�� U��u� �U�� # ��4�� R�� z��*�%�*��?_�� z�vn�� w��f��%�� # ��) ��H��u��v�� %�, ��F�� ,��w��z ��T��u��u��)�*�\+��v��,�� # ��w�� # )�� #��*�*�� u��)�*�� u��e��l�� i�\A�v�� v��v��5 ��u��*��u�� w�,�T�*�,+

yig 3jg 3�.\9T2�!

2" BC>�# ;��%$'&aD �� >

H�[ bj[YUfej]^V [ d [ BBejU*BBej] [Yb?o Z;] �]<B[Ydodo[TXdKB[ BBej[YSnZ;UfSBUl] dnSBUf\M[TZ;] G�hiRS`a@�>P<)(+*[O�<B@�R . ��g/5a]CmdD<JBJBnV�bjV ]oSAZ;]DBn[�SAZ;[YS0bj[Y\M[YS0bN[ ZBV43�iaZB[ ' V*g bj[Yd cLV%<�V)BAV eNbN]oeN[JZBV d

1959[ ZBVs3�iJ_Za�$�] [�e

SA[Yd1962

�fXdKB[�do[ <;bN]odo]C$8$�V ejUfSAUJSA[YdodoV ej[�Vfd ]D$�$4V%$Y]oUfSA[OZ;[�]ngN]ogLbN[�\M] ' R ` eN]^gNBn[YbLbN]D5�V \M[YS0bj[ZB[YdodoV4A�[YSAV*<Bd b�[TZ;[�d d^V ' ] bNejU-,YS9i�maU*d UcSA[Yd

1970Aci]qAUfejeN[4g b?gjX_U�BBe)�9doVlX_UfSASB[�gjgL]oUfSA[ bNehV

]od(doVV5*UfejUJZ;]�_Za�$�] [�eH[ ]>B(Ufdo]oSBUf\M]nZB]�_O[YejSAgLbN[�] S �0XdKB[TZ;]>. V�bNbNUlgNUfSBU d^Vr(AV*gL[!Z;[Ydod [!XE<Be?5f[ZB]`_Za�$Y]o[Ye4i

(L�� $#$%'�-, # �'#/.+&1�% � �!&�#$%

m;UfSBUkZ;[ �ASB] bN]FZAV��

Bni (t) =

(n

i

)

ti(1 − t)n−i i = 0, . . . , n

[)Bn[�eTX_U*S]5*[YSJ$�] U*SB[+BnU*SB]oVf\MUBn

i (t) ≡ 0gN[

i < 0U

i > ni ��[4Z;]oVf\MUWVfZ;[4gNgNUkV d^XE<BSB[

BAeNU�BBeN]o[_bdoMXdKB[gKAV SBSAU��]<B[4g bj] BnU*d ]oSBU*\ ]vi : SBSAVfSJ$Y] bN<BbLbNU B(U*gjgN]oVf\ UMV�!,[Yej\kV ej[JXdKA[n∑

i=0

Bni (t) = 1

XY] U/�c]�BnU*d ]oSBU*\ ]/Z;]�_�[�eNSng bj[Y]oS b NJRST]< O�<BHJCZNJC9H'jek hiCD<MT i -/V do[rBBeNU�BBej] [Yb?okgN]�ej]oX�VV5uVMZBVfd doUgN5a] dD<JBJB(UWZ;] " [�� bjUfS �;]oS=. V�bNbN]

1 = 1n = (t + 1 − t)n =

(n

i

)

ti(1 − t)n−i

q/]^gNgjV�bjUt ∈ [0, 1]

ej]^g/<Bd bjV XdKB[ ]Bn

i

gNUfSBU-SBUfS SB[�I0V�bN]D5a]�� ]<B] SnZ;]gBnU0gNgNUfSBU [�gjgL[�eN[<BbN]od ]D$�$4V�bN]FX_Uf\M[ X_Ua[�0kXY] [�S0bN]�B([Ye Z;[ �ASB]oeN[r<ASAVMX_Uf\4(B]oSAV%$�] U*SB[JXYUfS�5f[4gNgjVBi

� S�L V d bNehV4BAeNU�BBeN]o[_bdol]oS0bN[�eN[4gNgjV S0bN[ Z;] �]<B[4g bj]�BnU*d ]oSBU*\ ]��cXdKA[+B(U*gjgNUfSBU [�gjgL[�eN[ Z;[ �JmSA] bj]`B([Ye�ej[�XE<AejgN] U*SB[ ZBV dodoV4. U*eN\4<BdoV

Bni (t) = (1 − t)Bn−1

i (t) + tBn−1i−1 (t)

� 9

� @ �� D �p�!�� ?�

-/V do[)BBejU*BAeN]o[_bdokgL]�Z;]o\MU*gLbNehV Bn[�e!]oSAZ=<>$Y]oUfSB[ g/<ni Y �Ji8� 3HejU�5a]oVf\MU�# XYUfS

n�ngjgjV�bNU/%

XdKB[ V*Ifdo]D[�gLbNej[Y\M]{

Bn0 (t) = (1 − t)n

Bnn(t) = tn

Bn0 (t) =

(n

0

)

t0(1 − t)n = (1 − t)n = (1 − t)(1 − t)n−1

Bnn(t) =

(n

n

)

tn(1 − t)0 = tn = t · tn−1

�E�� "!ehV��;ej]^X_UfehZBVfSAZ;Uld^V6BBejU*BBej] [Yb?o Z;[Y]FX_Ua[E0WX_]o[YS0bj]�(B]oSBUf\M]^V do](

n

i

)

=

(n − 1

i

)

+

(n − 1

i − 1

)

"TbNbN[YSA]oVf\ U

Bni (t) =

(n

i

)

ti(1 − t)n−i =

(n − 1

i

)

ti(1 − t)n−i +

(n − 1

i − 1

)

ti(1 − t)n−i

A�]^X_U*ejZAV SAZ;UMXdKB[

Bn−1i (t) =

(n − 1

i

)

ti(1 − t)n−i−1 e Bn−1i−1 (t) =

(n − 1

i − 1

)

ti−1(1 − t)n−i

]od�BAVfgjgNV*I*I*] U B([YeJU bNbN[YSA[Yej[ d^Vkbj[�gN] � ]o\ \M[4Z;]oV bNUni+3?eNU�5a]^V \MU�VfZB[�gjgLUWd^V BBeNU�BBej] [Yb?oWZ;]gN] \M\M[_bjeN]^V��BXY] U/�

Bni (t) = Bn

n−i(1 − t)]oS=. V�bNbN]

Bni (t) =

(n

i

)

ti(1 − t)n−i , Bnn−i =

(n

n − i

)

(1 − t)n−i(t)i

[�gN]oX�X_Uf\M[ (ni

)=(

nn−i

) doV!bN[4gL](� 5*[Yej].�nX�V�bhVBi ' VfdoXYUfdo]oVf\MU!VfZB[�gjgLUJdoV Z;[�eN]D5�V�bhVJZB]Bn

i (t)[ Z;[YbN[Yej\M] SA]oVf\ UM]�B><BS0bN]�Z;]�\MV*gNgN]o\ UdBn

i (t)

dt=

(n

i

)[iti−1(1 − t)n−i + (−1)(n − i)ti(1 − t)n−i−1

]=

=

(n

i

)

ti−1(1 − t)n−i−1 [i(1 − t) − (n − i)t] =

=

(n

i

)

ti−1(1 − t)n−i−1 [i − nt]

W <B]oSAZ;]�]Bn

i

KnV SBSBU \kVfgjgL]o\M]/doU;XYVfd ]/SB[Y]�BJ<BS0bj]ti = i/n, i = 0, . . . , n

i ' U0g��X_U*\ [V*(J(B]^V \MU). V bLbjU Bn[�e

Bni

[�gNBBej] \M]^V \MU d^VMZ;[Yej]D5uV bjV ]oS . U*eN\kV eN]^X_U*ejgN]D5uV!�dBn

i

dt=

d

dt

(n

i

)

ti(1 − t)n−i =

=i n!

i!(n − i)!ti−1(1 − t)n−i − (n − i)n!

i!(n − i)!ti(1 − t)n−i−1 =

= n(n − 1)!

(i − 1)!(n − i)!ti−1(1 − t)n−i − n

(n − 1)!

i!(n − i − 1)!ti(1 − t)n−i−1 =

= n

((n − 1

i − 1

)

ti−1(1 − t)n−i −(

n − 1

i

)

ti(1 − t)n−i−1

)

=

= n(Bn−1

i−1 (t) − Bn−1i (t)

)

�D �� c�� ?� �� F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

B04(t)

B14(t)

B24(t)

B34(t)

B44(t)

q/]CI�<BejV � i ! ��3�Ufdo]oSBUf\M]�Z;]�_�[�eNSng bj[Y]oS Z;] I*ejV*Z;U4]oS

[0, 1]

� U*gLbNej]oVf\ U6<BS�L V d bNehV6BBejU*BBej] [Yb?o ] \B(UfeNbjVfS*bj[ Z;[Y]�B(Ufdo]oSBUf\M]�Z;]�_�[�eNSng bj[Y]oS �n∑

i=0

i

nBn

i (t) ≡ t# � i ! %

W <A[�gLbjV6BBejU*BBej] [Yb?o gL]FZ;] \MU0g bjejV4Bn[�e ]oSAZ=<J$�] U*SB[ g/<ni

Y �JiO� _ V SnV do\ [�S0bN[B1

1(t) = t�� m=<JBJB(UfSB]^V \MU-XdKA["d^V # � i ! % 5�V dDI*V Bn[�en − 1

�TZB] \MU*gLbNej]^V \MUfdoU B([Yeni

� bN]od ]D$�$4V SAZBUkdoV4. Ufej\kV eN[4XE<BehgL]D5�V ZB]Bn

i (t)V*(J(B]^V \MU!�

n∑

i=0

i

nBn

i (t) =

n∑

i=0

i

n

[(1 − t)Bn−1

i + tBn−1i−1

]=

=n − 1

n(1 − t)

n−1∑

i=0

i

n − 1Bn−1

i (t) + tn − 1

n

n−1∑

i=1

i

n − 1Bn−1

i−1 (t) =

=n − 1

n(1 − t)t +

n − 1

nt

n−1∑

j=0

j + 1

n − 1Bn−1

j (t) =

=n − 1

n(1 − t)t +

n − 1

nt

n−1∑

j=0

j

n − 1Bn−1

j (t) +1

n − 1

n−1∑

j=0

Bn−1j (t)

=

=n − 1

n(1 − t)t +

n − 1

nt

[

t +1

n − 1· 1]

=

=n − 1

n(1 − t)t +

n − 1

nt2 = t

: S��ASB[��BBejU�5a]oVf\MUJd^VgBBejU*BBej]o[_b?ogBA]��W]o\BnU*eLbhV S0bN[ ZB[Y]>B(Ufdo]oSBUf\M],Z;]W_O[YejSAgLbN[�] S �0X_]oU/�TXdKB[[4gNgN] . U*eN\kV SAU4<BSnV4(nVfgN[ ZB]

Πni A�]^gN<Bd bhVM] SJ. V�bLbj]��

Bni (t) =

(n

i

)

ti(1 − t)n−i =

(n

i

)

tin−i∑

j=0

(n − i

j

)

(−1)n−i−jtn−i−j

� & �� D �p�!�� ?�

\kV ] d`BBeN[4X_[4Z;[YS0bN[ Vfd bjeNU SBUfS�� XdKB[ dML [4g/BAeN[4gNgN] U*SB[J[�gNBBd ]^X_] bjVMZ;[�d�BBejU;Z;U bNbNU��/�

Bn0 (t)

iii

Bnn(t)

=

1 ∗ ∗ ∗ ∗i i i∗ ∗ ∗(ni

)∗ ∗i i i

∗1

1tiiiiii

tn

Z;U�5*[ ]HbN[Yej\M] SA]�] SAZB]oX�V�bN]�X_UfS ∗ gLU*SBU SBUfS S�<Bdod ]viWR 0kSAXdK��M] Bni

. U*eN\M]oSBU <BSAV (AV*gL[��d^Vk\kV�bjeN]^X_[4BBeN[4X_[4Z;[YS0bN[ Z;[�5*[c[�gjgL[�eN[lSBUfS"gN]oSJIfU*doVfeN[*i � V �)<ASAVk\kV�bjeN]^X_[lbNej]oVfSJIfU*doVfeN[gN<JB([Yej] U*eN[�� ]<B] SnZ;]�] d�Z;[_bj[Yej\ ]oSAVfS*bj[��WZBV�bjU ZAV dodoV BBejU;Z=<;bLbjUfej]oV Z;[8Ifdo]�[Ydo[Y\M[YS0bj]?g/<Ad d^VZ;]^V%I*UfSAVfd [��0bN<;bNbN]�Z;]D5f[�ejgN]9ZBV

0�AejV*If]oUfS B([YeTX�<B]9]

Bni

. U*eN\kV SAU6<ASAV6(AVfgN[ Z;]Πni

m�<JB>BnU*SB]oVf\MU ZB]�VV5*[Yej[)<BSnV . <BS>$Y]oUfSB[f : [0, 1] → R

X_U*S0bN]oS]<nVBis3�U*gjgN]oVf\ U Bn[�S=mgjV ej[lZ;]�V%B>BBeNU0gNgN]o\MVfeN[

fbjejVf\M] bj[)<BS BnU*d ]oSBUf\M]oU Z;]�IfehVfZBU

n[�gNBBeN[4gNgNUWSB[�d d^V (nVfgN[ Z;]

_O[YejSAgLbN[�] S9i�XY] U/�Pn(t) =

n∑

i=0

f

(i

n

)

Bni (t)

" V�bN<AejVfd \M[YS0bj[�FZAV dod\L ] S0bj[Ye?5uVfd doU[0, 1]

gL]pBJ< � BAV*gNgjV ej[kVfZ <BS�]oS*bj[Ye?5�V dod U[a, b]

<;bj] do]D$Em$4V SAZ;UM] B(Ufdo] SAUf\M]�Z;]`_O[�eNSAgLbN[�] S If[�SB[YehV do]D$�$�V bN]��

Bni (t) =

(n

i

)(t − a)i(b − t)n−i

(b − a)n

XYUfSt = a + (b − a)τ

[τ ∈ [0, 1]

i -�<;bLbhVV5a]oV!�=5uVfd [ ]od�gN[�I*<A[YS0bN[�� >�� S 8��;:=<BN

f(x)hiCZN �)h�O�<BHJCZ@�G[HPCDT]<MCDhDN�E)h

[0, 1] V � jMj HPReN b @�RcHer CD< x ∈ [0, 1]@^Her CD<_<MC=Tm@�ReH b HJE)<MT]<M`JH nES<!fqN��

|f(x) − Pn(x)| ≤ 9

4ω(f ; n−1/2)

>�HJ`J@w(x)

��<Mj3IQH\>Jhij H'>J<?G)HPCDT]<MCDhi<MT >J<f(x)

<MC[0, 1] V

W <B[4g bjUkgL]DIfSB]��nX�V XdKA[ doVMX_U*S�5f[Ye?If[�SJ$�V,� \MUfd bNUkdo[YS0bjVAi ��[�Z;]^V \MUkVfZB[�gjgLU<BS�L <Bd bN[�eN]oUfej[BBejU*BAeN]o[_bdo ZB[Y]`B(Ufdo]oSBUf\M]�SA[YdodoV (nVfgN[cZ;]�_O[YejSAgLbN[�] S �(XdKB[rBBej[YSAZB[c] dFSBUf\M[cZ;] `JNJRS<BNJT]<BHJC>P<MI�<MC=<FElfi<MC7r i ��Wjt�WiO�� S 8 S :=<BN

N<Mj3C=hiIQ@�ReH >P<_ReN >P<BG$<>J<

Pn(t) ∈ Πn �G)HPC

Pn(t) =

n∑

i=0

ciBni (t)

G)HJC=TmNPTm@�G)HJC j NLILHJj T @ b j <BG�<MT 0@^G[fD@�E�HJCZH <MC [0, 1] V � jMj HJReN

N ≤ V (c0, c1, . . . , cn)

>�HJ`J@VR[N b b Re@$E$@�CDT N�<MjDCDhiIL@$ReH�>P< `aNPRS<BN�O�<BHJC=<�>P<ZE$@mr C9H�>P< c0, . . . , cn V�� C b NJRST]<BG)HJj NPRe@ �

N = V (c0, c1, . . . , cn) − 2k, k ∈ N V�� "!$#$% 3�[YelZ;]o\MU*gLbNehV ej[-��<A[�gLbjV BBejU*B(U*gN]C$�] U*SB[<;bN]od ]D$�$�]oVf\MUdoV ej[�I*Ufd^V Z;]' VfeLbj[�gN] U�� <MjDCDhiIL@�ReH�>J< R[N >J<BG�J< hiC b HPj <MC9HPI�<BHPn �

@SE b R[@SE)E�HgCZ@$jMj N �$HJRSILN G)NPC9HJC=<BG)N � ES<BN @$E)E�H N � RS<FEShij T N N ≤ V (a0, . . . , an)@)>

�D �D �p�� ?� � 1

<MC b NJRST]<BG)HJj NPRe@ N = V (a0, . . . , an) − 2ki : S BAejV bN]^XYV 5*U*Ifdo]^V \MU[�gLbN[�SAZ;[Yej[ �]<B[4g bjU

bj[YU*eN[�\MVMVfSAXdKB[ V ] B(Ufdo] SAUf\M]D[�gNBBej[�gjgL]DSB[�d d^V6(AVfgN[ Z;]`_O[YejSAgLbN[�] S9i�3�[Ye ZB[ �ASB]D$Y]oUfSA[�

Pn(t) =

n∑

i=0

ciBni (t) =

n∑

i=0

ci

(n

i

)

ti(1 − t)n−i

3�U*SB]oVf\MUt = u

1+u

�AXYUfSu ∈ (0, +∞)

�BVfd doUfehV

Pn(t(u)) =

n∑

i=0

ci

(n

i

)ui

(1 + u)i

1

(1 + u)n−i=

1

(1 + u)n

n∑

i=0

ci

(n

i

)

ui

W <A[�gLb8L <Bd bj] \MU$� <BS B(Ufdo] SAUf\M] Ur[4g/BBej[�gjgNU SB[�d d^V (AV*gL[XYVfSBUfSA]oX�V #�(AVfgLbjV B(UfejeN[ai =

ci

(ni

) % i : S0��<A[�gLbN[!XYUfSAZB]C$�] U*SB]WB(U*gjgN]oVf\ U4<;bN]odo]C$8$�V ej[Td^Vcej[�I*Ufd^VcZ;] ' V eNbN[4gL]oUc[JdoVcbN[4gL] �5*[Yej].�(XYV�bhVBi

�

(L�[( * ��1 ��& ��#/.��"4#�&1��[4Z;]^V \MU�UfehVrX_Uf\M[ <;bj] do]C$8$�VfeN[ ] B(Ufdo] SAUf\M]TZ;]+_�[�eNSng bj[Y]oS B([Ye�ZB[ �ASB]oej[d [ X�<BeN5*[Z;]_Za8$Y]o[Ye4i5 �76��XS 8Dh X�C9N�G�hiRS`aN->J< (+*[O�<B@�R��hiCZN�G$hiRS`JN b NJR[NJIL@$T]RS<BG)N-> @UA C=<MTmN�> N n+1b hiCDT]<�� >J<?G)HJC=TUReHPjMj H�� bi ∈ E

d, i = 0, . . . , n

X(t) =

n∑

i=0

biBni (t) t ∈ [0, 1]

: d=B(Ufdo]DIfUfSAUTXdKB[ gL]aU bNbN]o[YS0bN[ X_U*SJIf]D<BSJI*[YSAZBU ]=BJ<BS0bN];Z;];XYUfS0bNejUfdod U BBeN[�SAZ;[O] daSBU*\M[OZ;] b H��j < r�HPC9HL>J<�G)HJCDT]ReHPjMj H i ��[�5a]^Z;[YS0bj[TXdKB[�dML U*ejZ;]oSB[ Z;[�]>BJ<AS*bj]nZ;](X_U*S*bjeNU*d doU � Z;[_bj[Yej\M] SAVfS0bN[B([YeTdoVMZ;[ �ASB]D$Y]oUfSB[cZ;[�d d^VMXE<Be?5�VBi

' U*\M] SAXY]oVf\MU X_U*SWdML [4gNVf\ ]oSAVfeN[ BA]�� V%BJBBejU%. U*SAZ;] bjVf\ [�S0bN[Tdo[+BBeNU�BBej] [Yb?ocZ;[�d do[ XE<Be?5f[ZB]p_Za�$�] [�e�i H�V XE<Be?5�V

X(t)gNVfe?o X_U*\BBeN[4gNV SB[�d dML ] S�5a] dD<JBJB(U X_U*S�5f[�gjgNU Z;[�]

biB(Uf]^XdK��

]tBnU*d ]oSBUf\M]HZ;] _O[�eNSAgLbN[�] SrZ;[ �ASB]^gNXYUfSBU <BSAV X_U*\)(B]oSAV*$Y]oUfSB[MX_U*S]5*[�gjgNVAi � U0g bjeN]^V \MU�d^VBAeNU�BBeN]o[_bdoMXdKB[gBBeN[�SAZ;[ ]od9SBUf\M[ Z;] @�C9> � b HP<MCDT<MCDT @$R b HJj NJT]<BHJC �

X(0) =n∑

i=0

biBni (0) = b0

X(1) =

n∑

i=0

biBni (1) = b1

�]<B[4g bjU gN]CI*SB]��nXYVcXdKA[Td^VlXE<AeN5�VlZB]�_Za�$�] [�e?]oS0bN[Ye?B(Ufd^Vc]�BJ<BS0bN]DZ;],X_U*S*bjeNU*d doUlV*Ifdo]A[�gLbNej[Y\M]vi� S�L V d bNehVM] S0bN[�eN[4gNgjV S0bj[+BAeNU�BBeN]o[_bdo��cd\L <MCD`JNJRS<BNPC O\N b @�R�T]ReNaE �$HPRSIQN\O$<BHPCD<N � CD< iZ: S=. V�bNbN]��ZB[ �ASB] b?o<ASAV bNehVfg/. Ufej\kV%$Y]oUfSA[

Φ : Ed → E

d�AZB[YdDbj]CB(U

Φ(p) = Ap + v�AeN]^g/<Ad bhV

Φ(X(t)) =

n∑

i=0

Φ(bi)Bni (t)

[��]<B[�gLbNU��)I*]C<AgLbN]��nX�V�bNU ZBV d�. V�bNbNU XdKB[l]Bn

i (t)]oSAZ;]D5a]oZJ<AV SBU <BSAV X_Uf\4(B]oSAV%$�] U*SB[s(AV%m

ej]^X_[YS0bjeN]^XYVAitH9[cX�<BeN5*[cZ;]`_Za�$�] [�eTgNUaZAZ;]og/. V SAUMVfSAXdKB[ d^VBBejU*BBej] [Yb?oMZ;] E)<MI^IL@$T]RS<BN �(X_]oU ��

, 7 �� D �p�!�� ?�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Curva di BezierPoligono di controllo

q/]DI*<BehV � i � � P SAZ=mMB(Uf]oS0b ] S0bN[�eNB(Ufd^V�bj] U*S

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Curva di BezierPoligono di controlloCurva di Bezier ruotataPoligono di controllo ruotato

q/]CI�<BehV � i ,!��: S]5�VfeN]^V SJ$4VsB([Ye�bjejV*g . U*eN\kV*$Y]oUfSB],V�0kSB]��eNUfbjV*$Y]oUfSB[ Z;]π

�D �D �p�� ?� , !

B(U*gLbNU

C(s) =

n∑

i=0

bn−iBni (s)

ej]^g/<Bd bjVC(s) = X(s)

i W <B[�gLbNUkgN]DeN]^XYVV5�V ZBV d . V bLbjUMXdKA[

Bni (s) = Bn

n−i(1 − s)

]oS=. V bLbN]

C(s) =

n∑

i=0

bn−iBnn−i(1 − s) =

n∑

j=0

bjBnj (1 − s) = X(1 − s)

H�[cXE<Be?5f[ Z;]`_Za8$Y]o[Ye gNUaZAZ;]og/. V SAUMVfSAXdKB[ d^V4BAeNU�BBeN]o[_bdo ZB] j <MC9@)NPRS<MT �nX_]oU/��BZBV bN]

X(t) =n∑

i=0

biBni (t) , Y(t) =

n∑

i=0

ciBni (t)

ej]^g/<Bd bjVαX(t) + βY(t) =

n∑

i=0

(αbi + βci)Bni (t)

� S�L <Bd bj[Yej] U*eN[gBBejU*BAeN]o[_bdo ZB[Ydod [ XE<AeN5*[ ZB]`_Za�$Y]o[Ye � d^V b Re@)G�<FES<BHJC9@�j <MCZ@)NJRe@ i�ma[ ] bigNUfSBU

Vfd do]oSB[�V bN]bi = b0 +

i

nv i = 0, . . . , n

RTd doUfehV doVMX�<BeN5�VMZ;]`_Za8$Y]o[Ye � doV ej[_bNbjVkgN< X�<B]�If]^VfX�X_]oUfSBU��a] S=. V bLbj]

X(t) =

n∑

i=0

biBni (t) =

n∑

i=0

(b0 +i

nv)Bn

i (t) =

= b0 +

(n∑

i=0

i

nBn

i (t)

)

v

= b0 + tv

��[4Z;]^V \MUkUfehV XYU*gjVMg/<AX�X_[4Z;[ gL[��AZBV�bhV6<BSAVkXE<AeN5�VX(t)

�ngNBnU0g bj]oVf\MU6<AS B><BS0bNUbji

` [ �nSB]oVf\MU Bn[�eLbhV S0bNU

bj = bj + ∆v bi = bi, i 6= j

RTd doUfehV doVMX�<BeN5�VX�cZ;[ �ASB] bjVMZBV

X(t) =n∑

i=0

biBni (t) =

∑

i6=j

biBni (t) + (bj + ∆v)Bn

j (t) = X(t) + Bnj (t)∆v

-/V do[ BBejU*BBej]o[_b?o Z;[YbLbhV G)HJCDT]ReHPjMj H b E�@$hD>�H�� j H\G)NJj @ ��\ U0g bjejVXdKA[kd U gNBnU0g bhV \M[YS0bjU ZB] <BSB><BS0bNUZ;]?XYUfS0bNejUfdod U ] S��><B[YS>$�V bN<;bNbjV d^V X�<Be?5uV [�Z ]oS BAVfeLbj]oXYUfd^V eJ\MUaZBU dML ] S0bNU*eNSAU Z;[�dB><BS0bNU!gNB(U*gLbjV�bjUAi�3?eNU�5a]^V \MU�XdKB[ B([Ye�do[?X�<BeN5*[HZ;]�_pa8$Y]o[Ye�5�Vfd [Hd^V BBejU*BAeN]o[_bdo�Z;] `JNJRS<BNPTU<BHPC>P<MI�<MC=<FEef <MC7r �AXdKB[!V%(J(B]^V \MUrI*]DocZB] \MU*gLbNehV�bjUr5�V do[Yej[ Bn[�e?]�BnU*d ]oSBUf\M](Z;]�_O[�eNSAgLbN[�] S�i�ma]^V

X(t) =

n∑

i=0

biBni (t)

, � �� D �p�!�� ?�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q/]DI*<BehV � i 2!� m�BnU0g bhV \M[YS0bjU�Z;]�<BS BJ<BS0bNU Z;[Yd�BnU*d ]DIfU*SBUWZ;]/X_UfS0bjeNU*d doUWZ;]`<ASAVWXE<AeN5�VWZ;]_Za�$�] [�e

[r : αx + βy + γ = 0

<ASAV I*[YSB[�eN]^XYV ej[_bLbhV ZB[Yd BB]^V SBUni)3HejU�5a]oVf\MU XdKB[ ] d�S�<B\M[�eNUZ;]�] S0bj[YehgL[8$Y]oUfSB] #

N% bjejV

X(t)[

r�W\M] SBU*eN[kUX<JI*<AVfd [WZB[YdOS]<A\ [�eNU ZB]?]oS*bj[YehgL[8$Y]oUfSB]

. ehVW]od�B(Ufdo]DIfUfSAU�Z;]�X_UfS0bjeNU*d doUW[r�,]oSXBAVfeLbj]oXYUfd^V ej[ SB[ Z;] !,[Yej]ogjX_[4Bn[�e <BS S�<B\M[YejU BAV ej]vi

3�U*gjgN]oVf\ U B([YSngNVfeN[�XdKA[�d^V ej[_bLbhV Z;]D5a]oZBV"] dZBB]^V SBU"] S Z=<B[ gN[Y\M]DBB]^V SB]�[�Z�V*gNgN[�I*SAV \MUV*Z <BS�gL[�\M]CBB]^V SAU ] dOgL[8IfSBU

+[�Vfd dML V d bNejU] dOgL[8IfSBU − i 3�[Yel\MU0g bjejVfeN[M]od�S�<B\M[YejU Z;]]oS0bN[YehgN[�$Y]oUfSA]/Z;[�d d^V XE<Be?5�V�XYUfS doV eN[YbLbhV��D\M[_bNbN]^V \MUV gN]ogLbN[�\kV B([Yeg5f[4Z;[Yej[MgL[ ] BJ<BS0bN]

Z;[�d d^VMXE<Be?5�VMgLU;ZBZ;]^g/. V SBUMdML [3��<nV%$Y]oUfSA[JZ;[�d d^VMej[_bLbhVBi�R (>(B]oVf\MUMXdKA[

X(t) =

(x(t)y(t)

)

=

( ∑

i bixBni (t)

∑

i biyBni (t)

)

�]<B]oSAZ;]α∑

i

bixBni (t) + β

∑

i

biyBni (t) + γ = 0

XdKB[gB(U*gjgL]^V \MUMgNXYeN]D5f[�eN[ X_U*\M[

∑

i

αbix + βbiy + γ︸︷︷︸

ci

Bn

i (t) = 0

R �]<B[�gLbNU BJ<BS0bjU B(U*gjgN]oVf\ U <;bj] do]C$8$�VfeN[ doV BBejU*BBej]o[_b?o Z;] `JNJRS<BNPTU<BHPC >J<MI^<MCD<FEefi<MC7r Z;[Y]B(Ufdo] SAUf\M]/ZB]�_O[YejSAgLbN[�] S[4Z V%(J(A]oVf\ U 5*[Yej].�(XYV�bjUWd^V BBejU*BBej] [Yb?okXY[YehXYV�bhVBi P gNBBdo]oXY] bj]oVf\MUU*ejV doV4. U*eN\kVMZ;[Ydod^V ZB[Yej]C5�V�bhVMZ;]�<BSAVMX�<BeN5�VMZ;]`_Za8$Y]o[Ye4i

X(t) =

n∑

i=0

bidBn

i (t)

dt=

= n

n∑

i=0

bi

(Bn−1

i−1 (t) − Bn−1i (t)

)=

�D �q �� ?� � �i�� ,*,

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−3 −2 −1 0 1 2 3 4−7

−6

−5

−4

−3

−2

−1

0

1

−7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4−14

−12

−10

−8

−6

−4

−2

0

Poligono di controllo

Curva

Poligono di controlloHodograph I

Poligono di controlloHodograph II

Poligono di controlloHodograph III

q/]CI�<BejV � i 9�� ' <AeN5�VMZ;]`_O[�$�] [�e [ eN[�doV bN]D5f[ � U;Z;U*I*ejV*BJK

= n[Bn

0 (t)(b1 − b0) + . . . + Bnn−1(bn − bn−1)

]=

= n

n−1∑

i=0

∆biBn−1i (t)

HFVWZ;[�eN]D5�V�bjVM[4g/BAeN[4gNgjVM] S �]<B[4g bOL <Ad bj] \kV . U*eN\kVBBej[YSnZ;[c]od�SBU*\ [ Z;] G�hiRS`JN^fDH�>�Her ReN b f i` V-�]<B[�gLbjV4. U*eN\4<BdoV*$Y]oUfSB[ gN]9eN]^XYVV5�V!�

X(0) = n∆b0

X(1) = n∆bn−1

W <A[�gLbNUMgL]DIfSB]��nX�VlXdKB[JUfd bNej[!V ] S0bj[Ye?BnU*doVfeN[!] d�BBej] \MU [!dML <Bd bj] \MUBJ<BS0bjU!�adoV XE<AeN5�VsBnV eNbN[[4Z VfeNej]D5uV�bjVfSJIf[�S0bN[MV d B(Ufdo]CI*UfSBU Z;]HX_U*S*bjeNU*d doUAi�� gN[Y\BBdo]^X_[kZ;[ �nSB] ej[6<BS�L [�gNBBeN[4gNgN]oUfSB[B([YeTdoVMZ;[�eN]D5�V�bjV

jm&[�gN] \kV��

X(j)(t) = n(n − 1) . . . (n − j + 1)

n−j∑

i=0

∆jbiBn−ji (t)

(L� � & ��1�&�&�& $& �p ��!#$��%

m=<JBJB(UfSB]^V \MU Z;]AVV5*[Ye�Z;]ogN[�I*SAV�bjU <ASAV XE<Be?5�VJZB]J_pa8$Y]o[Ye�Z;]JI*ejV*Z;Uni��U*I*d ]^V \MU V%I�If]D<BS=m

I*[Yej[r<BS <Bd bN[Yej]oUfej[+<Ad bj[Yej] U*eN[gBJ<BS0bjUWZ;]FXYUfS0bNejUfdod UMV d �ASB[lZ;]FVV5f[Yej[ \kV%I*I*] U*e�X_UfS0bjeNU*d doU!�

,*2 �� D �p�!�� ?�

b

b

b

bc

c

c

c0

1

2

3

0

1

2

3

4c

=

=

qF]DI*<AejV � i @��R I*If]D<BS0bjVMZ;] <AS B><BS0bNUWZ;]�XYUfS0bNejUfdodoU

\kV65fU�Ifdo]oVf\ U d^VfgjX_]^V ej[T]oSAV d bN[�ejV bjV doV4. U*eN\kVMZ;[Ydod^V X�<Be?5uV!�BZ;[�5*[ ��<A] SAZB]9eN]^gN<Bd bhV ej[�

n∑

i=0

biBni (t) =

n+1∑

i=0

ciBn+1i (t)

� U*d bj]CBAd ]^XYVfSAZ;UM] d�\ [�\)(BejUMgL]oSB]^g bjeNUB([Ye(1 + (1 − t))

U bNbN[�SB]oVf\MU

[t + (1 − t)]

n∑

i=0

biBni (t) =

n+1∑

i=0

ciBn+1i (t)

n∑

i=0

bi

(n

i

)[ti(1 − t)n+1−i + ti+1(1 − t)n−i

]=

n+1∑

i=0

ciBn+1i (t)

m�5a]odC<>BJBAV SnZ;UMdoVMgN[Yej] [ [cX_U*\BAV ehV SAZBUl]�XYU0[�0WX_]o[YS0bN]9Z;]ti(1 − t)n+1−j

U bLbj[YSB]^V \MU��

ci

(n + 1

i

)

= bi

(n

i

)

+ bi−1

(n

i − 1

)

XY] U/��ci =

i

n + 1bj−1 +

(

1 − i

n + 1

)

bi i = 0, . . . , n + 1

�HVJSBUfbjV�bjUJXdKA[�]od=5�V doUfej[�Z;]c0[cn+1

ej]ogN<Bd bjV!eN]^g/B([_bNbN]D5�V \M[YS0bN[b0[bni�R�Zl[4gL[�\6BA] U��

gN[?X_U*SAgN]oZ;[�eN]^V \MU d^V Z;[8Ifej[Y[H[�d [85�V�bN]oUfSrB([Ye/d^VTXE<AeN5�V XYUfSrBJ<BS0bj]aZB]0XYUfS0bNejUfdod Ub0, . . . ,b3

��]<B[�d do]�Z;[�d d^V S]<AU�5uV XE<AeN5�VMgNVfejVfSBSBU��

c0 = b0

c1 = b0 +3

4(b1 − b0)

c2 = b1 +1

2(b2 − b1)

c3 = b2 +1

4(b3 − b2)

c4 = b3

(Q�� 1�#)��, � ��# �'& *+ � �!& ��

HtL VfdCI*Ufej] bj\ U�XdKB[�Z;[�gjX_ej]C5*[Yej[Y\MU2�H]od*BB]��c]o\6B(UfeNbjVfS0bN[H[ (nVfgN] d^V ej[�Bn[�e�]od0Z;]^gN[�IfSAU Z;]*X�<BeN5*[Z;]�_pa8$Y]o[Ye�[4Z��cgNUfe?BBej[YSAZB[YS0bN[�\ [�S0bN[ gL[�\6BAd ]^X_[*i�m0bNU*eN]^XYVf\M[YS0bN[ X_U*S ��<A[�gLbNUkV dDIfU*eN] bN\MU]oSB]D$Y] � ]od9d^VV5fU*eNU g/<Bd ' R� ` Z;]9Z;[ ' V*g bj[Yd cLV%< �;SB[�d

1959i

�D �D �c� �� ?�� Q� ,�9

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

q/]CI�<BejV � iGF(��m;] \4<BdoV*$Y]oUfSB[ Z;[8Ifej[Y[ [Ydo[�5�V�bj] U*S ej]DBn[YbN<;bhV

1−tt

1−t

t

1−t

t

b

b

b

b

b

b

b b

b

b

0

0

0

00

1

1

1

0

2

21

1

0

1

2

3

30

qF]DI*<AejV � i &��Hma]o\)<Bd^V%$�] U*SB[ Z;[YdodML VfdCI*Ufej] bj\ UMZ;]�ZB[ ' V*g bj[Yd cLV%<

,�@ �� D �p�!�� ?�

` V�bN],Z;[�]>BJ<BS0bj]nZ;](X_UfS0bjeNU*d doUbi ∈ E

d, i = 0 . . . n[ �ngjgjV�bNUr<AS

t ∈ [0, 1]�*gN]A[4gL[8I*<B[

]od�XYVfdoXYUfdoU ej]^X_UfehgN]C5*U{

b0i = bi i = 0, . . . , n

bri (t) = (1 − t)br−i

i + tbr−1i+1 i = 0, . . . , n − r r = 1, . . . , n: S BAVfeLbj]oXYUfd^V ej[�0d\L <Bd bN]o\MVM[4g/BAeN[4gNgN] U*SB[+BJ< �M[�gjgL[�eN[ gjX_ej] bNbjVMX_U*\M[

bri (t) = [(1 − t)I + tE]br−1

i (t)RTd�5�V ej]oVfeN[ Z;]

t��<A[Ydod UkXdKB[ UfbLbN[�SJIfU�� ]od�Z;]^gL[8IfSBUMZ;[Ydod^V X�<Be?5uV!�BX_]oU/�r5uVfd [ ]od�gN[�I*<A[YS0bN[��

� ��>�� S 8

bri (t) =

r∑

j=0

bi+jBrj (t) r = 1, . . . , n i = 0, . . . , n − r

�� "!$#$% 3HeNU;XY[�Z;]^V \MU6Bn[�e ] SnZ=<J$Y]oUfSA[ gN<r�

Y �JiO�� B([Yer = 1

eN]^gN<Bd bhV

b1i (t) = (1 − t)bi + tbi+1 =

1∑

j=0

bi+jB1j (t)

# � i � %

�E�� g/<JB>BnU*SB]oVf\MU d^V bN[4gL]�5*[YehV)B([Yer − 1

�BZ;]o\MU*gLbNej]oVf\ UMXdKB[ �r5f[�ejV4B([Yer�

bri (t) = (1 − t)br−1

i (t) + tbr−1i+1 (t) =

= (1 − t)

r−1∑

j=0

bi+jBr−1j (t) + t

r−1∑

j=0

bi+1+jBr−1j (t) =

= (1 − t)

i+r−1∑

k=i

bkBr−1k−i (t) + t

i+r∑

k=i+1

bkBr−1k−i−1(t) =

=

i+r∑

k=i

[(1 − t)bkBr−1

k−i (t) + tbkBr−1k−i−1(t)

]=

=

i+r∑

k=i

bkBrk−i(t) =

=r∑

j=0

bj+iBrj (t)

�q/]^gNgN]oVf\MU�UfehVt = 1/2

[kgL]o\)<Ad ]^V \MU��]<B[4g bjU VfdCI*Ufej] bj\MU V SnV do] bj]oX�V \M[YS0bN[FI �]<B[YdodoU XdKB[UfbLbN[�SB]^V \MU��

b1i = (1 − t)b0

i + tb0i+1 =

1

2(b0

i + b0i+1) i = 0 . . . 2

b2i = (1 − t)b1

i + tb1i+1 =

1

2(b1

i + b1i+1) i = 0 . . . 1

b30 = (1 − t)b2

0 + tb21 =

1

2(b2

0 + b21)

: d�XYU*gLbNUMX_U*\6B><;bjV*$Y]oUfSAVfd [ Z;[YdodML VfdCI*Ufej] bj\ U ej]ogN<Bd bjVn∑

r=1

n−r∑

i=0

3d = 3d

n∑

r=1

(n − r + 1) = 3d

[n(n + 1)

2

]

�D �� c� �� ,]F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

q/]DI*<BehV � i 1�� ' <Be?5uV ZB]W_O[�$�] [�eHU bNbN[YS�<;bhVJbNehV \M] bN[TZ;[ ' VfgLbN[Yd cLV%<M[�ej[Yd^V�bj]C5�V X_U0g bjeN<>$Y]oUfSB[

(L�� -��1�#S��, ��# � ��. �'# �L# �3#$�-%

' KA]oVf\ ]^V \MUX(t)

doVMX�<BeN5�VMX_U*S B><BS0bN]�Z;]FX_UfS0bjeNU*d doUb0, . . . ,bn

[t ∈ [0, 1]

�BX_]oU/�

X(t) =

n∑

i=0

biBni (t) t ∈ [0, 1]

3?eN[�SAZ;]^V \MUt ∈ (0, 1)

[ X_[�ejXdKB]^V \MU <ASAV ehV%BJBAeN[4gL[�S*bhV%$�] U*SB[WZ;] _�[8$Y]o[YeMZ;[Y] Z;]�Z=<B[B([�$8$Y]�ZB]�XE<Be?5�VMZ;[ �ASA] bj]�ZBV

XL(t) =

n∑

i=0

ciBni (s) s =

t

t∈ [0, 1]

[

XR(t) =

n∑

i=0

diBni (u) t ∈ [t, 1] , u =

t − t

1 − t

��U�Ifdo]oVf\MU 5f[Yej]��nXYVfeN[XdKB[ci = bi

0(t)B([Ye

i = 0, . . . , ni # ["Z;] X_U*SAgN[�I*<A[YSJ$4V XdKA[

di = bn−ii (t)

%_i W <B[4g bjU gL]DIfSA].�nX�V�XdKB[ B([YeFZB[ �ASB]oej[�]*Z=<B[tBn[8$�$Y]*Z;]*X�<BeN5�V�gN]fej]C<BbN]od ]D$�$4V SBU] BnVfgjgL](] S0bj[Yej\ [4Z;]DZ;[YdodML VfdCI*Ufej] bj\ U Z;]DZ;[ ' V*g bj[Yd cLV%<9i�R�Z�[�gN[Y\BB]oU!�a] S eN]C. [Yej]o\ [�S0bNUMV dodoV�WI*<BehV � i &��9Z;U�5*[��kZ;]^gN[�IfSnV�bjV <BSAV X�<BeN5�V Z;]t_Za8$Y]o[Ye Z;]HU*ejZ;]oSB[

3�9dML V dDIfUfej] bN\MU ZB]?Z;[

' V*g bj[Yd cLV%<��/If[YSA[YehV��Jd^V \kV bNej]oXY[

b00 b1

0 b20 b3

0

b01 b1

1 b21

b02 b1

2

b03

,�& �� D �p�!�� ?�

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Poligono di controllo di XL

Curva di Bezier XL

Poligono di controllo di XR

Curva di Bezier XR

Poligono di controllo di X

Poligono di controlloCurva di Bezier X

q/]DI*<BehV � i !87 �/RTdCI*Ufej] bj\ UkZ;]9gN<J(,Z;]D50]^gN] U*S ]oSt = .3

P Z ] BJ<AS*bj] Z;]TXYUfS0bNejUfdod U Z;[Ydodo[ X�<Be?5f[XL

[XR

gNUfSBU BBejU*BBej] U��]<B[Ydod ]�[85a]oZ;[�SJ$Y]^V�bj] i3�[Ye!Z;] \MU0g bjejVfeNdoU!�aej]CBAVfejVf\M[_bNej]D$�$Y]^V \MU doVkX�<Be?5uV

XL(t)BBej[YSnZ;[YSAZBUMXYUf\M[rBAVfejVf\ [YbNejU

s = tt

[cXdKB]oVf\M]oVf\ U bjVfd [ X�<BeN5�VC1i A�]^gN<Bd bhV

C1(t) =n∑

i=0

bi0(t)B

ni (s) =

=

n∑

i=0

[(1 − t)I + tE

]ib0 ·

(n

i

)

si(1 − s)n−i =

=n∑

i=0

(n

i

)[s(1 − t)I + stE

]i(1 − s)n−iIb0 =

=[s(1 − t)I + stE + (1 − s)I

]nb0 =

=[sI − stI + stE + I − sI

]nb0 =

=[stE + (1 − st)I

]nb0

B(UfSB[�SAZ;UM]oS(�ASB[s = t/t

U bNbN[�SB]oVf\MU6BAeNU�BBeN]oU

C1(t) = [(1 − t)I + tE]n b0 t ∈ [0, t]

: S �]<B[�gLbNU�\MU;Z;Urd^V�XE<Be?5�V�g/<AZAZ;]C5a]^gNVrVV5aedoX<BS�S�<BU�5*U B(Ufdo]DIfUfSAUrZB]TX_U*S*bjeNU*d doUAi HFVgN[�XYUfSAZBV4BAVfeLbj[ ZB[YdodoVkX�<BeN5�V gNVfe?o ]oS]5*[�XY[�

C2(s) =

n∑

i=0

bn−ii Bn

i (s)

�D �� p�� ?� ��p� �� o�o�� ,*1

x

x

x

P

1

0

0

1

2

=( 1,x,y

X=(x0 , ,x 1 x 2)

)

q/]DI*<BehV � i !*! � ' UaUfehZ;]oSAV�bj[!U*\MU*If[�SB[Y[

(L� � * ��1 ��& ��#/.��"4#�&1 1� �"#$��%' $#

H�[ X�<Be?5f[ ZB] _pa8$Y]o[Ye+5a]ogLbN[,�ASBU*ejV B(U*gjgNUfSBU [�gjgL[�eN[4<;bN]od ]D$�$4V�bj[4B([Ye ehV%BJBBej[�gN[YS0bhV ej[r<BSAV5�V*g bhV 5�V ej]o[_b?o Z;]�XE<Be?5f[��D\kVWSBU*S gNUfSBU ] S IfehVfZ;U Z;]�ejV*BJBBej[�gN[YS0bjVfeN[cgN[�$�] U*SB]�X_U*SB]oXdKA[fi3�[�e BnUfbN[�eHejV*BJBBej[�gN[YS0bjVfeN[�d [ gN[�$�] U*SB]BX_U*SB]^XdKB[ ] S0bNejU;Z=<AXY]oVf\ U Vfd doUfehVTdo[ G�hiRS`a@p>J<)( *)O�<B@$RR[N�O�<BHJCZNJj < ip3/[�e bNehV�bNbjV ej[ ��<A[�gLbNU bN]DBnUWZB]9XE<Be?5f[ � SB[�XY[�gjgNVfeN]oU)<AS V*XYXY[YSBSBUMV d9XYUfSAXY[_bNbNUZB]�X_UaUfehZ;]oSAV�bj[ Uf\MU*I*[YSB[�[fi

�� " � �� % ��&� � � �� &��

' [�ejXdKA]oVf\ U Z;]AZAV ej[Z<AS�L ]oS0bN[�eNBBej[_bhV%$�] U*SB[pI*[YU*\ [YbNej]oX�V!VfdAX_U*SAX_[YbLbjU ZB]AX_UaUfehZ;]oSAV�bj[�U*\MU%mI*[YSB[�[MSB[Yd BB]^V SAU

E2i4: \M\kV%I*] SB]^V \MU XdKB[ ] d BB]^V SAU

E2gN]oV ]oSAgL[�eN] bNU SB[�d BB]oVfSBU

E3] S

\MU;Z;U XdKB[kdML Ufej]CI*] SB[kZ;]E

2SBU*SrXYUf]oSAX_]^ZBVXYUfS d\L Ufej]CI*] SA[MZB]

E3i R dodoUfehV��9U�IfSB] B><BS0bNU

PZ;]

E2BJ< � [4gNgN[Yej[+<BSB]^XYV \M[�S*bj[ ]oZ;[�S0bN]��nXYV bNUkX_U*S�d^V do]oSB[�V

OPi : S ej]C. [Yej] \M[YS0bjUMVfd d^V

�WI*<BehV�# � i !*! % �D]od BJ<BS0bjUP = (1, x, y)

�kVfgjgNUaXY]oV bNU VX = λ(1, x, y) = (x0, x1, x2)XYUfS

λ ∈ R+ − {0} i�-[�5a]^Z;[YS0bN[JXdKB[ ZBV�bjU)<BS I*[YSB[�eN]^X_U)BJ<BS0bNU SB[YdodoU gNBAV%$�] U �+(AVfSAV do[ bjeNU�5�V ej[�]od�B><BS0bNU

XYUfejeN]^gNBnU*SAZ;[YS0bj[!SB[�d�BB]^V SBU��;] SJ. V�bLbj]��

(x0, x1, x2) → (1,x1

x0,x2

x0) x0 6= 0

' U*S bjV do[ BBej] SnX_]DBB] U gN] BBejU;X_[4Z;[Yej[kV SAVfd U�I*V \M[�S*bj[ VfSAXdKB[ B([YelZB[ �ASB]oej[kd [WXYU0U*ejZB] SAV bN[U*\MU*If[�SB[Y[ V

E3i

�� ! "��& �� &��

HtL ]oS0bNejU;Z=<J$Y]oUfSA[ XdKB[ V%(J(A]oVf\ Ug. V bLbjUrBn[�eHd [!X_UaUfehZ;]oSAV�bj[OU*\MU*If[�SB[Y[2� gLbjV bjVgBBej[�gN[YS0bjV bjVB([YehXdKJaZ5f[4Z;eN[�\MU d [ XE<AeN5*[�Z;]=_Za�$�] [�eFehV%$�] U*SAV do]fSB[�d�BB]oVfSBU

E2XYUf\M[tBBejUf]o[�$�] U*SB]aZ;];XE<Be?5f[

ZB]�_pa8$Y]o[YeZBnU*d ]oSBU*\ ]^V do],] SE

3# ]oS I*[YSB[�ejVfd [ <BSAV XE<Be?5�VlZ;]�_Za8$Y]o[YeOejV*$Y]oUfSAVfd [!] S

Ed�!d^V

BAeNU*] [8$Y]oUfSB[lZB]�<BSnV�XE<AeN5�VWZ;] _Za�$�] [�e B(Ufdo] SAUf\M]oVfd [l] SE

d+1% i ` V�bjV V dod U*ejV<ASAV�X�<BeN5�V

ZB]`_O[�$Y]o[Ye B(Ufdo]oSBUf\M]^V do[

X(t) =

n∑

i=0

BiBni (t)

2 7 �� D �p�!�� ?�

q/]CI�<BejV � i ! � ��3HeNU*] [8$Y]oUfSB[ X�<BeN5�VMZ;]`_Za8$Y]o[Ye BnU*d ]oSBU*\ ]^V do[

XYUfSt ∈ [0, 1]

�Bi ∈ E

d+1[

Bi =

(βi

βibi

)

XYUfSβi > 0

[bi ∈ E

d��UfbLbN[�SB]^V \MU doV gN[�I*<A[YS0bN[4. Ufej\)<Bd^V Bn[�es<BSAV XE<AeN5�V Z;] _Za�$�] [�e

ehV%$�] U*SAV do[�

X(t) =

n∑

i=0

βibiBni (t)

n∑

i=0

βiBni (t)

: S BAVfeLbj]oXYUfd^V ej[�BgN[β0 = β1 = . . . = βn

�]<B[Ydod UWXdKA[cU bNbN[YSA]oVf\ UWgNUfSAUkS]<AU�5uVf\M[YS0bN[ do[X�<BeN5*[ ZB]�_O[8$Y]o[YegBnU*d ]oSBU*\ ]^V do]vi R SAVfd U�I*V \M[�S*bj[ V0�]<AV S0bjU . V bLbNU B([Ye �]<B[�gLb8L <Bd bN]o\MUbj]CB(U Z;]�X�<BeN5*[��5f[4Z;]oVf\MUWd [4BBejU*BBej] [Yb?oWXdKA[cd [MXE<AeN5*[lZ;]�_�[8$Y]o[YeJejV*$Y]oUfSAVfd ]FgLU;ZBZ;]^g/. V SBUniHFV4BAeNU�BBeN]o[_bdo ZB]9[YSAZ=mMB(Uf]oS0b ]oS0bN[�eNB(Ufd^V�bj] U*S 5�V do[ V SAXYUfehV��a]oS=. V�bNbN]��

X(0) =β0b0

β0= b0

X(1) =βnbn

βn= bn

R SAXdKA[Jd^V6BBejU*BBej] [Yb?oMZ;]9gN] \M\M[_bjeN]^V65�V do[cV SAXYUfehV��0] S=. V bLbj]��WBnU0g bjUs = 1− t

ej]ogN<Bd bjV

X(s) =

∑ni=0 βn−ibn−iB

ni (s)

∑ni=0 βn−iBn

i (s)=

�D �� p�� ?� ��p� �� o�o�� 2 !

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

qF]DI*<AejV � i ! ,!� ' <AeN5�VMZ;]`_Za�$�] [�e ejV*$Y]oUfSAVfd [

q/]DI*<BehV � i ! 2!��3HejUf]o[�$�] U*SB[ XE<Be?5�V Z;]�_pa8$Y]o[Ye B(Ufdo] SAUf\M]oVfd [

2 � �� D �p�!�� ?�

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q/]DI*<BehV � i ! 9(� ' <Be?5�V ZB]`_Za�$Y]o[Ye�ehV%$�] U*SAV do[

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Poligono di controlloCurva

q/]DI*<BehV � i ! @�� P SAZ=mMB(Uf]oS0bT] S0bj[Ye?BnU*doV bN]oUfS

�D �� p�� ?� ��p� �� o�o�� 2�,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Poligono di controlloConvex hullCurva di Bezier razionale

q/]DI*<BehV � i ! F(� ' U*S]5*[��m K]<Ad d

=

∑ni=0 βn−ibn−iB

nn−i(1 − s)

∑ni=0 βn−iBn

n−i(1 − s)=

=

∑nj=0 βjbjB

nj (t)

∑nj=0 βjBn

j (t)

�HVfd [cVfSAX_U*ejV d^V4BAeNU�BBeN]o[_bdo ZB]�X_U*S]5*[��m K]<Ad d��a] S=. V bLbj]�BnU*SB[YSnZ;U

Hi(t) =βiB

ni (t)

∑nj=0 βjBn

j (t)i = 0, . . . , n

3�U0gNgN]oVf\MU [4g/BAeN]o\M[Yej[ <BSnV X�<Be?5uVMZ;]�_pa8$Y]o[Ye�ehV%$�] U*SAV do[!XYUf\M[�

X(t) =n∑

i=0

biHi(t)

' U*\M[ �g. V*X_]od [ SBUfbjVfeN[��0do[g. <BSJ$�] U*SB]Hi(t)

gLU*SBUlb?<;bLbj[rBnU0gL] bN]D5f[ [ d^V d U*eNUMgLU*\M\MV45�V do[1�;X_U*gLbN] bN<A]ogjX_U*SBU��]<B]oSAZ;]�<BSAV X_Uf\4(B]oSAV%$�] U*SB[TX_U*S�5f[�gjgjV Z;[Y]

bi�;ZBV XE<B](doV4BBejU*BBej]o[_b?olZ;]

XYUfS�5f[��mMK�<Bdod i�: SBUfd bNej[�bhV do[ BBejU*BBej] [Yb?o ]o\BBd ]^XYVcd\L ] S�5�V ej]^V SJ$4VgBn[�eHbNehVfg/. Ufej\kV%$�] U*SB]BV�0kSB]��

T (p) = Ap + q → T (X(t)) =

n∑

i=0

T (bi)Hi(t)

3�[�e�d [�XE<AeN5*[�Z;]J_Za�$�] [�e�ejV*$Y]oUfSAVfd ]0SBUfS65�V do[�d^V+BBejU*BBej] [Yb?o!Z;];d ]oSB[4V ej] bdo �]<AVfSAZ;U Z=<B[�XE<BeNm5*[gKAV SBSAU6B([�gN]�Z;]D5f[�ejgN] # X�V \)(A]oV ]od�Z;[YSAUf\M] SnV�bNU*eN[ %H[ SAUfS 5uVfd [ SB[4V SAXdKA[Jd^V6BBejU*BBej] [Yb?oZB]�BBej[�X_]^gN] U*SB[ d ]oSB[�VfeN[��a] S=. V bLbj]��WBBeN[4gL]

bj = b0 + jnv�Aej]ogN<Bd bjV

X(t) =

n∑

j=0

(

b0 +j

nv

)

Hj(t) = b0 +

n∑

j=0

(j

nHj(t)

)

v

2�2 �� D �p�!�� ?�

\kVkd^V �]<AVfS*bj] bdo. ehV BAV ej[YS0bj[�gN]��BBejU*BBej] UWV�XYV%<ngNVWZB[Y]�Bn[4gL]�� ] SXIf[YSA[YehV do[cZ;]C5*[YehgNV�ZBVti ' UfS0bNejUfdodo]oVf\ U"UfehV gL[ d [ XE<Be?5f[ Z;] _pa8$Y]o[Ye ejV*$Y]oUfSAVfd ]�gLU;ZBZ;]^g/. V SBU doVXBBeNU�BBej] [Yb?o Z;]5�V ej]^V�bN]oUfS Z;]o\ ]oSB]^g/KA] SJIni ` V bjV<BSnV ej[_bNbjV

r : ax + by + c = 0

' VfdoXYUfdoU d^VMXE<Be?5�V ZB]`_Za�$Y]o[Ye�ehV%$�] U*SAV do[!gN<BdodoV ej[_bNbjV

a

∑ni=0 βibixBn

in∑

j=0

βjBnj (t)

︸︷︷︸

w(t)

+ b

∑ni=0 βibiyBn

in∑

j=0

βjBnj (t)

︸︷︷︸

w(t)

+ c = 0

W <B]oSAZ;]��

a

n∑

i=0

bixHi(t) + b

n∑

i=0

biyHi(t) + c

n∑

i=0

Hi(t)

︸︷︷︸

1

= 0

ehVfX�X_U*I*d ]o[YSnZ;Ul]Dbj[Yej\M] SB]�U bLbj[YSJI*Un∑

i=0

(abix + bbiy + c) Hi(t) = 0

: S BAVfeLbj]oXYUfd^V ej[ �]<B[�gLb8L <JI�<AV%I*d ]^V SJ$4V �r5*[Yej].�nX�V�bhV gNUfd^V \M[YS0bj[JgN[

n∑

i=0

βi (abix + bbiy + c)︸︷︷︸

di

Bni (t)

= 0

W <B[4g bOL <Ad bj] \kV [�gNBBej[�gjgL]oUfSB[ �r<BSAVMX�<BeN5�VMZ;]�_Za�$�] [�e BnU*d ]oSBU*\ ]^V do[�BZB]�XE<B]FV%(J(B]^V \MU6If] oBBejU�5�V�bjU d^V65uVfd ]^Z;] b?oMZ;[�d d^V5�V ej]oV bN]oUfS ZB] \M]oSB]ogNKB]oSJI!�

N ≤ V (d0, . . . ,dn) = V (r0, . . . , rn)

XYUfSri = abix + bbiy + c

ic��[�Z;]^V \MUWU*ejV ��<A[Ydod U�XdKA[ gN<AXYXY[�Z;[c\MU;Z;]��nXYVfSAZ;U <ASXB([�gNUZ;] <ASAV X�<Be?5uV Z;] _Za�$�] [�e!ehV%$�] U*SAV do[fi ` V <BSAV If[�SB[Yej]oX�V

X(t)If[�SB[YejU

X(t)X_U*S

βi =βi + ∆βi

iHRTd doUfehV

X(t) =

∑nj=0 βjbjB

nj + ∆βibiB

ni

n∑

j=0

βjBnj + ∆βiB

ni

︸︷︷︸

w(t)

��U�Ifdo]oVf\ U 5f[�ZB[Yej[cdoV 5�V ej]oV*$Y]oUfSB[cej]^g/B([_bNbNU Vfd d^V�X�<Be?5uV � U*eN]DIf]oSAV do[�� nXY] U/�X(t) − X(t)

iA�]^gN<Bd bhV[

X(t) −X(t)]

w(t) =n∑

j=0

βjbjBnj + ∆βibiB

ni

︸︷︷︸

X(t)w(t)

−n∑

j=0

βjbjBnj −X(t)∆βiB

ni (t) =

= ∆βiBni (t) [bi −X(t)]

�D �� p�� ?� ��p� �� o�o�� 2]9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q/]CI�<BejV � i ! &!� � U;Z;].�(XYVMZ;]�<BS B([�gNU

3�[�eTXE<B]X(t) −X(t) = ∆βi

Bni (t)

w(t)[bi −X(t)]

W <A[�gLbNU!gN]DIfSB]��nXYVTXdKB[?XYVf\)(B]^V SnZ;U <BSsB([�gNU!gN]0\ U;Z;]��nX�V dML ]oS0bN[YehV X�<BeN5�VAi`: SsBAVfeLbj]oXYUfd^V ej[�d^VkZ;]oeN[8$Y]oUfSB[ Z;[�d doUkg/B(U*gLbjVf\ [�S0bNUkZ;]DU*I*SB] BJ<BS0bjU

tZ;]DBn[�SAZ;[cZBV

bi −X(t)[4Z ]od�5*[YehgLU

ZB[Ydod UWgNBnU0g bhV \M[YS0bjU Z;]DB([YSAZ;[cZAV∆βi

�

• gN[∆βi

�gBnU0gL] bN]D5fUMd^V X�<Be?5uVMgN]�VV5�5a]oXY] SAV Vbi

• gN[∆βi

� SB[�I0V�bN]D5fU d^VkXE<Be?5�V gN]�V dodoUfS0bjVfSAV ZAVbi

��[4Z;]^V \MU gN[c[�gN]^g bj[)<BSAV�eN]DBAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[ V \M\M]ogjgN]C(B]odo[)B([Ye VV5f[Yej[β0 = βn = 1

i3�U*SB]oVf\MU

t =(1 + α)u

1 + αu︸︷︷︸

φ(u)

con α ∈ R, u ∈ [0, 1]

]oS��]<B[4g bjU \MUaZBUTV SAXdKA[t ∈ [0, 1]

i ` [Yej]D50]^V \MUφ(u)

#�Bn[�e/gLbN<AZB]oVfeNSB[tBnU*]fd^V�\MUfSBUfbNUfSA]oV!�gN[Ye?5f[gB([Ye BBejU�5�V ej[JXdKA[Jd^V eN]DBAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[ � Vf\M\ ]^gjgL]D(B]od [ % �

φ′(u) =1

(1 + αu)2[(1 + α)(1 + αu) − α(1 + α)u] =

=1 + α

(1 + αu)2# � i ,�%

W <A[�gLb8L <Bd bj] \kV [3��<nV%$Y]oUfSA[,�B(U*gN] bj]C5�V gL[M[kgNUfdoU gN[α > −1

i4: S BAVfeLbj]oXYUfd^V ej[lU*gjgN[Ye?5fU]oS0bjVfS*bjUkXdKB[

1 − t =1 − u

1 + αu

2]@ �� D �p�!�� ?�

A�]DBAVfejVf\ [YbNej]C$8$Y]^V \MU Vfd doUfehV doVMXE<AeN5�V

X(t(u)) =

∑ni=0 βibi

(ni

) (1+α)iui

(1+αu)i

(1−u)n−i

(1+αu)n−i

∑ni=0 βi

(ni

) (1+α)iui(1−u)n−i

(1+αu)n

=

=

∑ni=0 βibi

Bni (u)

︷︸︸︷(

n

i

)

ui(1 − u)n−i(1 + α)i

∑ni=0 βi

(n

i

)

ui(1 − u)n−i

︸︷︷︸

Bni(u)

(1 + α)i

=

=

∑ni=0 βibi(1 + α)iBn

i (u)∑n

i=0 βi(1 + α)i

︸︷︷︸

βi

Bni (u)

=

∑ni=0 βibiB

ni (u)

∑ni=0 βiBn

i (u)

: \B(UfSB[�SAZ;UMdoVkXYUfSAZB]C$�] U*SB[βn = 1

U bLbj[YSJI*U

βn(1 + α)n = 1 ⇒ α =1

n√

βn

− 1

[(AVfSAV do\M[YS0bN[β0 = (1 + α)0β0 = β0

�/XdKB[kg/<>BJBnU*SB]^V \MU If] o <JI�<AV do[kV1[ �]<B[4g bjU

gN[YSJ$4VBn[�ejZB] bhVkZ;]`I*[YSB[�ejVfd ] b?o!�;] S=. V bLbj]FgNUfSBUM]o\6B(UfeNbjVfS0bN]9]FejV*BJBnU*eLbj] . ehV ]�Bn[4gL]�BB]�� XdKB[] B([�gN]!gLbN[4gNgN]�# gL]DIfSB]��nX�V XdKA[ gL[\MU;Z;]��nX_U BBejU*B(Ufe?$Y]oUfSnV do\ [�S0bN[ ] Bn[4gL]!doV�X�<Be?5uV SBU*SX�V \)(A]oV % i H9[ XE<Be?5f[ Z;]�_��$�] [�e ejV*$Y]oUfSAVfd ],X_U*S ] B([�gN]9V%I*d ],[�gLbNej[Y\M]�<JI�<AV do]DV

15f[�SJIfU*SBU

Z;[YbLbj[ G�hiRS`J@ >J<�(��[O�<B@�R�EST NJCZ> NJR[> i)3?eNU�5a]^V \MU�U*ejV XdKB[ d^V XE<Be?5�V�Z;] _O[�$�] [�e � gN[Y\BBej[bhV SJI*[YS0bN[ V d B(Ufdo]DIfUfSAU Z;]FX_UfS0bjeNU*d doUAi ` V bjV<BSnV X�<Be?5uV ehV%$�] U*SAV do[ I*[YSB[�eN]^XYV!�

X(t) =C(t)

w(t)⇒ C(t) = X(t)w(t)

` [Yej]C5�VfSAZ;UkgL]DUfbLbj] [�SB[C′(t) = X(t)w(t) + X(t)w′(t)

XY] U/�X(t) =

C′(t) −X(t)w′(t)

w(t)m;V*BJBB]^V \MUkXdKB[ doVMZ;[�eN]D5�V�bhV ZB]�<BSAVMXE<AeN5�VMZ;]`_Za�$�] [�e BnU*d ]oSBUf\M]^V do[

C(t)eN]^gN<Bd bhV��

C(r) = n(n − 1) . . . (n − r + 1)

n−r∑

i=0

∆r(βibi)Bn−ri (t)

RTSAV doU*I0V \M[YS0bN[ gN]9XYV d^X_U*doV d^VMZ;[Yej]C5�V bjV Z;]w(t)

�

w(r)(t) = n(n − 1) . . . (n − r + 1)

n−r∑

i=0

∆rβiBn−ri (t)

�D ��a �� 2�F

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

0 0.5 1 1.5−2

0

2

4

−15 −10 −5 0 5 10 15−30

−20

−10

0

10

−2500 −2000 −1500 −1000 −500 0 500−2000

−1000

0

1000

0 0.2 0.4 0.6 0.8 10

500

1000

1500

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1TorsioneCurvatura

dddX

ddX

dX

q/]DI*<BehV � i ! 1�� ' <Be?5�V Z;]�_pa8$Y]o[YeFejV*$Y]oUfSAVfd [�[�eN[�doV bN]D5f[ KBU;Z;U�IfehV%BJK �uXE<AeN5�V�b?<BehV [HbNU*ejgN]oUfSB[

RTd doUfehV

X(t) =n[∑n−1

i=0

(∆(βibi)B

n−1i (t)

)−(∑n−1

i=0 ∆βiBn−1i (t)

)

X(t)]

w(t)

' VfdoXYUfd^V SAZBU d^VBBej[�X_[4Z;[YS0bj[JV*Ifdo]D[�gLbNej[Y\M]DU bNbN[YSA]oVf\ U��

X(0) = n(β1b1 + β0b0) − (β1 − β0)X(0)

β0= n

β1

β0∆b0

X(1) = n∆n−1

βn∆bn−1

W <A[�gLbNU gN]CI*SB]��nXYV XdKA[�doV XE<Be?5�V Z;]_pa8$Y]o[Ye ehV%$�] U*SAV do[$��bhV SJI*[YS0bN[ Vfd)B(Ufdo]CI*UfSBU Z;]XYUfS0bNejUfdodoU V*Ifdo]D[�gLbNej[Y\M] i

(L� � � &/2��%�#�2 �'&

3�[�e BnUfbN[�e] S0bNejU;Z=<BejeN[ do[�[ �]<AV%$�] U*SB]cXdKB[�Z;[4gNXYeN]D5fU*SBU X�<Be?5f[rX_UfSA]oXdKB[��SB[4X_[4gNgjV ej] UV*BJBBejU%. U*SAZ;]oeN[J] dFX_UfSnX_[_bNbNUkZ;]�XYUaUfehZ;] SnV�bN[+(AVfeN]^X_[�S*bjeN]^XdKB[*i

�� " � �� % ��&� � ��&�� &�

' U*SAgN]oZ;[�eN]^V \MU bNej[ BJ<BS0bN]b0,b0,b0

SBU*S�V dod ]oSB[�V bN] SA[Yd BB]^V SAUAi H�VrXYUfS�5f[��]m K�<Bdod Z;]�]<B[4g bj] BJ<BS0bN]2� doV g/B([�$8$�V�bhV"XdKB[ d ] <BSA]ogjX_[ [ Z;[�d ]o\M] bhVX<AS�bNej]^V SJI*UfdoUAi � S I*[YSB[�eN]^X_UB><BS0bNU

PV*BJBAV eNbN]o[YSA[JVfdnbjeN]^V S>IfUfdoU gN[ �JZB[_bN[�eN\M]oSAV%(A] do[JbNehV \M] bj[!d^V bNej]CBAdoV

(u0, u1, u2)bhV do[cXdKB[{

P = u0b0 + u1b1 + u2b2

u0 + u1 + u2 = 1

2]& �� D �p�!�� ?�

b

b

bP

r

x

y

2

1

0 ∆2

∆ 1 ∆0

q/]DI*<BehV � i �%7 � ' UaUfehZ;]oSAV�bj[g(AV ej]oXY[YS0bNej]^XdKB[

HFV bNej]CBBd^V(u0, u1, u2)

ehV%BJBAeN[4gL[�S*bhV d [ XYU0U*ejZB] SAV bN[ (AVfeN]^X_[�S0bNej]oXdKB[Z;]Pi ��[4Z;]oVf\MU

XYUf\M[ BnVfgjgNVfeN[ ZBV dod [rX_UaU*ejZ;]oSAV bN[rXYV eNbN[4gL]^V SA[ Z;]PVfd do[rX_UaUfehZ;]oSAV�bj[ (AVfeN]^X_[�S0bNej]oXdKB[*i

` U*(J(A]oVf\ UMej]ogNUfdD5f[�eN[!] dFgL]^g bj[Y\kV3 × 3

x0 x1 x2

y0 y1 y2

1 1 1

︸︷︷︸

A

u0

u1

u2

=

xy1

Z;U�5*[HV%(J(A]oVf\ U B(U*gLbNUbi =

(xi

yi

) iFR 0MSnXdKJaHd^V�eN[�doV*$Y]oUfSB[�. ehV�do[?X_UaUfehZ;]oSAV�bj[HXYVfeLbj[�gN]oVfSB[�[do[TX_UaU*ejZ;]oSAV bN[ (AVfeN]^X_[�S*bjeN]^XdKB[ gN]oVg(A]C<BSA]C5*UaX�V d^V \MV bNej]oXY[

AZB[�5f[�[4gNgN[Yej[�SBU*SWgL]oSJIfU*doVfeN[*i

� V ] d Z;[_bj[Yej\M] SAVfS0bN[ ZB[YdodoV\kV bNej]oXY[A��]oSAZ;]^XdKB]^V \MUfdoU"X_U*S

∆��do[�I*V bNU Vfd dML VfeN[4VZ;[Yd

bjeN]^V SJI*UfdoU!�a]oS=. V�bNbN]

area(T ) =1

2|∆| =

1

2|(b1 − b0) × (b2 − b0)|

W <B]oSAZ;]A] d∆� gN<Bdod UcgN[ [ gNUfdoU gN[ dML VfeN[4V ��S]<Ad d^V��0X_UfSnZ;]C$�] U*SB[ XdKB[ gN]J5f[�eN]��nX�VJgNUfd^V \M[�S*bj[

gN[ ]�BJ<BS0bN]bigNUfSBUlVfd do]oSB[�V bN]A[JXdKB[TVV5*[�5�V \MU [�gjX_dD<AgNUc]oS BAV eNbN[�SJ$�VAi ` [ �ASA]oVf\ U V dod U*ejV!�

∆0(x, y) = det

x x1 x2

y y1 y2

1 1 1

∆1(x, y) = det

x0 x x2

y0 y y2

1 1 1

∆2(x, y) = det

x0 x1 xy0 y1 y1 1 1

` V ]�BAeN[4X_[�ZB[YS0bN]��BX_U*S d^V eN[8IfUfd^V Z;] ' ehV \M[Ye�gN]DeN]^XYVV5�V

ui =∆i(x, y)

∆: S BAVfeLbj]oXYUfd^V ej[��]∆i

ejV*BJBBej[�gN[YS0bjVfSBUJd [TVfeN[�[�Z;[Y]AbNej]oVfSJIfU*d ]AXdKA[ gL]nU bLbj[YSJI*UfSBU XYUfSJI*]C<BSJmI*[YSAZ;U ] 5f[�eLbj]oXY]HV d�BJ<BS0bj]

Pi,��. V*X_]od [MSBUfbjV ej[�D]oS eN]C. [Yej] \M[�S*bjUV dodoV �>I*<AejV � i �%7 �9XdKA[

�D ��a �� 2�1

b

b

b

y

2

1

0

− + −

+ − −

+ − +

− − +

− + +

+ + −

+ + +

x

q/]CI�<BehV � i �J! ��ma[�I*SB]�Z;[8Ifdo]ui

U�IfSB] BJ<AS*bjUPV%BJBAVfeLbj[YSB[�S0bN[�Vfd d^V"ej[_bNbjV

rKAV"doU gLbN[4gNgNU

u2BnU*]oXdK!� dML VfeN[4V

∆2SBUfS

X�V \4(B]oVAi : SBUfd bNej[��gL[PgLb?o g/<Bd d^V�bjU

b0b1] dObNej]^V SJI*UfdoU

∆2KnV VfeN[4V S�<BdodoV1��<A] SAZB]

u2 = 0i�: S ej] . [�eN]o\M[YS0bNUMV dod^V��WI*<BehV � i �J! B(U*gjgL]^V \MU ] SnZ;]C5a]^Z=<AVfeN[J]9gL[8IfSB]DZ;]

u0, u1, u2]oS (AVfgN[ V dod\L V ej[�V Z;]�V*BJBAVfeLbj[YSB[�SJ$�V Z;]Pi

�� ! � ��&& �� %&� � � �� &��

' U*SAgN]oZ;[�eN]^V \MU <ASAV I*[YSB[�eN]^XYV�XE<Be?5�VWZ;] _Za�$Y]o[YeJehV%$Y]oUfSnV do[lZ;]�IfehVfZ;U2] SX. Ufej\kV�gLbjV eNm

ZAV ehZ��X(t) =

b0B20(t) + b1B

21(t) + b2B

22(t)

B20(t) + β1B

21(t) + B2

2(t)︸︷︷︸

w(t)

# � i 2 %

3�U*SB]oVf\MU!�

u0(t) =B2

0(t)

w(t)=

(1 − t)2

w(t)

u1(t) =β1B

21(t)

w(t)=

2β1t(1 − t)

w(t)

u2(t) =B2

2(t)

w(t)=

t2

w(t)RTd doUfehV6BnU0gNgN]^V \MU [4g/BAeN]o\M[Yej[JdML [ �]<AV%$�] U*SB[ � i 2MXYUf\M[X(t) = u0(t)b0 + u2(t)b2 + u2(t)b2XYUfS

u0(t)+u1(t)+u2(t) = 1[�Z ]oS,�]<B[4g bjUT\MU;Z;U KBUJ[�gNBBej[�gjgLU�d^VTX�<Be?5uVT] S X_UaUfehZ;]oSAV�bj[

(nV ej]oXY[YS0bNej]oXdKA[fi ` V��]<B[4g bOL <Bd bN]o\kVM[ �]<AV*$Y]oUfSB[��0[�d [85�V SAZ;UkVfd �]<AVfZ;ehV�bjU gN]DU bNbN]o[YSB[u2

1(t) = −4β21t2(1 − t)2 = −4β2

1u0(t)u1(t) = 0 ∀tA ]oXYUfehZBV SAZBU6B(Uf]Dd^VMZ;[ �nSB]C$�] U*SB[cZ;]

ui�

∆21(x, y) − 4β2

1∆0∆2 = 0W <A[�gLb8L <Bd bj] \kV��4<AS�L [3�]<AV%$�] U*SB[ VfdCI*[�(Bej]oX�VkZ;]�gL[4X_U*SAZ;U I*ejV*Z;U!� �]<B]oSAZ;]�<BSAV X_UfSA]oX�VBig: dbj]CB(UWZ;[�d d^VkX_U*SB]oX�V ZB]CB([YSAZB[lZBV

β1[4Z�]oS BAV eNbN]^X_U*doVfeN[��

β1 < 1 → ellisseβ1 = 1 → parabolaβ1 > 1 → iperbole

9 7 �� D �p�!�� ?�

0.5 1 1.5 2 2.5 3 3.5

2.5

3

3.5

4

4.5

qF]DI*<AejV � i � � � ' ]oehX_UfSJ. [Yej[YSJ$4V

W <B[�d doUWXdKB[cgN]9U bNbN]o[YSB[cZB]ogN[�I*SAV SAZBU��]<B[4g bj[lXE<AeN5*[ �r<AS E$@mr IQ@�CDT H >P<G)HJC=<BG)N iZ: S BnV eNmbj]oXYUfd^V ej[ B([Ye

β1 = cosφ�AZ;U�5*[

φ� d\L V S>IfUfdoU4. ejV

b1 − b0[b2 − b0

�;U bNbN[YSA]oVf\ U<BSnVXY] ehX_U*S=. [Yej[YSJ$4VBi�" 5�5a]^V \M[YS0bN[ SA[YdFX�VfgNUkZ;[YdodoVBnV ehV%((Ufd^V [ Z;[Ydod\L ]DBn[�eN((Ufdo[ � ]o\6B(U*gjgN]C(B]odo[]od�Z;]^gL[8IfSBUWZB[YdodoVkX�<BeN5�VkXYUf\BBdo[_bjV!�B\kVBn[�e dML [�d do]ogjgN[ �rBnU0gNgN]C(A] do[cZ;]^gL[8IfSAVfeN[ X_U*\BBd [YbjV%m\M[YS0bj[ d^V XE<Be?5�V��ZB]ogN[�I*SAV SAZBU U*d bjeN[kV dHgN[�I*\M[YS0bNU Z;]?XYUfSB]^XYV d^V BnV eNbN[ eN[4g bhV S0bN[*i63�[�e. Vfe X_] �6(nVfgLbjVMgL[�\BBd ]^X_[�\M[YS0bN[ XYVf\)(B]^V ej[J] d�gN[�I*SBUkZ;]

β1i

H�VJSAV bN<BehVJZ;]β1SBU*S)KAV+<BSAV XdKB]^V ehV ]oS0bN[8BBeN[YbjV*$Y]oUfSB[ZIf[�Uf\M[_bjeN]^XYV!� �]<B]oSAZ;]BgN]=BBej[E. [�m

ej]ogjX_[ gNBn[4gNgNU<;bN]od ]D$�$4V ej[!]oS ] S>BJ<;b ]od BJ<BS0bNUP = X(1/2)

i�: SAZ;]^XYVfSAZ;UkX_U*SM

] d B><BS0bNU\M[�ZB] UkZ;[�d�gL[8If\M[YS0bjU

b0b2ej]ogN<Bd bjV

P = X(1/2) = M +β1

1 + β1(b1 −M)

: dBehV%B>BnU*eLbjUs = β1/(1+β1)

]oSAZ;]D5a]oZ=<nVg<BSB]D5fU;XYVf\ [�S0bN[�]odJBJ<BS0bNU g/<BdngL[8If\M[YS0bNUMb1

��]<B]oSAZ;]�g/. e?<;bLbhV SAZ;UMdo[cX_U*SAgL]^Z;[�ejV*$Y]oUfSB]W. V�bNbN[cgN<

β1V%(J(A]oVf\ U

s < 1/2 → ellisses = 1/2 → parabolas > 1/2 → iperbole

�D ��a �� 9 !

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

q/]CI�<BehV � i � ,��3�V ehV%((Ufd^V

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40.2

0.4

0.6

0.8

1

1.2

1.4

1.6

q/]CI�<BehV � i � 2!��3�V ehV%((Ufd^V

9 � �� D �p�!�� ?�

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

q/]DI*<BehV � i � 9�� P dodo]ogjgL[

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

q/]DI*<BehV � i � @!� P dodo]ogjgL[

�D ��a �� 9%,

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

q/]CI�<BejV � i � F(��: Bn[�eN((Ufdo[

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

q/]CI�<BejV � i � &!��: Bn[�eN((Ufdo[

9%2 �� D �p�!�� ?�

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Poligono di controlloPunti di controlloEllisseParabolaIperbole

q/]DI*<BehV � i � 1�� ' U*SB]oXdKA[

� & �� u��*��u�� i�� i� � �� T$�� *�� U�� T��%$�� \+ &(��*� �u��*��u�� n� �� u� ��l� � ��*��)�*�i��%$��f ��X�*�� U��4 ��w��*�%�\�#��\+ & �� u�� u�� n� �� d��%� �\�# �� %$�� *�� U��= ��,��)�*�\+

��g��V.\1 :�W

3" BC>�# ��

H�[XE<Be?5f[ Z;] _Za�$Y]o[Ye�# B(Ufdo]oSBUf\M]^V do]OU ehV%$Y]oUfSnV do] %lgLU*SBU <BbN]od ]D$�$4V�bN[ B([YeMejV*BJBBej[�gN[YS0bjVfeN[<ASAVs5�VfgLbjVsI*V \M\kV Z;],XE<Be?5f[�� b?<;bLbhVV50]^VcXY](gNUfSBUcgN] bN<AV*$Y]oUfSB]nSB[Yd ' R� ` ]oS�X�<B]Ad^VlX�<BeN5�VZAV ehV%B>BBeN[4gL[�S0bjV ej[ �kX_U*\6BAd [4gNgjV��,[�Z"] d�\MU;Z;[�d doU ZB]/ehV%B>BBeN[4gL[�S0bjV%$�] U*SB[lZ;] _Za8$Y]o[YeJSBUfS�MV%B>BBeNU�BBej]oV bNUAig-/V do[l]oSAVfZ;[8I*<AV bN[8$�$�V � Z;U�5�<;bhV�gNU*BBehV�bNbN<;bNbNU V d�BBejUf]D(B] bj]C5*U�V%<A\ [�S0bNUZB[YdZIfehVfZ;UZ;[�d d^V XE<AeN5�V Z;]Z_O[�$Y]o[Ye3��XdKB[�gN]oX�<BejVf\M[YS0bN[ B(UfeNbjVV*Zr[YejeNU*eN]HZB]pBBej[�XY]ogN]oUfSB[ZB]�\MV*XYXdKB]oSAV . V�bjVfd ]�B([Ye d^V EefqH\HJT f CZ@SE)E Z;[YdodoV XE<Be?5�V ZAV�ehV%BJBAeN[4gL[�S*bhV ej[fi : S*bjeNU;Z=<nX_]^V�m\MU �]<B]oSAZ;] <BS`L Vfd bjejV . V \M]DIfdo]oV Z;]?X�<BeN5*[�!d [WX�<BeN5*[ E b j <MC9@ �FXdKB[MU*eN\kV ] �WZ;]D5f[�S*bhV�bjU ]odgN]oSBUfSB]o\MU ZB] X�<Be?5f[ B(Ufdo] SAUf\M]oVfd ]�V bNehV�bLbj] i -/V do[ . Vf\M]CI*d ]^V"Z;]�X�<BeN5*[ XY]pB([Yej\M[_bLbj[Z;]\kVfS*bj[Yej[Yej[ � (nVfgjgLU � ]od`I*ejV*Z;UWZ;[Yd�B(Ufdo] SBU*\M] U B([Ye!ejV*BJBBej[�gN[YS0bjVfeN[ d^VWXE<AeN5�V��n\kVk] d�. V�bNmbjU Z;] <;bj] do]C$8$�VfeN[ bNehV�bNbN]HZ;]�. <ASJ$Y]oUfSB]tBnU*d ]oSBUf\M]^V do]�SB[�XY[�gjgL] bjV Z;[�d do[WX_U*SAgL]^Z;[�ejV*$Y]oUfSB]�Bn[�e�]<AVfS0bNUMeN]DI*<nV ehZBVl]od �LejV*XYX_U*ejZBU �!Z;] �]<B[4g bj],bNehV�bNbN]vi

�� * ��1 ��& . ��"#$&1� � � �#$%�&

: SA]C$�]oVf\ UMd^V bNehV�bLbhV%$�] U*SB[ Z;[Ydod [ X�<BeN5*[ g/BBdo]oSB[ eN]DBBej[YSAZ;[�SAZ;U-�]<AV S0bjU45a]^g bjU SA[Yd�X�V%BB] bNU*d UBAeN[4X_[�ZB[YS0bN[��/XY] U/� <;bj] do]D$�$Y[�eN[�\MU d [ XE<Be?5f[WZ;]Z_Za�$Y]o[YecbNehVfZ;]D$Y]oUfSAVfd ]tBn[�ecejV*BJBBej[�gN[YS0bjVfeN[]HbNehV�bLbj]?Z;[�d d^V XE<Be?5�V gNBBdo] SA[fi W <B[4g bjU bj]CB(UZ;]?X�<BeN5*[BBeN[�SAZ;U*SBU V*BJBJ<BS0bjU ]odHSBU*\M[WZ;]_Za8$Y]o[YeNm&gNBBdo] SB[*iXq/]ogjgjV�bNUX<BS�]oS*bj[Ye?5�V dod U

[a, b]��X_UfSngL]^Z;[Yej]^V \MU <BSAVXBAV eNbN]D$Y]oUfSB[ Z;]?bhV do[

]oS0bN[�eN5�V dodoU!�a = u0 < u1 < . . . < um = b

' KA]oVf\ [�eN[�\MU ]oSgL[8I*<B] bNU6Ifdo]uiX_U*\ [ CZH�>P< i � SAVkX�<BeN5�V_Za�$�] [�e/m g/BAd ]oSB[ �cZ;[ �ASA] bhV ZAV

X : [u0, um] → Ed

XYUfSX(u) = Xi(u)

B([Yei = 0, . . . , m−1

[u ∈ [ui, ui+1]

i �Jd ]Xi(u)

gLU*SBU B(Ufdo]oSBUf\M]��XdKA[ ejV*BJBBej[�gN[YS0bN[�eN[�\ U SB[Ydod^V). U*eN\kV

Xi(u) =

n∑

i=0

bkiBnk (

u − ui

hi)

9*9

9*@ �� 4� �q �p�� i�?� ��

b

b

b

b

00

10

20

30= b01

b11

b21

b31

b

= b02

b

12

22

b32

q/]DI*<BehV4,Ai ! � P gL[�\BB] UWgNBBdo] SA[�BehVfX�X_UfehZ;UC0

XYUfShi = ui+1 − ui

�i = 0, . . . , m − 1

i : S ��<A[�gLbNUW\MU;Z;U KBUWZB[ �ASB] bNUmX�<BeN5*[�B\kV

5*U*Ifdo]^V \MU)5*[�Z;[�eNSB[gI*ejV��nXYVf\ [�S0bN[ gLU*d bhV S0bNU6<BSAV!��<A] SAZB]9] \B(UfSB]^V \MUMd^VkX_UfSnZ;]C$�] U*SB[

Xi(ui+1) = Xi+1(ui+1): S BAV eNbN]^X_U*doVfeN[��Bn[�eJdo[BBejU*BBej] [Yb?o Z;]�[YSAZ=mMB(Uf]oS0b ]oS0bN[�eNB(Ufd^V�bj] U*S Z;[�d do[

Xi�5f[Yej]��nXYV bjV

gN[ [cgLU*d UMgL[bni = b0 i+1 i = 0, . . . , m − 2

W <B[4g bjU�bj]CB(U�Z;] X_UfSnZ;]C$�] U*SB[ gN<Bdod [rgNBBd ]oSB[ 5a] [�SB[ ZB[_bLbjU ReN G)G)HJRe>�H >J<^TU< b H C0i � S

[4gL[�\6BA] U �T\ U0g bjejV bNU SB[�d d^V �WI*<BehVr,Bi ! i�:�ejV*XYX_U*ejZB]AZ;]nbN]DBnUC0

gNUfSBUcgN]oX�<BehV \M[YS0bN[ <;bN]od ]B([YelZB]ogN[�I*SAV ej[kXE<Be?5f[kgNBBd ]oSB[W]oSr\kV SB]o[YehVgL[�\BBd ]^X_[��9b?<;bLbhVV50]^V��B([YelVV5*[Yej[6<ASAVXE<Be?5�VBB]�� eN[8IfU*doVfeN[��0gN]9XdKB]o[�Z;[cgNBn[4gNgNUMXdKB[ ]DejV*XYXYUfehZ;]DgL]^V SBUkZ;],bj]CB(U

C1�AX_]oU/�

dXi

du(ui+1) =

dXi+1

du(ui+1)

-/V do[ X_UfSnZ;]C$�] U*SB[g5�V do[ gN[ [cgLU*d UMgN[1

hi

dXi

dt(1) =

1

hi+1

dXi+1

dt(0)

#�,Ai ! %HFV ZB[Yej]C5�V�bhVMZ;[�I*d ]

Xi� SBU bhV��

dXi

dt(ui) = n

n−1∑

k=0

∆bkiBn−1k (

u − ui

hi)

W <B]oSAZ;]�d\L [ �]<AV*$Y]oUfSB[g,Bi ! Z;]C5*[YS0bjV∆bn−1 i

hi=

∆b0 i+1

hi+1

�JehV"�nX�V \M[YS0bj[ �]<B[�gLbNU gL]DIfSB]��nX�V XdKB[�]odZBn[�S]<Ad bj] \MU SBU;Z;U Z;[�dpB(Ufdo]DIfUfSAU"Z;]�X_U*S*bjeNU*d doUZ;[�d d^VkXE<AeN5�V

imv[4gL]o\kV��B]od9SBU;Z;UWZ;]FejV*XYXYUfehZ;U [ ]od�BBej]o\ UkSAUaZBUWZ;[YdodoVkX�<Be?5uV

i + 1mv[4gL]o\kV

Z;[85fU*SBU gLbjVfeN[ g/< <BSAV ej[_bLbhVBi -/V do[ X_UfSnZ;]C$�] U*SB[ Bn[�e � SBUfS X_] V*gNgN]oX�<BehV dML V*gNgN[YSJ$4V"Z;]�NgNVfd bj] � Z;[YdodoVkX�<Be?5uV bN<BehVBi��[�ZB]oVf\ UB([YehX_] �Md [cXYUfSAZ;]D$Y]oUfSA]9g/<B]�ejV*XYX_U*ejZB](Z;]9bN]DBnU

C2�

d2Xi

d2u(ui+1) =

d2Xi+1

d2u(ui+1)

W <B[4g bOL <Ad bj] \kV65�V do[cgL[ [cgNUfdoU gN[

1

h2i

d2Xi

d2t(1) =

1

h2i+1

d2Xi+1

d2t(0)

�q �� p�� ?�� ?� �� 9�F

b

bb

D

h

h h

h h

bh

n−2, i

n−1, i

n,i = b0, i+1

1,i+1

2,i+1b

i+1

i

i i+1

i+1

i+1

i

q/]CI�<BejV,Ai � ��ATVfXYXYUfehZ;UC2

U��B[ �]<B]D5�V do[YS0bN[�\M[YS0bN[��

1

h2i

∆2bn−2 i =1

h2i+1

∆2b0 i+1 i = 0, . . . , m − 2

m=50]odD<JBJBB]^V \MU ��<A[�gLb8L <Bd bj] \kVT[ �]<AV%$�] U*SB[tB([Ye�Z;[YbN[Yej\M] SnV ej[?X_U0gNV!X_Uf\B(UfeNbjV I*ejV��nXYVf\M[YS=mbj[ �]<B[�gLbNU<Bd bN[�eN]oUfej[g50]oSAXYUfdoUAi

h2i+1 [(bn i − bn−1 i) − (bn−1 i − bn−2 i)] = h2

i [(b2 i+1 − b1 i+1) − (b1 i+1 − b0 i+1)]

m;[Y\BBdo].�nX�V SAZBUkU*BJB(UfeNbN<BSnV \M[YS0bN[

h2i+1(bn i−bn−1 i)+h2

i (b1 i+1−b0 i+1) = h2i+1(bn−1 i−bn−2 i)+h2

i (b2 i+1−b1 i+1)

` ]D50]^Z;]^V \MU �]<B[�gLb8L <Bd bN]o\kVM[ �]<AV%$�] U*SB[+Bn[�eh2

i+1

[ g/. eN<;bNbN]^V \MUBnU*]9d^V XYUfSAZB]C$�] U*SB[C1

(bn i − bn−1 i) +hi

hi+1(bn i − bn−1 i) = (bn−1 i − bn−2 i) +

h2i

h2i+1

(b2 i+1 − b1)

� Ufd bN]DBBd ]^XdKB]^V \MUMUfehV Vf\)((UM]9\M[Y\)(AeN] B([Yehi+1/hi

(b1 i+1 − b0 i+1) + (bn i − bn−1 i) = hi+1/hi(bn−1 i − bn−2 i) + (b2 i+1 − b1)

m�. eN<;bNbjVfSAZ;Ukd^V XYUfSAZB]C$�] U*SB[C0

U bLbj[YSB]^V \MU

bn−1 i +hi+1

hi(bn−1 i − bn−2 i)

︸︷︷︸

Di+1

= b1 i+1 +hi

hi+1(b1 i+1 − b2 i+1)

W <A] SAZB]�BnU0gNgN]^V \MU gjX_ej]D5f[Yej[

b1 i+1 − b2 i+1 =hi+1

hi(Di+1 − b1 i+1)

�JehV"�(XYV \M[�S*bj[ doVMgN] b?<AV%$�] U*SB[ �J\MU0g bjejV bjV SB[Ydod^V��WI*<BehV4,Ai � i

9*& �� 4� �q �p�� i�?� ��

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

b00

b10

b20

b30

b40

= b01

b11

b21

b31

b41

= b02

b12

b22

b32

b42

q/]DI*<BehV6,Bi ,�� _Za�$Y]o[YeNm g/BBdo]oSB[ X_UfS ehVfXYXYUfehZ;UC0

−1 −0.5 0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

q/]CI�<BejV6,Bi 2!� ` [Yej]C5�V bjVMXE<Be?5�V_pa8$Y]o[YeNm&gNBBdo] SA[+,Ai ,

�q �� p�� ?�� ?� �� 9%1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

b00

b10

b20

b30

b40

= b01

b11

b21 b

31

b41

= b02

b12

b22

b32

b42

qF]DI*<AejV,BiG9(� _Za�$�] [�e/m gNBBd ]oSB[ X_U*S ehVfX�X_UfehZ;UC1

−1.5 −1 −0.5 0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

q/]DI*<BehV,Bi @�� ` [�eN]D5�V�bhV4BAeN]o\kV ZB[YdodoVMXE<AeN5�V_Za�$Y]o[YeNm g/BBdo]oSB[+,Ai 9

@ 7 �� 4� �q �p�� i�?� ��

−4 −3 −2 −1 0 1 2 3 4−6

−5

−4

−3

−2

−1

0

1

2

3

4

q/]CI�<BehV4,AiGF(� ` [Yej]D5uV bjVMgN[�X_U*SAZBVMZ;[�d d^VMXE<Be?5�V6_Za8$Y]o[YeNm&gNBBdo] SB[g,BiG9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

b00

b10

b20

b30 D

1

b40

= b01 b

11

b21

b31

D2

b41

= b02

b12

b22

b32

b42

q/]DI*<BehV6,Bi &�� _Za�$Y]o[YeNm g/BBdo]oSB[ X_UfS ehVfXYXYUfehZ;UC2

�q �� p�� ?�� ?� �� @ !

0.8 1 1.2 1.4 1.6 1.8 2−2

−1

0

1

2

3

4

q/]DI*<BehV,Bi 1�� ` [�eN]D5�V�bhV4BAeN]o\kV ZB[YdodoVMXE<AeN5�V_Za�$Y]o[YeNm g/BBdo]oSB[+,Ai &

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

4

5

6

q/]DI*<BehV4,Ai !O7 � ` [Yej]C5�V�bhVMgL[4X_UfSnZBV ZB[YdodoVMXE<AeN5�V_Za�$Y]o[YeNm g/BBdo]oSB[r,Bi &

@ � �� 4� �q �p�� i�?� ��

−15 −10 −5 0 5 10 15−5

0

5

10

15

20

q/]CI�<BehV4,Ai !�! � ` [�eN]D5�V�bhVlbj[Ye?$�VMZ;[�d d^VMXE<Be?5�V6_Za8$Y]o[YeNm&gNBBdo] SB[g,Bi &

�)( *+��%��!#$% �'#)�� &��-, &�31�#$2

R (J(B]^V \M] U��ASBU*ejV XYUfSAgN]^Z;[YehV�bjU X_Uf\M[ X_UfSnZ;]C$�] U*SB[ B([Ye6<BSAV"eN[8IfUfd^V�bj] bdoZ;[Yd ehVfX�X_UfehZ;Ud^V X_UfS0bj] S�<B] b?o X_d^VfgjgL]^XYV!�

Ci i qBU*eN\kVfd ]D$�$�]oVf\ U <BSrV d bNejU \MU;Z;UZ;] B(UfejeN[65a] SAXYUfdo]Hg/<Ad d^Vej[�I*Ufd^V ej] bdo��*g/. e?<;bLbhV SAZ;UMX_U*SAX_[YbLbj]9Z;],bN]DBnUBB]�� If[YU*\M[_bNej]^X_U!�FdoV G)HJC=TU<MC=hi<MT �r7@)HJIL@$T]RS<BG)N i5 �76�� h38��;:=<�>J<BG)@ G)fq@g>PhD@nG$hiRS`J@ b NJReNPIQ@�TURS<BG)fq@ X,Y

ES<cR[N G)G)HJRe>�NJC9H G)HPCG)HJC=TU<MC=hi<MT �r�@)HPIQ@�TURS<BG)N

Gr <MC hiC b hiCDT H PE�@Q@SES<FESTm@-hiC9N RS H b H j NLRS�H

Cr V' U*SAgL]^Z;[�eN]^V \MU �]<B]oSAZ;];Z=<B[ XE<AeN5*[tBAVfejVf\ [YbNej]oXdKA[

X(u)[Y(t)

�fXYUfSX : [u0, u1] → E

d[Y : [t1, t2] → E

di?ma]^V

P<BS BJ<AS*bjUkZ;]9ehVfXYXYUfehZ;Ul]oSXE<B]

X(u1) = Y(t1)i

A�]DBAV ehV \M[YbNej]C$8$Y]^V \MUMdoVWX�<Be?5uVYXYUfS

u = u(t)�u : [t1, t2] → [u1, u2]

i R 0kSAXdK>abhV do[Jej]DBAV ehV \M[_bjeN]D$�$4V%$�] U*SB[!gN]oVMVf\ \M]^gNgN]D(B] do[ Z;[�5*[ eN]^g/<Ad bhV ej[

du/dt > 0i W <B]oSAZ;]

dX

du(u1) =

dY

du(u1)

5�V do[ gL[ [cgNUfdoUkgL[

dX

du(u1) =

dY

dt(t1)

dt

du(u1)

︸︷︷︸

w1>0

= w1dY

dt(t1)

W <B[4g bOL <Ad bj] \kV [4g/BAeN[4gNgN] U*SB[-��d^V"XYUfS0bN]oS�<B] b?o I*[YUf\M[YbNej]oX�VG1�WgN]�]o\BnU*SB[�]oS BBehV�bj]oX�V

XdKB[ ]od 5f[�ejgNUfej[ bhV SJI*[YS0bN[ gN]oV6<JI�<AV do[ ZBVMZ;[�gLbNehVl[ ZBVkgN]oSB]ogLbNehV0# X_U*S <AS [85a]oZ;[�S0bN[ gNVfd bjUB([Ye �]<AVfS*bjU eN]DI*<AVfejZAVT]odB\MU;Z=<BdoU % �*] SJ. V�bLbj]AgN]=BAV ejdoV Z;]nX_UfS0bj] S�<B] b?o

G1X_U*\M[ G)HPCDT]<MCDhi<MT

>�@$j `J@�R)E$HPRe@�TmNPC7r7@$C=Tm@ i ` [ �ASB]^V \MUkUfehV doVMX_U*S0bN]oS]<A] b��G2

# XdKB[rBBej[�gN<JBJB(UfSB[ d^VG1% �

d2X

d2u(u1) =

d2Y

d2u(u1)

�q �D �� ^�� c� � �p�� @*,

[ �]<B[4g bjU5uVfd [ gN[ [ gLU*d UkgN[

d2X

d2u(u1) =

d

du

(dY

dt

dt

du

)

=

=d2Y

d2t

dt

du

dt

du+

dY

dt

d2t

d2u︸︷︷︸

w2

=

=d2Y

d2t(t1)w

21 +

dY

dt(t1)w2

W <A[�gLbjV [�gNBBej[�gjgL]oUfSA[MZB[ �ASB]^gjX_[Wd^VX_U*S*bj] S�<B] b?o If[YU*\M[_bNej]^XYVG2i ` [_bN[�eN\M]oSB]^V \MU X_U0gNV

XYUf\B(UfeNbjV doV XYUfS0bN]oS�<B] bdo If[YU*\M[_bNej]^XYV ] S bN[�eN\M]oSB]HZ;]?X�<Be?5uV bN<BehV [ BB]oVfSBU U*gjXE<Bd^V�bjUfej[�XYUfSAgN]^Z;[Yej]oVf\MU ]od�5f[�ejgNUfej[ (A] SBU*eN\kVfd [ gN] SB]^gLbNejU [cZ;[4g bjeNU

e−3 =N−|N−|

con N− =dX

du(u1) ∧

d2X

d2u(u1)

e+3 =

N+

|N+|con N+ =

dY

dt(t1) ∧

d2Y

d2y(t1)

A ]ogN<Bd bjVB([Ye��

N− = w1dY

dt(t1) ∧

(

w21

d2Y

d2t(t1) +

SBUfS ZJoMX_U*S*bjeN]D(J<;bjU︷︸︸︷

w2dY

dt(t1)

)

= w31

dY

dt(t1) ∧

d2Y

d2t(t1) = w3

1N+#\,Bi � %

�]<B]oSAZ;]e−3 =

w31N+

| w31

︸︷︷︸

>0

N+|=

N+

|N+|= e+

3

XY] U/�cV*(J(B]^V \MUMdoVMX_U*S0bN]oS]<A] bdo ZB[Yd�5*[YehgLU*eN[ (B] SAUfej\MVfd [*i ' UfS0bNejUfdod ]^V \MU gL[ d^VMXE<Be?5�V�bN<AejV�cXYUfS0bN]oS�<AV

κ− =|N−|

∣∣dX

du (u1)∣∣3 =

w31 |N+|

w31

∣∣dY

dt (t1)∣∣3 = κ+

��! �� " �� &�

R BJBBdo]oXdKA]oVf\ U"d [�XYUfSAgN]^Z;[YehV%$�] U*SB] . V�bNbN[�gN<BdodoV"X_UfS0bj] S�<B] b?o I*[YUf\M[YbNej]oX�VV dtB(Ufdo]DIfUfSAU Z;]XYUfS0bNejUfdodoU Z;[�d d^VIf[�SB[Yej]^XYV XE<Be?5�V4_Za�$�] [�e/m gNBBd ]oSB[��

Xi(u) =

n∑

k=0

bkiBnk

(u − ui

hi

)

XYUfSu ∈ [ui, ui+1] i = 0, . . . , m − 1

i�: \6B(UfSA]oVf\ UkV��]<B[�gLbjV d^VG1�

dXi

du(ui+1) = w1i

dXi+1

du(ui+1)

XY] U/��BeN]DBAVfejVf\ [YbNej]C$8$�VfSAZ;U1

hi

dXi

dt(1) = w1i

1

hi+1

dXi+1

dt(0)

@*2 �� 4� �q �p�� i�?� ��

b

b

b

b

20

30= b01

11

b10

00 b21

b31=b02

q/]DI*<BehV6,Bi ! � � A V*XYXYUfehZ;UG1

XdKB[ gN]�BJ< �kgjX_ej]C5*[Yej[JXYUf\M[1

hi∆bn−1 i = w1i

1

hi+1∆b0 i+1

: S6BAejV bN]^XYV ��<A[�gLb8L <Bd bj] \kV gN]CI*SB]��nXYVJXdKB[ ]=BJ<BS0bN]bn−1 i,bn i = b0 i+1,b1 i+1

gLbjV SASBUgN<BdodoV g bj[�gjgNV ej[_bLbhVl[ d^VMZ;]ogLbjVfSJ$�V Z;]

b0 i+1ZBV d BJ<AS*bjUkX_[YS0bjejVfd [ �

w1i5fU*d bj[

∆bn−1 igN[ ]�BJ<BS0bN]FgLU*SBU [3�]<B]ogNBAV*$Y]^V�bN]vi��[�Z;]^V \MUWV*Z;[�gjgLUMX_U0gNV X_U*\BnU*eLbhV d^VMX_U*S*bj] S�<B] b?oG2�

d2Xi

d2u(ui) = w2

1i

d2Xi+1

d2u(ui+1) + w2i

dXi+1

du(ui+1)

ej]CBnV ehV \M[_bjeN]D$�$4V SAZ;U U bNbN[YSA]oVf\ U

1

h2i

d2Xi

d2t(1) = w2

1i

1

h2i+1

d2Xi+1

d2t(0) + w2i

1

hi+1

dXi+1

dt(0)

" [Y]Dbj[Yej\ ]oSB]9Z;[�d�B(Ufdo]CI*UfSBUkZ;]9XYUfS0bNejUfdod U��]<B[�gLb8L <Bd bN]o\kVkgL]9gjX_ej]C5*[ X_Uf\M[��n(n − 1)

h2i

∆2bn−2 i = w21i

n(n − 1)

h2i+1

∆2b0 i+1 + w2in

hi+1∆b0 i+1

#�,Ai , %

3�UfSA]oVf\ U

a = ∆bn−2 i

b = ∆bn−1 i

c = ∆b0 i+1

d = ∆b0 i+1

3�U*gjgN]oVf\ U ej]ogjX_ej]C5*[Yej[Td^V,Bi , XYUf\M[

(b − a)n − 1

h2i

= w21i

n − 1

h2i+1

(d− c) + w2ic

hi+1

�q �q �p� �� ?�� @�9

va

b c

d\\

\\

qF]DI*<AejV,Bi ! ,!� A VfX�X_UfehZ;UG2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

b00

b10

b20

b30

b40

= b01

b11

b21

b31

b41

= b02

b12

b22 b

32

b42

qF]DI*<AejV6,Bi ! 2��_Za8$Y]o[YeNm&gNBBdo] SB[ X_U*S ehVfX�X_U*ejZ;UG1

` [ �ASB]^gjX_U <BS 5f[YbLbNU*eN[v ⊥ b

[ d UM\MUfd bN]DBBdo]oXYUMVfd dML [ �]<AV%$�] U*SB[+BBeN[4X_[4Z;[YS0bN[ UfbLbN[�SB[YSnZ;U

dv = w−21i

h2i+1

h2i

(−av)

: S ej] . [�eN]o\M[YS0bNU V dodoV �>I*<AejV ,Bi ! ,��<A[�gLb8L <Bd bj] \kV gL]DIfSA].�nX�V XdKA[ Z;[85fU*SBU�[4gNgN[Yej[4BAeNU�BnU*e/m$�] U*SAV do]Ddo[cX_Uf\B(UfSB[�S0bN]DdD<BSJIfU

vZ;]

a[di

� ��%'"#$��%'# � � $#$%'&

�J[�SB[YehV do]D$�$Y]^V \MU �]<AV SnZ;U Z;[�gjX_ej] bNbNU �ASBU*ejV gN<Bdod [ gNBBdo] SB[*i qF]^gjgL]^V \MUg<BSM] S0bj[Ye?5uVfd doU[a, b][cgN<��<A[�gLbNUM] S0bj[Ye?5uVfd doUkX_U*SAgL]^Z;[�eN]^V \MU)<ASAVBAV eNbN]D$Y]oUfSA[

τZ;[_bNbjV `J@$T]T HJRe@^> @�<_C9H\>J< bjV do[

XdKA[a = τ0 < τ1 < . . . < τL = b[rB(UfSA]oVf\ U

Ii = [τi, τi+1]Bn[�e

i = 0, . . . , L − 1i

@�@ �� 4� �q �p�� i�?� ��

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

q/]CI�<BejV,Ai ! 9(��[YehgLU*eN[!bjVfSJIf[�S0bN[ Z;[YdodoVkX�<Be?5uV_Za�$�] [�e/m gNBBd ]oSB[g,Bi ! 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b00

b10

b20

b30 b

40

b50

= b01

b11

b21 b

31

b41

b51

= b02

b12

b22

b32

b42

b52

q/]DI*<BehV4,Ai ! @�� _Za�$�] [�e/m g/BAd ]oSB[ X_UfS ehVfXYXYUfehZ;UG2

�q �q �p� �� ?�� @]F

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

q/]CI�<BejV6,Bi ! F(� ' <BeN5�V bN<BehV ZB[YdodoVkX�<BeN5�V6_Za�$Y]o[YeNm g/BBdo]oSB[r,Bi ! @

5 �76�� h38 S �!HLE b N�O�<BHn>J<T]hiTUT @�j @ �)hiC O�<BHJC=J<�rPR[N > H m>�@BA?CD<MT @

EShτ�

Pm,τ = {f : [a, b] → R, ∃p0, . . . , pL−1 ∈ Πm|fIi= pi i = 0, . . . , L − 1}

HFVlZ;]o\M[YSAgN]oUfSB[JZ;]'�]<B[4g bjUlgNBAV%$�] U �L(m+1)

BnU*]oXdK!� ]WB(Ufdo]oSBUf\M]�KAV SBSAUm+1

IfehVfZ;]ZB]9d ]D(([YeNb?oM[gBnU0gNgNUkgNXY[�I*d ]o[YejSB[J] SAZB]CB([YSAZB[YS0bN[�\ [�S0bN[

Li

5 �76�� h38Dh �!H E b N�O�<BH >�@$jMj @0E b j <MCZ@gG�j NaE)ES<BG[fD@ >P<r ReN >�H G m>�@BA?CD<MT @'ESh

τ�

>�NJT H�>�N

Sm,τ = {f : [a, b] → R, ∃p0, . . . , pL−1 ∈ Πm|fIi= pi i = 0, . . . , L−1}∧Cm−1([a, b])

HFVlZ;]o\M[YSAgN]oUfSB[ Z;]nbjV do[!gNBAV*$Y]oU��L + m

] S0�]<AV S0bNU gLU*SBUl]o\BnU0g bj[!gN<B],SBU;Z;](L− 1)mXYUfSAZB]C$�] U*SB] i H L ] S0bN[�ejgN[�$�] U*SB[ �XYUfS

Cm−1B([YehXdK�� X_UfS

mVV5aeN[�\ U"d\L ] S0bj[Ye?BnU*doV*$Y]oUfSB[

B(Ufdo]oSBUf\M]^V do[ X_d^VfgjgL]^XYVAi � . VfXY] do[ 5*[Yej].�nX�V ej[OXdKA[ 5uVfd [TdoV gN[�I�<B[YS0bN[Z. ehV g/BnV%$Y]>BnU*d ]oSBUf\M]^V do][cgNBAV*$Y]9Z;]�gNBBdo] SA[�

Πm ⊆ Sm,τ ⊆ Pm,τ

�� %&� �� $� � �� &��

` [ �nSB]oVf\MU!�9U*ejVWXdKA[ V*(J(B]^V \MU . U*eN\kVfd ]D$�$4V�bNU�d U g/BnV%$Y]oU ZB[Ydod [kgNBBd ]oSB[��DdoV (AVfgN[ ZB]�bjV do[gNBAV*$Y]oUAi�m0bNU*eN]^XYVf\M[YS0bN[��doV BBej] \kV (nVfgN[ VfZ [�gjgL[�eN[ <;bN]od ]D$�$4V�bhV�� g bhV�bhV d^V K)NaE�@g>�@$jMj @b HJT @�C O\@�T]ReHPC9G)NJT @ i H�[gBnUfbN[YS>$Y[ bNejUfSnXYV�bj[JgNUfSAUMZB[ �ASB] bN[cZAV

(t − τi)m+ =

{(t − τi)

m se t ≥ τi

0 altrimentii = 1, . . . , L − 1

ma]^XYXYUf\M[(t−τi)

m� <BSB(Ufdo] SBU*\M] UcZ;]WIfehVfZ;U

mKnV V dJBB]��

m−1Z;[�eN]D5�V�bj[�SBUfSMS�<Bdod [��

K�� N�6#1 :�.\]A8�cn:%8 c�8�6�h o�y/-P1#cU0U8�Q�8 2�1nY�.\9T6O.\9%8 2,8�6U`�9%.\2�N�2,8�6#6O.zW�Y,6#1#c�8(2,1�Y�.\9T6O.z2�1��wkk = m+1

g

@�& �� 4� �q �p�� i�?� ��

a bτ i = τL

= τ 0

q/]CI�<BehV4,Ai ! &��3�U bN[�SJ$Y[JbjeNU*SAXYV bN[

�]<B]oSAZ;]�V*BJBAVfeLbj] [�SB[JVCm−1([a, b])

i HFV4(nVfgN[ ZB]Sm,τ

� ��<A] SAZB]

{1, t, . . . , tm, (t − τ1)m, . . . , (t − τL−1)

m}��[�eN]��nXdKB]^V \MU"XdKB[ � <BSAVX(AVfgN[�Z;] Z;]o\M[YSAgN] U*SB[

L + m. V*X_[YSnZ;UfSB[�doV X_Uf\4(B]oSAV%$�] U*SB[

do] SA[�V ej[�m∑

i=0

aiti +

L−1∑

i=1

bi(t − τi)m = 0 ∀t ∈ [a, b]

' U*SAgL]^Z;[�eN]^V \MU ]9gL]oSJIfU*d ]9gNU bNbNU*] S0bN[�eN5�Vfd do]��

• SB[Ydod\L ]oS*bj[Ye?5�V dod U[a, τ1] = [τ0, τ1]

d^V (AV*gL[?ZB[Ydod [ B(U bj[YSJ$�[�bNejUfSnXYV�bj[ �HS�<BdodoV!��ej] \kVfSB[��<A] SAZB]

m∑

i=0

aiti = 0

� V-�]<B[�gLb8L <Bd bN]o\MV��r5*[Yej].�nX�V�bhV gNUfd^V \M[YS0bj[JgN[ai = 0

i

• SB[Ydod\L ]oS*bj[Ye?5�V dod U[τ1, τ2]

V%(J(A]oVf\ UrB([Ye?]CB(U bj[�gN]AXdKB[ Ifdo]aigNUfSBUJb?<;bLbj]nS�<Bdod ]��]<B]oSAZ;]

eN]o\kV SB[β1(t − τ1)

m = 0XdKB[��65f[�eN]��nX�V�bNU gLU*doVf\ [�S0bN[)B([Ye

β1 = 0icR SAVfd U�I*V%m

\ [�S0bN[cgN]9U bNbN[YS>IfUfdoU6Ifdo]9V d bNej]9XYVfgN]viHFV (nVfgN[ ZB[Ydod [ B(U bj[YSJ$�[ bNejUfSAX�V�bN[ SBU*S � <;bN]od ]D$�$4V�bhV SA[Yd ' R� ` Bn[�ek[�5a]oZB[YS0bN] BAeNU*m(Bdo[Y\M]OZ;]OXYUfSAZB]C$�] U*SAV \M[YS0bjUAi H L <;bN]od ]D$�$�U"Z;]?bjV do[ (AV*gL[ X_Uf\B(UfeNbjV ]oSBUfd bNej[Wd^V Bn[�ejZB] bhVZ;[�d dML ] S0bN[�eNBAeN[YbjV%$�] U*SB[+If[YU*\M[_bNej]^XYV Z;[Yd�B(Ufdo]DIfUfSAU Z;]FX_UfS0bjeNU*d doUAi

� � � �! � � � � % � �� &�

` [�gjX_ej]C5a]^V \MU <BScVfd bjeNU bj]CB(U!Z;]�(AVfgN[ B([YeFU�5]5a]^V ej[�V ]*BAeNU�(Bd [�\M]fZ;[Ydod^V (AVfgN[?Z;[�d do[ B(U bj[YSJ$�[bjeNU*SAXYV bN[fi�q/]ogjgjV�bNU ]od 5f[_bNbNU*eN[JZ;[Y]DSBU;Z;]

τ = {τ0, τ1, . . . , τL}ZB[ �ASB]^V \MUM]od `J@�TUT HJRe@�> @�<

CZH�>P<?@SEST @SE�H

T = {t0, . . . , tk−2, tk−1, . . . , tn+1, tn+2, . . . , tn+k}:MSBU;Z;] Z;]

TgLU*SBU�X_U0g��lZ;[ �ASA] bj]��

tk−1 = τ0, . . . , tn+1 = τL[�Z ] CZH�>P<�NJh ES<Mj <BNPRS<

t0, . . . , tk−2[

tn+2, . . . , tn+kgNUfSBU bjVfd ]�XdKA[�

t0 ≤ t1 ≤ . . . ≤ tk−1 = a < b = tn+1 ≤ . . . ≤ tn+k−1 ≤ tn+k

�q �q �p� �� ?�� @*1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

q/]DI*<BehV,Bi ! 1��_ VfgN[Ni 2

` [_bj[Yej\M] SB]^V \MUn�

L + 1 = (n + 1) − (k − 1) + 1 = n − k + 2 + 1 = n − k + 3

�]<B]oSAZ;]n = k + L − 2

i H�VMZ;]o\ [�SAgN] U*SB[ Z;[Ydod UkgNBAV%$�] U��gB([YeNbjV S0bjU

dim Sm,τ = L + m = n + 1

` ]^V \MU <BSAV"Z;[ �ASA]C$�] U*SB[�Z;[�d d^V (AVfgN[�Z;[�d do[ _pm g/BBdo]oSB[k[�Zr[�d [�SAXdKB]^V \MUfSB[kdo[ BBejU*BAeN]o[_bdo��VfSAV doU*I0V \M[YS0bj[ V��<nV S0bNUg. V bLbNU)Bn[�eH]WB(Ufdo]oSBUf\M]AZ;]�_O[YejSAgLbN[Y]oS9i�H�Vg(AV*gL[ ZB[Ydod [ _pm g/BAd ]oSB[�cZB[ �ASB] bjVMej]^X_UfehgN]C5�V \M[�S*bj[T]oS (AVfgN[ V dod\L UfehZ;]oSB[

k�

{Ni k(t), i = 0, . . . , n}` [ �nSB]oVf\MUrVfd doUfehV��]<B[�d do[ Z;]�UfehZ;]oSB[

1# �]<B]oSAZ;] IfehVfZ;U

0��gNUfSBU . <BSJ$�] U*SB]�X_U*gLbjVfS0bN]�V

bjejV bLbj] %

Ni 1(t) =

{1 se ti ≤ t ≤ ti + 10 altrimenti

: S I*[YSB[�ejVfd [��

Ni k(t) =t − ti

ti+k−1 − tiNi k−1(t) +

ti+k − t

ti+k − ti+1Ni+1 k−1(t)

� [�5a]^Z;[YS0bj[ ]oS&�]<B[4g bOL <Ad bj] \kVX. Ufej\)<Bd^V dML V*<B\M[YS0bNUrZ;[�dZI*ejV*Z;U"Z;[�d dMLNi k

BBeN[4X_[4Z;[YS0bN[*i" [Yd�X�VfgNU

k = 2V*(J(B]^V \MU!�

Ni 2(t) =

t−ti

ti+1−tise ti ≤ t ≤ ti+1

ti+2−tti+2−ti+1

se ti+1 ≤ t ≤ ti+2

0 altrimenti

HFV. <BSJ$�] U*SB[Ni2(t)

�s<ASAVWgNBBd ]oSB[cdo]oSB[�VfeN[��JKAVWXYUfS0bN]oS�<B] bdoC0

[4Z KAVkdML V*g/B([_bNbNUWZ;]�<BS� bN[YbLb?<AXYXY] U �hi : SBUfd bNej[ � <BSnV . <BSJ$�] U*SB[ V"g/<JB>BnU*eLbjU"X_U*\6BnV�bLbjU [�Z ��[�5a]^Z;[YS0bj[ XdKB[�gN[]tigLU*SBU [ �]<B]^Z;]ogLbjVfS0bN]�]?X�V d^X_Ufdo]�gL]�gL[�\6BAd ]��nXYVfSBU SBU bj[�5fU*d \M[�S*bj[ # ]od?Z;[YSAUf\M] SnV�bNU*eN[��

gN[Y\BBej[ d UWgLbN[4gNgNU %_i " [YdFXYVfgNUk = 3

V%(J(B]^V \MU ] S�5*[�X_[��

F 7 �� 4� �q �p�� i�?� ��

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q/]CI�<BehV4,Ai �*7 � _ VfgN[Ni 3

Ni 3(t) =

(t−ti)2

(ti+2−ti)(ti+1−ti)se t ∈ [ti, ti+1]

(t−ti)(ti+2−t)(ti+2−ti+1)(ti+2−ti+1)

+ (t−ti)(ti+3−t)(ti+2−ti+1)(ti+3−ti+1) se t ∈ [ti+1, ti+2]

(ti+3−t)2

(ti+3−ti+1)(ti+3−ti+2)se t ∈ [ti+2, ti+3]

0 altrimenti

: S0��<A[�gLbNUMXYVfgNUld^V4. <BSJ$�] U*SB[Ni3(t)

KAV X_U*S*bj] S�<B] b?oC1

[�Z KAV d\L VfgNBn[YbLbjU Z;] <ASAV �NXYVf\6mBAVfSAV � i ' U*\ [ BBeN]o\kV 5�V dDIfU*SBU do[ BBejU*BBej]o[_b?oZB] ESh b b HJRST H G)HJI b NPTUT H [ Z;]OeN]^Z=<J$�] U*SB[Z;[�]FX�V d^X_Ufdo]�gN[c]

tigLU*SBUk[ �]<B]^Z;]^g bhV S0bN]vi : S If[YSA[YehV do[�AZBV bNU <BS

kI*[YSB[�eN]^X_U��

Ni k(t)KnV

gN<JBJB(UfeNbNUsupp Ni k(t) = (ti, ti+k)

: SBU;Z;] X_U*\BBeN[4gL]?] S(ti, ti+k)

5*[YSJI*UfSBU ZB[_bLbj] CZH�>P<pNPTUT]<M`�< i � S <Bd bj[Yej] U*eN[ BBeNU�BBej] [Yb?oZ;[�d do[

Ni k(t)� doV C9HPC CZ@]r7NJT]<M`\<MT �AX_]oU/� gNUfSAU gN[Y\BBej[gBnU0gL] bN]D5f[��BX_Uf\M[+. VfXY] do[Y\M[�S*bj[ gL]

ej]oX�VV5�V ZBVfd d^V4. Ufej\)<Bd^V eN]^X_U*ejgN]D5uVAi H�V6BBejU*BBej] [Yb?o XdKB[rB([Ye��MXY]9]oS*bj[Yej[�gjgNV \kV%I�If]oUfej\ [�S0bN[� d^V b NPRSTU< O�<BHJCZ@Q> @�jMjlk hiCD<MT �

n∑

i=0

Ni k(t) = 1 ∀t ∈ [tk−1, tn+1]

` ] \MU0g bjeN]^V \MUfd^V4Bn[�e�] SAZJ<J$Y]oUfSB[cgN<k�

T �JiO��JiO�� B([Yek = 1

d^V bN[4gL]��r(nV SAVfd [��=50]o[YSA[ ZAV dodoVMZ;[ �ASB]D$Y]oUfSB[ gLbN[�gjgjV� jp�>�O�]iO�t�\��8��U � � g/<JB>BnU*SB]oVf\MU d^V bN[4gL]�5*[YehV �nSBUkV

k − 1

�Ji8iO� ��`�8�8� U�� BBeNU�5a]^V \MUMXdKB[ doV bN[4gL]`5uVfd [ VfSAXdKB[gBn[�ek�

n∑

i=0

Ni k(t) =

n∑

i=0

[t − ti


ti+k − t


]

=

=t − t0

tk−1 − t0N0 k−1(t) + N1 k−1(t)

[t − t1tk − t1

+tk − t

tk − t1

]

+ . . .

. . . + Nn k−1(t)

[t − tn

tn+k−1 − tn+

tn+k−1 − t

tn+k−1 − tn

]

+

+tn+k − t

tn+k − tn+1Nn+1 k−1(t) =

�q �q �p� �� ?�� F !

=

n∑

i=1

Ni k−1(t)

W <B[4g bOL <Bd bN]o\kV [�gNBBej[�gjgL]oUfSB[ B([Ye�]CB(U bj[�gN]B] SnZ=<;bLbj]C5�V)KAV gLU*\M\MV+BAVfeN]>KAV1� ��<A] SAZB]

d^V4BAeNU�BBeN]o[_bdo ZB]�BAV eNbN]D$Y]oUfSA[ ZB[Ydod\L <BSB] b?o�� Z;]o\MU*gLbNehV�bjVAiHFVkZ;[Yej]D5uV bjV Z;]

Ni kej]^g/<Bd bjV!�

dNi k

dt= (k − 1)

[Ni k−1(t)

ti+k−1 − ti− Ni+1 k−1(t)

ti+k − ti+1

]

-/V do[TZ;[ �ASB]D$Y]oUfSB[!gL[�eN5*[ Bn[�etBBejU�5�V ej[ XdKB[ Ifdo]Ni k(t)

gLU*SBU d ]oSB[4V ej\ [�S0bN[ ]oSAZ;]DBn[�SAZ;[�S*bj] i' U*SAgN]oZ;[�eN]^V \MU ] SA]C$�]oVfd \M[YS0bj[ ] d9X�VfgNU

Ni1[gB(UfSB]^V \MU

s1(t) =

n∑

i=0

aiNi 1(t)

` U*(>(B]oVf\MU BAeNU�5�V ej[ XdKA[s1(t) ≡ 0 ⇐⇒ ai = 0 i = 0, . . . , n

i � V ]oS <BSrgN] SJI*UfdoU]oS0bN[�eN5�V dodoU

[tj , tj + 1]d\L <BSB]^X_U

Ni 1(t)XdKB[ ZJoMX_UfS0bjeN]D(J<;bjU,� ]od

jmv[4gL]o\MU!�;XdKB[r5�V do[

1]oS

b?<;bLbjU"dML ]oS0bN[�eN5�V dodoU!� �]<B] SnZ;]�Z;[85f[W[4gNgN[Yej[aj = 0

Bn[�elU*I*SB]j = 0, . . . , n

� ��<A[Ydod U"XdKA[5*Ufdo[�5�V \MUMZ;]o\ U0g bjejVfeN[*i " [YdFXYVfgNU

Ni 2(t)eN]^gN<Bd bhV ] S�5f[4X_[

s2(t) =

n∑

i=0

aiNi 2(t)

[ BAeNU�5a]^V \MU XdKA[WgL]HVfSBS�<Bd d^V gL[k[WgNUfdoUgL[I*d ]aigNUfSBU b?<;bLbj]?S�<Bdod ]vi ma[WXYUfSAgN]oZB[Yej]oVf\ U

<AS gL]oSJIfU*d UMgNU bLbjUf]oS0bN[Ye?5�V dod U[ti, ti+1]

�Bd [r<ASB]oXdKA[Ni 2(t)

XdKB[ ZBVfSBSBUWX_U*S0bNej]C(J<BbNUkgLU*SBUd^V

(i − 1)mv[�gN]o\MVl[ doV

imv[4gL]o\kVBi�: S BAV eNbN]^X_U*doVfeN[��*]oS

tiKBUM] d9XYUfS0bNe?<J(J<BbNUMZ;[YdodoV

Nj−1 2�

�]<B]oSAZ;]/Z;[�5*[ eN]^g/<Ad bhV ej[aj−1 = 0

�=B([Yej = 1, . . . , n + 1

�AZAVkXE<B]9d^V bN[4gL]vi : S If[�SB[YehV do[�B([Ye

k ≥ 3Z;U*(J(A]oVf\ UMZ;]o\MU*gLbNehV ej[JXdKB[

s(t) =

n∑

i=0

aiNi k(t) = 0 ⇐⇒ ai = 0 i = 0, . . . , n

� VkgL[s(t)

gN]�V SBS�<Bdod^V��;Z;[�5*[cV SBS�<Bdod^V ehgL]DV SnXdKB[ doVMZ;[�eN]D5�V�bjV!��]<B] SnZ;]

ds(t)

dt=

n∑

i=0

aidNi k(t)

dt= 0 ∀t ∈ [tk−1, tn+1]

m=50]odD<JBJBB]^V \MU �]<B[�gLb8L <Bd bN]o\kV [3�]<AV%$�] U*SB[�

ds(t)

dt= (k − 1)

n∑

i=0

ai

[Ni k−1(t)

ti+k−1 − ti− Ni+1 k−1(t)

ti+k − ti+1

] #\,Bi 2 %

3�U0gNgN]oVf\MUTVfgjg/<A\ [�eN[ ��<A[�gLb8L <Bd bj] \kV!X_Uf\M[�]CB(U bj[�gN]*]oSAZ=<;bNbN]D5�VT[ Z;]o\ U0g bjejVfeN[HXdKA[�V SnXdKB[?]odNi k

gNUfSBU�do] SA[�V ej\M[YS0bN[�]oSAZ;]DB([YSAZ;[�S0bN]vi�: SsBnV eNbN]^X_Ufd^V ej[��g/5a]odC<JB>BAV SAZBU doV ,Ai 2�UfbLbj[YSB]^V \MU

ds(t)

dt= (k − 1)

[

N1 k−1(t)a1 − a0

tk − t1+ . . . + Nn k−1(t)

an − an−1

tn+k−1 − tn

]

=

= (k − 1)

n∑

i=1

ai − ai−1

ti+k−1 − tiNi k−1(t)

F � �� 4� �q �p�� i�?� ��

' U*SAZ;]D$Y]oUfSB[ B([YeMXE<A]?d^V BBej[�XY[�Z;[�S*bj[ . Ufej\)<AdoV gN]�VfSBS�<Bdod ]��ai = ai−1 i = 0, . . . , n

i: \B(UfSB[�SAZ;U bjVfd [cXYUfSAZ;]D$Y]oUfSA[J]oS

s(t)[rBnU*SB[YSAZBU

a = ai

s(t) = a

n∑

i=0

Ni k(t)

� V6B(Uf]^XdK�� d^V gNUf\M\kVkZ;[8Ifdo]Ni k

�1�;dML <BSB]^X_Uk\MU;Z;UWXdKB[ doVBAeN[4X_[�ZB[YS0bN[ gN]�V SBS�<Bdodo]��

a = 0�AXY] U/�

ai = 0B([Ye U�IfSB]

ii

� � � �� &�� $� � �

�J[�SB[YehV do]C$8$Y]^V \MU d UkgNBAV%$�] UMZ;[Ydodo[cg/BBdo]oSB[rBBej[�gN[YS0bjV bNU,�ASBUfehVBiHma]^V

M = {m1, m2, . . . , mL−1}XYUfS

mi ∈ N�1 ≤ mi ≤ k

igH�[mi

5f[YS>IfUfSAUWXdKB]oVf\kV�bN[ IQHPj T @ b j <BG$<MT Z;[�dFSAUaZBU�[MVfZU�IfS�<BSAV Z;]([�gjgN[�!V*gNgNU;X_]^V�bNUg<BS�SBU;Z;Uτii ` [ �ASB]^V \MU��]<B]oSAZ;](doVlg/BAV*$Y]oUlZ;[�d do[ _pm g/BBdo]oSB[

I*[YSB[�ejVfd ]D$�$4V�bN[

Sm,M,τ = {f : [a, b] → R, ∃p0, . . . , pL ∈ Πm|fIi= pi,

p(l)i−1(τi) = p

(l)i (τi), l = 0, . . . , k − mi − 1 i = 1, . . . , L − 1}

HFV ZB] \M[YSngL]oUfSB[ Z;]9bjV do[ g/BnV%$Y]oU��

dimSm,M,τ = (m + 1)L −L−1∑

i=1

(k − mi) = (m + 1)L − k(L − 1) +

L−1∑

i=1

mi =

= k +

L−1∑

i=1

mi = k + µ

A�]^Z;[ �ASB]^V \MU Bn[�eLbhV S0bNU d^V (AVfgN[� {Ni k(t), i = 0, . . . , n} X_U*S n = k + µ − 1[�Z ] d

5*[_bLbjUfej[![4g bj[�gNUkZ;[Y]9SAUaZB]

T = {t0, . . . , tk−1︸︷︷︸

a=τ0

, . . . , tn+1︸︷︷︸

b=τL

, . . . , tn+k}

XYUfSt0 ≤ t1 ≤ . . . ≤ tn+1 ≤ . . . ≤ tn+k

i :kSAUaZB]J]oS0bN[�eNSB] gNUfSBU�[4g/BAeN[4gNgN]!XYUfS\MUfd bN[8BBdo]oXY] bdo��BX_]oU/��

{tk, . . . , tn} = {τ1, . . . , τ1︸︷︷︸

m1

, . . . , τL−1, . . . , τL−1︸︷︷︸

mL−1

}

` [ �ASA]oVf\ UkVfd doUfehV)I*d ]Ni r(t)

� B([Yer = 1

V%(>(B]oVf\MU!�;X_U*\ [ SB[�d�XYV*gLU6BBej[�XY[�Z;[�S*bj[�

Ni 1(t) =

{1 ti ≤ t < ti+1

0 altrimenti

#�,Ai 9�%

� [�S0bNej[J]oS If[�SB[YehV do[

Ni r(t) =

0 se ti = . . . = ti+rti+r−t

ti+r−ti+1Ni+1,r−1(t) se ti = . . . = ti+r−1 < ti+r

t−ti

ti+r−1−tiNi,r−1(t) se ti < ti+1 = . . . = ti+r

t−ti

ti+r−1−tiNi,r−1(t) + ti+r−t

ti+r−ti+1Ni+1,r−1(t) altrimenti

�q �q �p� �� ?�� F�,

iO� � j��\� h38 3�[�e

k = 4[

L = 3V%(J(B]^V \MU

τ = {τ0, τ1, τ2, τ3}V dod U*ejV

1 ≤ m1, m2 ≤ 4i 3Hej[YSAZB]oV%m

\MU X_Uf\M[c\MU*d bj[�BBdo]oXY] bdoM = {m1 = 1, m2 = 2} i A�]^gN<Bd bhV n = 4 + 3 − 1 = 6

XY] U/�dimS3,M,τ = 4 + 3 = 7

ir: d 5*[_bLbjUfej[l[4g bj[�gNU Z;[�]/SAUaZB] a ehV%BJBAeN[4gL[�S*bhV�bjUWSB[Ydod^V �>I�<BehV,Ai �J! i ' V d^X_U*d ]^V \MUMV d^XE<BSA]

Ni r(t)��SB[�d�XYV*gLU

r = 1

tt t t t t t t t t0 1 2 3 4 5 7 8 9 10

||t 6

a bτ τ

ττ|| ||

0

1 2

3

qF]DI*<AejV,Bi �=! �!��[YbLbjUfej[cZ;[Y]DSBU;Z;]�[�gLbN[�gNU

N5 1(t) = 0

�]<B[4g bjUkgL]9ej]^XYVV5�V)(nV SAVfd \M[YS0bj[JU0gNgN[Ye?5�V SAZBUl]od�g/<>BJBnU*eLbjUAi " [�d�XYV*gLUk = 2

]oS]5*[�XY[�

N5 2(t) =t7 − t

t7 − t6N6 1(t) =

{ t7−tt7−t6

se t ∈ [t6, t7)

0 altrimenti

" [Yd�X�VfgNUk = 3

gN]�<;bj] do]D$�$�V d^V6. Ufej\)<Bd^V eN]^X_U*ejgN]C5�V X_U*\6BAd [YbjV��

N5 3(t) =t − t5t7 − t5

N5 2(t) +t8 − t

t8 − t6N6 2(t) =

=

(t−t5)(t7−t)(t7−t5)(t7−t6) + (t8−t)(t−t6)

(t8−t6)(t7−t6) se t6 ≤ t < t7(t8−t)2

(t8−t6)(t8−t7) se t7 ≤ t < t80 altrimenti

' VfdoXYUfdo]oVf\MU4B(Uf]��

N4 3(t) =

(t−t4)2

(t6−t4)2 se t4 ≤ t < t5 = t6(t7−t)2

(t7−t5)2 se t5 = t6 ≤ t < t70 altrimenti

HFVgL] bN<nV%$Y]oUfSA[�� BAeN[4gL[�S*bhV�bhV I*ejV��nXYVf\M[YS0bN[ SB[Ydod^V �WI*<BehV ,Bi � � i � SAVX_U*SAgN]oZ;[�ejV*$Y]oUfSB[]o\BnU*eLbhV S0bN[cZAV6. V ej[ � SBU bhV ej[cXdKB[cgL[

τl ∈ {τi, . . . , τi+k}[�Z ]oS bjV do[ SBUfbNU KAVM\MUfd bN[�m

BAd ]^X_] b?orV dod U*ejV

Ni k(t)KAVJ] S bjVfd [ SBU;Z;U ej[�I*Ufd^V ej] bdo

Ck−1−ri ' Uf\M[�XYV*gLU+BAV eNbN]^X_U*doVfeN[

−0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N

33(t)

N43

(t)N

53(t)

qF]DI*<AejV6,Bi � � � P gL[�\6BA] UkZ;]�(nVfgN[g_pm g/BAd ]oSB[

F�2 �� 4� �q �p�� i�?� ��

XYUfSAgN]oZB[Yej]oVf\ U��]<B[YdodoU ]oS XE<B]��

{ti, . . . , ti+k} = {k

︷︸︸︷τl, . . . , τl, tl+1}

A�]^gN<Bd bhV6Bn[�eLbhV S0bNUNi k(t) =

ti+k − t


" U bN]^V \MUkXdKB[gB([YeNi+1 k−1

]od�5f[YbLbjUfej[![4g bj[�gNUMZB[Y]9SBU;Z;]Dej]^g/<Bd bjV

{ti+1, . . . , ti+k} = {k−1

︷︸︸︷τl, . . . , τl, tl+1}

B([Ye X�<B]

Ni k(t) =ti+k − t

ti+k − ti+1

ti+k−1 − t

ti+k−1 − ti+2Ni+2 k−2(t) =

=(τl+1 − t)k−1

(τl+1 − τl)k−1Ni+k−1 1(t) =

=

{(τl+1−t)k−1

(τl+1−τl)k−1 se τl ≤ t < τl+1

0 altrimenti

#�,Ai @ %

A�]^gN<Bd bhVk[�5a]^Z;[YSnZ;[cXdKB[cd^VBBej] \kVM[�gNBBej[�gjgL]oUfSA[ ZB[YdFgN]^g bj[Y\kV��sBBejU*BBej]oUBk−1

0 ( t−τl

τl+1−τl)i

� B(U*gjgL]D(B]od [ Z;]o\MU*gLbNehV ej[JXdKB[ gN[{

τl = ti = . . . = ti+k−1

τl+1 = ti+k = . . . = ti+2k−1

RTd doUfehV

Ni+j k(t) =

{

Bk−1j

(t−τl

τl+1−τl

)

j = 0, . . . , k − 1 t ∈ [τl, τl+1)

0 altrimenti

3�[Ye X�<B]Πm ⊆ Sm,τ ⊆ Sm,M,τ ⊂ Pm,τ

� � � �� "��

P gNBBej] \M[�eN[�\ U<BSnV X�<Be?5uV_pm g/BBdo]oSB[ X_UfS

X(t) =

n∑

i=0

diNi k(t)

��[4Z;]oVf\MU do[rBBejU*BBej]o[_b?o Z;] �]<B[4g bj[ XE<Be?5f[*iR SnV doU*I*Vf\M[YS0bN[ V dodo[ X�<Be?5f[ g/BAd ]oSB[ X_U*SAgL]^Z;[�ejV bN[ �ASBUfehV��=5�Vfd [ d^VBBeNU�BBej] [Yb?o Z;]9]oS]5�V%m

ej]oVfSJ$�V4B([Ye�bjejV*g . U*eN\kV%$�] U*SB],V�0kSB]vi: SBU*d bjeN[��X_U*\ [I*]Do BBejU�5�V�bNU ] S BBej[�XY[�Z;[�SJ$�V 5�V do[MdoV BBejU*BBej]o[_b?o Z;]HdoU;XYVfd ] b?o!�!do[ _pm

gNBBdo] SB[ SAUfS S]<Ad do[gBn[�et ∈ (tr, tr+1)

gNUfSAU

Nr−k+1 k(t), . . . , Nr k(t)

�q �q �p� �� ?�� F%9

d

d

d

d

d

d

0

1

2

3

4

5

q/]CI�<BehV4,Ai � ,��Hm0bNejUfS>I XYUfS�5f[�� K]<Ad d

W <A] SAZB]9] S bhV do[J]oS0bN[�eN5�V dodoUMgjV edo g/<=0WXY] [�S*bj[ XYV d^X_U*doVfeN[��

X(t) =

r∑

i=r−k+1

diNi k(t)

� S�L <Bd bj[Yej] U*eN[ BBeNU�BBej] [Yb?o B([Ye,�]<B[4g bjUbN]DB(U Z;] g/BAd ]oSB[ �Wd^V EST]ReHJC r G)HPCD`J@ed �Uf hijMj �M]odgN[�I*\M[YS0bNU"ZB] X�<Be?5uV"X_Ufejej]ogNBnU*SAZ;[�S*bj[WVfZ <AS gLUfbLbjUf]oS*bj[Ye?5�V dod U � SB[Ydod^V X_U*S]5*[��m K]<Ad dOZ;]kBJ<AS*bj]�Z;]�X_UfS0bjeNU*d doUX_UfSngL[4XE<;bj]C5a]vi�R�Z [�gN[Y\BB]oU!�`B([Ye

k = 3�FgN[

t ∈ (t2, t3)V dod U*ejV

]od?gN[�I*\ [�S0bNU Z;]HXE<Be?5�V gLb?o SB[�d�bNej]oVfSJIfU*d U ˆd0,d1,d2��\M[YS0bjeN[6Bn[�e

t ∈ (t3, t4)g bdo ] S

ˆd1,d2,d3i

: S�If[�SB[YehV do[XB([Ye do[ _tm gNBBd ]oSB[ SBUfS�5�V do[ d^V BBeNU�BBej] [Yb?o Z;] [YSAZ�m B(Uf]oS*b]oS0bN[Ye?B(Ufd^V�mbj] U*S �9\kV 5f[4Z;]oVf\MU XYUf\M[ ej[�X�<JB([YehV ej[cbjVfd [kX�V ehV�bLbj[Yej]ogLbN]^XYV!� �ngjgL]^V \MU ]�SBU;Z;]HV%<ngL]od ]^V ej]\4<Bd bN]DBBd ]��nXY] U/�

{t0 = . . . = tk−2 = tk−1(= τ0)tn+2 = . . . = tn+k = tn+1(= τL)W <A[�gLbNU�gL]DIfSA].�nX�V"XdKB[V*(J(B]^V \MU B(U*gLbNU Ifdo]�[�gLbNej[Y\M]�V \MUfd bN[�BAd ]^X_] b?o

ki : SQIf[�SB[YehV do[

Ni k(t)KnVkg/<JB>BnU*eLbjU ]oS {ti, . . . , ti+k}

�]<B]oSAZ;]FgL[

{ti, . . . , ti+k} = {τr, . . . , τr︸︷︷︸

k

, τr+1}

ej]^g/<Bd bjV

Ni k(t) =

{

Bk−10

(t−τr

τr+1−τr

)

se t ∈ [τr, τr+1]

0 altrimentiW <A[�gLbNUkgL]DIfSA].�nX�VMXdKB[ SB[Yd�X�VfgNU gNB([�X_]��nXYU ]oSXE<B]

{t0, . . . , tk} = {τ0, . . . , τ0︸︷︷︸

k

, τ1}

ej]^g/<Bd bjV

N0 k(t) =

{

Bk−10

(t−τ0

τ1−τ0

)

se t ∈ [τ0, τ1]

0 altrimenti�HVfdC<BbjV SnZ;U Bn[�eLbhV S0bNU d^V6. <BSJ$Y]oUfSA[J]oSτ0V%(>(B]oVf\MU

N0 k(τ0) = Bk−10 (0) = 1

RTd doUfehV ej]ogN<Bd bjVX(a) = X(tk−1) = d0

� X_]oU �-V%(J(B]^V \MU ej[�XE<>Bn[�ejV bNU�dML [�SAZ�m BnU*] S0b]oS0bN[�eNB(Ufd^V�bj] U*S Z;[Yd BJ<BS0bNU�]oSB]D$Y]^V do[fi 3�[Ye d\L V d bNejU�[�gLbNej[Y\MU gN] BAeNU;X_[4Z;[ B(Uf]J] S�\MU;Z;UVfSAV doU*I*UAi

F%@ �� 4� �q �p�� i�?� ��

qAV*XYXY]oVf\ UMVfZB[�gjgLUMV d^XE<BSA[JXYUfSAgN]oZB[YehV%$Y]oUfSA](gN<Bdod [ Z;[�eN]D5�V�bN[ Z;[�d do[r_pm&gNBBdo] SB[*iHm;V%BJBA]oV%m\MUkXdKB[�ASB[Yd9X�VfgNU ]oS XE<B]9]�SBU;Z;]�gNUfSBUMgL[�\BBd ]^X_]�[

k ≥ 3�

dNi k(t)

dt= (k − 1)

[Ni k−1(t)

ti+k−1 − ti− Ni+1 k−1(t)

ti+k − ti+1

]

W <B]oSAZ;]��AgN[ Z;[Yej]C5a]^V \MU6<BSAVkXE<AeN5�V6_tm gNBBd ]oSB[rI*[YSB[�eN]^XYV U bNbN[YSA]oVf\ U

dX(t)

dt=

n∑

i=0

didNi k(t)

dt=

= (k − 1)

n∑

i=0

di

[Ni k−1(t)

ti+k−1 − ti− Ni+1 k−1(t)

ti+k − ti+1

]

3�[Ye X�<B]�gN5a] dD<JBJBAVfSAZ;U-�]<B[�gLb8L <Bd bN]o\kV gNUf\M\kV��

dX(t)

dt=

d0

tk−1 − t0N0 k−1(t) + (k − 1)

n∑

i=1

[di

ti+k−1 − ti− di−1

ti+k−1 − ti

]

Ni k−1(t) +

+dn

tn+k − tn+1Nn+1 k−1(t) =

= (k − 1)

n∑

i=1

di − di−1

ti+k−1 − ti︸︷︷︸

d[1]i

Ni k−1(t)

W <B]oSAZ;]�] S If[�SB[YehV do[�

drX(t)

drt= (k − 1)r

n∑

i=r

d[r]i Ni k−r(t) e d

[r]i =

d[r−1]i − d

[r−1]i−1

ti+k−r − ti

B([Yei = r, . . . , n

i

� � � �� "�� &� � � �

H�[JXYUfSAgN]oZB[YehV%$Y]oUfSA]J. V�bNbN[ gN<Bd do[ Z;[Yej]C5�V bN[!XY]9gL[�eN5*UfSBU)Bn[�e ZB[�gjX_ej]C5*[Yej[ ] d�BBejU*(Ad [�\MV Z;[�d d^Vej[�Vfd ]D$�$4V%$Y]oUfSA[�Z;]�XE<Be?5f[ XdKB]C<ngL[*i ' UfS�do[ X�<Be?5f[ ZB] _Za8$Y]o[Ye3��doV eN[4V do]C$8$�V*$Y]oUfSB[WZ;] <BSnVX�<BeN5�VXdKB]D<AgjV VV5f[85�VX_Uf\M[ BAeNU�(Bd [�\kV doVSAUfS�X_U*S0bN]oS]<A] bdo g/<Ad?ehVfXYXYUfehZ;UZ;]�XdKA]C<AgN<BehV#�5f[�ZB]�d^V��>I*<AejV ,Ai � 2 % i ' UfSngL]^Z;[Yej]^V \MUkV dod U*ejV6<BSAV�XE<Be?5�V _pm&gNBBdo] SB[cZB[ �ASB] bjV ZBV

n + 1BJ<AS*bj]0ZB]0XYUfS0bNejUfdod Ud0, . . . ,dn

[�ZAVm+k+1

SBU;Z;] #k]oSAZ;]^XYV ] d�IfehVfZBU Z;[YdodoV _tm gNBBd ]oSB[ % i

��U�Ifdo]oVf\ U . V e�VfgjgN<B\M[Yej[�V d�BJ<AS*bjU Z;];XdKB]C<ng/<BehV!doV!\kVfgjgL]o\kV ej[�IfU*doVfeN] b?o BnU0gNgN]C(A] do[� XY] U/��

X(r)(τ0) = X(r)(τL) r = 0, . . . , k − r

P gNBBdo]oXY] bhV SAZBU ]FXYV d^X_U*d ]9VV5aej[Y\MU

X(τ0) = X(tk−1) =

k−2∑

i=0

diNi k(tk−1)

[

X(τL) = X(tn+1) =

n∑

i=n+2−k

diNi k(tn+1)

W <B]oSAZ;]��Bdo[ X_UfSnZ;]C$�] U*SB[gBn[�e�U bNbN[YSA[Yej[ doV ej[�IfU*doV bN] b?oleN]^XdKB]o[�gLbjVMgNVfejVfSBSBU��

�q �q �p� �� ?�� F*F

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

200CurvaturaPoligono di controllo

Curva di Bezier

qF]DI*<AejV,Bi � 2�� ' <AeN5�VMZ;]`_Za�$�] [�e XdKB]D<AgNV [ ej[Yd^V�bN]D5�VMXE<Be?5�V�bN<AejV

• gNU�50ehV%B>BnU*eNej[ ]�BBej]o\ ]k − 2

[rI*d ] <Ad bj] \M]k − 2

BJ<AS*bj]FZ;]9XYUfS0bNejUfdodoU ]oS \ U;Z;UkZBVVV5f[�eN[Jd^V XY]oXYd ]^X_] b?oMZ;[�d�B(Ufdo]CI*UfSBUMZ;]�XYUfS0bNejUfdod U

dn−k+2−i = di i = 0, . . . , k − 2

• Z;[ �ASB]oeN[ ]9SBU;Z;]FV%<AgN] do]^V ej](]oS \MUaZBUMXdKA[ gN]^V SBU<JI�<AV do]Dd [cZB]ogLbjVfSJ$Y[��BX_]oU �

hi = hn−k+2+i i = 0, . . . , k − 2

3�[�e��]<AV S0bNU ej]CI�<AV ehZBVJdo[�] S0bN[�ejgN[�$�] U*SB]BX_U*SM] dWBnU*d ]DIfU*SBU Z;]AX_U*S0bNejUfdod U��fV%(J(A]oVf\ U XdKB[��ZAV�bjV<ASAVMXE<Be?5�V6_pm g/BAd ]oSB[

X(t) =

n∑

i=0

diNi k(t)

ej]^g/<Bd bjV!�

• gN[(k − 1)

BJ<BS0bj],ZB](XYUfS0bNejUfdod U X_UfSngL[4XE<;bj]C5a],gLU*SBU Vfd do] SA[�V�bj](Vfd doUfehV d^VlX�<BeN5�VX(t)bNU;X�XYV ] dFgL[8If\M[YS0bNUMZBVM[4gNgN]D] SnZ;]C5a]^Z=<AV bNU!�;]oS=. V bLbN]

dj+r = dj + αrv r = 0, . . . , k − 2

�]<B]oSAZ;]

X(tj+k−1) =

j+k−2∑

i=j

diNi k(tj+k−1) =

=

j+k−2∑

i=j

(dj + αi−jv)Ni k(tj+k−1) =

= dj +

j+k−2∑

i=j

αi−jNi k(tj+k−1

• gN[k−1

B><BS0bN]nZB](XYUfS0bNejUfdod U XYUfSAgN[�X�<;bN]D5a]ngNUfSBUcXYUf]oSAX_]^Z;[YS0bj]nV dodoUfehVX(tj+k−1) =

dj�AX_]oU/�J]od�BJ<AS*bjU50]o[YSA[ ] S0bN[�eNB(Ufd^V�bjU

F%& �� 4� �q �p�� i�?� ��

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

d0

d7 = d

0

d1

d2

d3 d

4

d5

d6

d7d

8

d9

d1

d2

d3

d4

d5

d6

d8

d9

d7 = d

0d

8 = d

1

d9

d6

d2

d3

d5

d4

d7 = d

0

d8 = d

1

d3

d4

d5

d6

d9 = d

2

qF]DI*<AejV6,Bi � 9��_pm g/BAd ]oSB[ XdKB]D<AgNV

dj

dj+1

dj+2

q/]CI�<BejV,Ai � @��k − 1

BJ<BS0bN]FX_UfSngL[4XE<;bj]C5a]9V dodo] SB[4V�bj]

�q �q �p� �� ?�� F�1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q/]DI*<BehV4,Ai � F/� ' <Be?5�V6_pm g/BAd ]oSB[ Z;]�I*ejV*Z;U4XYUfS

3BJ<AS*bj]�X_U*SAgN[�XE<BbN]D50]9Vfd do]oSB[�V bN]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q/]DI*<BehV,Bi � &!� ' <Be?5�V_tm gNBBd ]oSB[ Z;]�I*ejV*Z;U4XYUfS SBU;Z;UWV \MUfd bN[�BAd ]^X_] b?o

3

& 7 �� 4� �q �p�� i�?� ��

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q/]CI�<BejV,Ai � 1�� ' <BeN5�V_pm&gNBBdo] SA[ ZB]�IfehVfZ;U4X_U*S

4BJ<BS0bN]FX_UfSngL[4XE<;bj]C5a]9V dodo] SB[4V�bj]

• gL[kBJ<BS0bj]�Z;]FXYUfS0bNejUfdodoU XYUfSAgN[�X�<;bN]D5a]�gLU*SBUkV dod ]oSB[4V�bN]FV dod U*ejV6<BSgN[�If\M[�S*bjUWZ;[Ydod^V

XE<Be?5�VIf]^VfX_[cgN<Bd�gN[�I*\M[YS0bNUWZBVM[4gNgN]9]oSAZ;]D5a]oZ=<nV�bNUnit: S=. V bLbj]/gN[t ∈ (tj+k−1, tj+k)V%(J(B]^V \MU

X(t) =

j+k−1∑

i=j

diNi k(t) =

j+k−1∑

i=j

(dj + αivj)Ni k(t) =

= dj +

j+k−1∑

i=j

αi−jNi k(t)

v#�,AiGF%

� � � �� %&� % � � � ��

HtL VfdCI*Ufej] bj\ UJZ;]AZ;[ _OUaUfe � <BSAVgI*[YSB[�ejVfd ]D$�$4V%$�] U*SB[�Z;[�d dML V dDIfUfej] bN\MU!Z;]nZ;[ ' VfgLbN[�d cLV*< Vfd do[_pm g/BBdo]oSB[�/] SJ. V�bLbj]�dML ]oZ;[4V Z;]2�]<B[�gLbNU �Wej]^Z=<BejeN[kdML U*ejZB] SB[ Z;[Ydod [ _tm gNBBd ]oSB[ B([YelbNejU�5�V ej[ ]BJ<AS*bj]�Z;[YdodoV XE<Be?5�VBi � bj] do]C$8$�VfSAZ;U�d^V . U*eN\kV�eN]^X_U*ejgN]C5�VB(U*gjgL]^V \MU gNXYeN]D5f[�eN[ dML [3�]<AV%$�] U*SB[Z;]`<BSAV6If[�SB[Yej]oX�V X�<Be?5uV_pm g/BBdo]oSB[ X_Uf\M[��

X(t) =

n∑

i=0

diNi k(t) =

=

n∑

i=0

di

[t − ti


ti+k − t


]

=

�q �q �p� �� ?�� & !

=

n∑

i=1

[t − ti

ti+k−1 − tidi +

ti+k−1 − t

ti+k−1 − tidi−1

]

︸︷︷︸

d[1]i

(t)

Ni k−1(t)

A ]oZ=<nX_[YSnZ;UkS�<BU�5�V \M[YS0bN[ dML U*ejZB] SB[ �]<B[YdodoUWXdKB[ U bLbj[YSJI*U��

X(t) =

n∑

i=1

d[1]i (t)Ni k−1(t) =

n∑

i=2

d[2]i (t)Ni k−2(t) = . . .

. . . =n∑

i=k−1

d[k−1]i (t)Ni 1(t)

#�,Ai & %

ZBU�5f[

d[j]i (t) =

α[j]i

(t)︷︸︸︷

t − titi+k−j − ti

d[j−1]i (t) +

ti+k−j − t

ti+k−j − ti︸︷︷︸

1−α[j]i

(t)

d[j−1]i−1 (t) i = j, . . . , n

RTd doUfehV��gN[t ∈ [tr, tr+1)

V*(J(B]^V \MUX(t) = d

[k−1]r (t)

i�: SsBAVfeLbj]oXYUfd^V ej[��B([Ye/U bNbN]o\M]C$8$�V ej[]od BBejU;X_[�ZB] \M[YS0bjU!� �ngjgNV bNU d\L ]oS*bj[Ye?5�V dod U

[tr, tr+1)gN]�X_UfSngL]^Z;[YehV SAU gNUfd^V \M[YS0bN[ ]

kSAUaZB]

V bLbj]C5a]��AX_]oU/��

X(t) =r∑

i=r−k+1

diNi k(t)

:gBAV*gNgN]HZ;[�d dML VfdCI*Ufej] bj\MU UfbLbj[YS�<;bN]OZBV �]<B[�gLb8L <Bd bN]o\kV . Ufej\)<Bd^V BnU0gNgNUfSBU [4gNgN[Yej[MejV*BJBBej[EmgN[YS0bhV�bN]�] S . Ufej\kVcbhV%(([YdodoVfeN[-# V SAVfd U�I*V \M[�S*bj[JV��]<AVfS*bjU). V bLbjUWX_UfS Z;[ ' V*g bj[Yd cLV%<�% �

dr−k+1 = d[0]r−k+1

dr−k+2 = d[0]r−k+2 d

[1]r−k+2iii

d[2]r−k+3iii i i i

dr−1 = d[0]r−1 d

[1]r−1 d

[2]r−1 . . . d

[k−2]r−1

dr = d[0]r d

[1]r d

[2]r . . . d

[k−2]r d

[k−1]r = X(t)

� S"[4gL[�\6BA] U ejV%0I�<BehV S0bN[ ]�BAVfgjgN]/Z;[�d dML V dDIfUfej] bN\MU-� \MU0g bjejV bNUkSB[�d d^V-�>I�<BehV ,Bi , 7 io�HVSAU bjV bNUTXYUf\M[�d^V BnU0gL]D$Y]oUfSA[HZ;[Y]

d[j]i

XYV \4(B]^V�ZBV gN[�If\M[�S*bjU V gN[�If\M[�S*bjU # V�\M[�SBU Z;]0SBU;Z;][3�]<B]ogNBAV*$Y]^V�bN] %�XYUfS0bNehV ej]oVf\M[YS0bN[ V-�]<AV S0bNUWV*XYX�VfZ;[85uVkXYUfSZ;[ ' V*g bj[Yd cLV%<9i -/V do[ bN[�XYSB]^XYV5a]o[YSB[cgNB([�gjgLUkXdKB]^V \kV�bhV G)HPRSC9@�R-G$hiT]TU<MC r i�: d�XYU*gLbNUMX_U*\6B><;bjV*$Y]oUfSAVfd [ Z;[YdodML VfdCI*Ufej] bj\ U��

k−1∑

j=1

r∑

i=r−k+j+1

(4 + 3d) = (4 + 3d)

k−1∑

j=1

(k − j) = (4 + 3d)k(k − 1)

2

W <A[�gLbNUkgL]DIfSA].�nX�VMXdKB[ ] d�XYU*gLbNUMZ;]DBn[�SAZ;[lZBVfd dML UfehZ;] SA[�aSBUfSZBVfdDS�<B\M[YejUMZB[Y]DSBU;Z;]vi

�� %&��

" [Yd Z;]^gL[8IfSBU-] S0bj[YehV�bLbj]C5*U �rg/B([�gjgLU-SB[�XY[�gjgNVfeN]oU�V%I�If]D<BSJI*[Yej[ <BS <Ad bj[Yej] U*eN[ B><BS0bNU Z;]XYUfS0bNejUfdodoUMV*Z <BSAVkXE<AeN5�VMgL[�SJ$�V \MU;Z;]��nXYVfeN[ d^VMg/<AV6. U*eN\kVAiHR (J(B]^V \MU I*]Do5a]^g bjUMXYUf\M[

& � �� 4� �q �p�� i�?� ��

d r−3(0)

d r−2

(1)

d(2)r−1

dr−2

(0)

d(1)

r−1

d r−1(0)

d r

(1)

d(2)r

X(t)

q/]DI*<BehV,Bi , 7 �/RTdCI*Ufej] bj\ UkZ;]FZ;[r_�UaU*e �k = 4

B(U bj[YeZ. V e X_] � XYUfS d [ XE<Be?5f[JZ;] _Za�$Y]o[Ye3��5f[4Z;]oVf\MU X_U*\M[ B(U bj[Yep. V ej[!doU gLbN[4gNgNU XYUfS d [g_pmgNBBdo] SB[*i��[�Z;ej[Y\MUWXdKB[��A]oS��]<B[�gLbNUWX�VfgNU!�nSBUfS �cSA[�X_[4gNgjV ej]oUMV*<B\M[YS0bjVfeN[ ]od`I*ejV*Z;UWZ;[�d d^VX�<BeN5�VAi HtL ]oSAgN[Yej] \M[�S*bjU ZB] <BS S�<BU�5fU BJ<BS0bNU�Z;] XYUfS0bNejUfdodoU

d∗i

�Z;[_bj[Yej\M] SAV bjVrZBVfd d^VB(U*gN]C$�] U*SB[ Z;[YdDSAUaZBU

t∗ ∈ [tk−1, tn+1]VfgjgLU;X_]^V�bjUAi�: SAgN[Yej]oVf\ U��]<B[�gLbNUMS�<BU�5*U SBU;Z;U6. ehV

[tr.tr+1]�;] d�5*[_bNbNUfej[ Z;[Y]DSAUaZB]9[�gLbN[4gLUkZ;]D5f[�S*bhV��

T ∗ = {t0, . . . , tr, t∗, tr+1, . . . , tn+k} = {t∗0, . . . , t∗n+k+1}Z;U�5*[

t∗i =

ti i = 0, . . . , rt∗ i = r + 1ti+1 i = r + 1, . . . , n + k + 1

m;XYeN]D5a]oVf\MU Vfd doUfehVldML [ �]<AV%$�] U*SB[ Z;[YdodoV _pm g/BAd ]oSB[ X_U*S d^V S]<AU�5uV6(AV*gL[��

X(t) =

n+1∑

i=0

d∗i N

∗i k(t)

HFV4(nVfgN[ ZB[Ydod [r_pm&gNBBdo] SA[ �J\MU;Z;]��nXYV bjVMgN[�X_U*SAZ;U

Ni k(t) =

{N∗

i k i = 0, . . . , r − kN∗

i+1 k i = r + 1, . . . , n

W <B]oSAZ;]�] S . Ufej\kV ej]^X_UfehgN]C5�V��

Ni k(t) = αi iNi k(t)∗ + αi i+1Ni+1 k(t)∗ i = r − k + 1, . . . , r

XYUfS

αi i =

{t∗−ti

ti+k−1−tii = r − k + 2, . . . , r

1 i = r − k + 1

αi i+1 =

{ti+k−t∗

ti+k−ti+1i = r − k + 1, . . . , r − 1

1 i = r

��[�Z;]^V \MUB([YeNbjV S0bjUMXYUf\M[g5�V ej]oVfSBU ]d∗

i

�

�q �q �p� �� ?�� &*,

d

d

d dd

d d

r−3 d= r−3*

r−2

r−1d

r =

r−2

r−1

r

r+1

*

*

*

*

q/]CI�<BejV6,Bi , ! ��R dDIfUfej] bN\MU ZB]`v SBU bNmv]oSAgN[YeNbN]oUfS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d0

d1

d2

d3

d4 d

5

d6

d7

d8

d9

d10 d

11

d12

d13

d0*

d1*

d2*

d3*

d4*

d5*

d6* = d*

d7*

d8*

d9*

d10*

d11* d

12*

d13*

d14*

qF]DI*<AejV,Bi , � ��v SAU b/m&]oSAgL[�eLbj] U*S Z;]`<BS SBU;Z;U

X(t) =

n∑

i=0

diNi k(t) =

r−k∑

i=0

diN∗i k(t) +

+

r∑

r−k+1

di(αi iN∗i k(t) + αi i+1N

∗i+1 k(t)) +

+n∑

i=r+1

diN∗i+1 k(t) =

r−k+1∑

i=0

diN∗i k(t) +

+

r∑

i=r−k+2

(αi idi + αi i+1di−1)N∗i k(t) +

n+1∑

i=r+1

di−1N∗i k(t) =

= X∗(t)

` V-�]<B[�gLbNUMUfbLbN[�SB]^V \MU

d∗i =

di i = 0, . . . , r − k + 1αi idi + αi i+1di−1 i = r − k + 1, . . . , rdi−1 i = r + 1, . . . , n + 1

' UfSAgN]^Z;[Yej]oVf\MU!VfZB[�gjgLU+<BSAV BBejU*BBej] [Yb?o!XdKB[�V%(J(B]^V \MU!bNehV d^VfgjX_]^V�bjU eN]^gNBn[YbLbNU Vfd d^V bjejV bLbhV�m$�] U*SB[ Z;[Ydod [OXE<Be?5f[OZ;]=_Za�$Y]o[Ye3�udoV `JNJRS<BNJT]<BHJC0>J<MI^<MCD<FEef <MC r i ' U*SAgL]^Z;[�eN]^V \MU

f ∈ Sk−1,M,τZB[YdDbj]CB(U

f(t) =n∑

i=0

ciNi k(t) t ∈ [tk−1, tn+1]

��U�Ifdo]oVf\MU BBeNU�5�VfeN[�XdKB[Vf ≤ V (c0, . . . , cn)

i�ma]^V SBUt1, . . . , tp

]*BJ<AS*bj]*XdKB[�]oS[tk−1, tn+1]KnV SBSBUMgN[�IfSAUlZ;]D5f[YehgNUAi � gjV SAZ;U ] d *aSBUfb/m&] SAgN[YeNbN]oUfS gL](eN]DB([_bNU*SBU bjV do]�BJ<BS0bN]

k− 15*Ufd bN[��

&*2 �� 4� �q �p�� i�?� ��

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q/]DI*<BehV,Bi ,*,!��v SBUfb/m&] SngL[�eLbj] U*S ej]CB([_b?<;bjV

XY] U/� X_ej[YUM]od 5f[_bNbNU*eN[ Z;[�]9SBU;Z;]D[�gLbN[4gLU��

T ∗ = T ∪ {t1, . . . , t1︸︷︷︸

k−1

,

k−1︷︸︸︷

t2, . . . , t2, . . . , tp, . . . , tp︸︷︷︸

k−1

}

3�U*gjgN]oVf\ U6B([YeNbjV S0bjU [4g/BAeN]o\M[Yej[!d^VMXE<Be?5�V XYUf\M[

f(t) =

n+p(k−1)∑

i=0

c∗i N∗i k(t)

RTd doUfehV ∃ij ∈ {0, . . . , n + p(k − 1)} bhV do[ XdKB[ f(tj) = cij

�j = 1, . . . , p

i W <B]oSAZ;]

Vf = V (c∗i1 , . . . , c∗ip

) ≤ V (c∗i0 , . . . , c∗in+p(k−1)

) ≤ V (c0, . . . , cn)

�� u��)�� U #��+ �"g��J1#6#1#Y,Y)8�6#6#1

4��=<+ > @ � D �)@ � l��B��ED �F@ ��

: d>BBeNU�(Bdo[Y\kV!ZB[Ydod\L V%BJBAeNU0gNgN] \kV*$Y]oUfSB[OZ;]=. <BSJ$�] U*SB]Bej]C5*[�gLbN[Z<BS�L ] \B(UfeNbjVfSJ$�V . UfSnZBV \M[YS0bhV�mdo[�SB[�d d^VJ\MV bN[�\MV bN]^XYV V%B>BBd ]^XYV bjV!� Z;U�5f[ g/B([�gjgNU gN]=BBej[�gN[YS0bjVfSBU . <BSJ$Y]oUfSA]AV SAVfd ] bN]^XYVf\M[YS0bN[XYUf\BBdo]oX�V�bj[�U Z;]BXE<A]BX_UfSAU*gjX_]^V \MUTgNUfd^V \M[YS0bN[p<BS ] SAgN]o[Y\M[OZB]ogjX_ej[_bjUJZ;]=5�Vfd U*eN]vi P gN]ogLbNU*SBUZB[Y]D\M[_bjUaZB]�XdKB[+Bn[�eN\M[_bNbNU*SBUkZ;]DeN]^X_U0g bjeN<A] ej[TdML VfSAZBVf\ [�S0bNUMZ;] <BSAV). <BSJ$�] U*SB[ V4BnV eNbN]oeN[ZAV ] ZBV bN]�]oS B(U*gjgL[4gNgNU [ gNUfSBUrV%B>BBd ]^XYV bN] BB]�� U \M[�SBU XYUfS g/<nXYX_[4gNgNU"]oS�Z;]DBn[�SAZ;[YS>$�VZB[YdodoV SAV bN<BehV ZB[Y] ZBV bN] ZBV g bj] \kVfeN[*i : d BB]��-]o\BnU*eLbhV S0bN[ \M[_bjUaZBU <BbN]od ]D$�$4V�bNU�� d\L ]oS=mbj[Ye?BnU*doV*$Y]oUfSB[ BnU*d ]oSBU*\ ]^V do[��XdKB[ gN]�ej[�Vfd ]D$�$4V X_U*S�]�\M[YbNU;Z;]�Z;] H�V*IfehV SJI*[kU " [ � bNU*S9i-�<;bNbjVV5a]oV!�HSB[�d dML Vf\)(B] bNU ' R� ` bjV do] \M[_bNU;Z;] SBU*S B(U*gjgLU*SBU"[4gNgN[Yej[�V%B>BBd ]^XYV bN] # B([Yejd U*m\M[�SBU�SBU*S"Z;]oeN[YbLbhV \M[YS0bN[ % B(Uf]^XdKJaldo[s. <ASJ$Y]oUfSB]�XdKB[ Z;U*(J(A]oVf\ U�] S0bN[�eNB(Ufd^V ej[cSBU*S"gNUfSBUgN[Y\BBdo]^X_]�eN[�doV*$Y]oUfSB]

y = f(x)Z;]�XE<B]�XYUfSBU0gNXY]oVf\ UWVfdoX�<BSB]�5uVfd U*eN]

(xi, yi)�,\kV�gNUfSBU�] S

I*[YSB[�ejVfd [ . <ASJ$Y]oUfSB]�BAVfejVf\ [YbNej]oXdKA[ Z;](X�<B],X_UfSAU*gjX_]^V \MU Z;[Y]�BJ<BS0bN]pi ∈ E

di�: BBejU*(Bdo[Y\kV

B([YeNbjVfS0bNU�� ]<B[Ydod UkZ;]9bNejU�5uVfeN[J]tiB([Ye X�<B]

X(ti) = pii

�� Z%��!&1 � � $ -"4#$�-%�& �0 �1� -, &�31�#$24

R (J(B]^V \MU V*XYXY[YSBSAV bNUW]oS BBej[�XY[�Z;[�SJ$�VM]od�BBeNU�(Bdo[Y\kVWZ;[�d dML ]oS0bN[�eNB(Ufd^V%$�] U*SB[sBAVfejVf\ [YbNej]oX�V��gLbjV*(B]od ]oeN[g<BSnVkX_Ufejej]ogNBnU*SAZ;[�SJ$�V

ti ⇐⇒ pi i = 0, . . . , n

: SAUfd bNej[s5fU*eNej[Y\M\MUWXdKB[c]ti. U0gNgN[YejUkUfehZ;]oSAV�bj]

t0 < t1 < . . . < tni H�VWgNUfdD<J$�] U*SB[cXdKB[

XYUfSAgN]^Z;[Yej[Yej[Y\MU Z L U*ejV!]oS6B(Uf]BgjV edo ��<A[YdodoV Z;]Bej]^gLU*dC5*[Yej[�gN[�BAVfejV bjV \M[�S*bj[�do[�X_U*\6B(UfSA[YS0bN]��XY] U/�cZBV bjV

X(t) =

(x(t)

y(t)

)

]od]BBeNU�(Bdo[Y\kV gN] �LeN]^Z=<Beje?o ��V*Z ] S0bj[Ye?BnU*doVfeN[�do[?X_U*\BnU*SB[YS0bN] # <ASB]oZB] \M[YSngL]oUfSAVfd ] %x(t), y(t)

iA [�gLbjV"XYUf\4<BS!�]<B[�]od BBejU*(Bdo[Y\kV Z;]�gjX_[8Ifdo] [�eN[W]odZBAV ehV \M[_bjeNU

tU*BJB(UfeNbN<BSnV \M[YS0bN[�� Bn[�e

X�<B] ]oS*bjeNU;Z=<AeNej[Y\MU�do[ T @)G$C=<BG[fD@ >J< b NPReNJIL@�TURS< O)O\N�O�<BHJCZ@ � [4Z <BS BBejU*(Bdo[Y\kV <Ad bj[Yej] U*eN[gjV edo �]<B[�d doU Z;] U0gNgN[Ye?5�V ej[WX_Uf\M[ doVX(nU*S*bdo"Z;[�d dML V%BJBBejU*gjgN] \kV%$�] U*SB[�Z;[�d do[ XYUf\B(UfSB[�S*bj]

&�9

&�@ �� 4� �D ��^��?��c��!�p� �� p� �p� ��

gN]�eN]DBn[�e �]<BUfbN[Mg/<Ad d^V . <ASJ$Y]oUfSB[BAVfejVf\M[_bNej]^XYV # ] S bN[�eN\M]oSB]�Z;]HVfX�XE<BehV�bj[�$8$�V��,ej[�I*Ufd^V ej] bdo��gN\MU0UfbNKBSA[�gjg �

. . .% i

� �)( � &�2%�#$2 �'& �'# �' �1� -, &�31�#$""4 -"4#$�-%�&

P do[YSnXdKB]oVf\MUkd [ bj[�X_SA]oXdKB[ Z;]�BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[+BBej] SAXY]CBnV do]��AZB[�gjX_ej]C5*[YSAZ;U*SB[ d [ XYV ehV�bNmbj[Yej]ogLbN]^XdKB[�� >�� doV bj[�XYSB]oX�VMZ;]�BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[!] S0bN<A] bj]C5�V \M[�S*bj[rBB]�� gN[Y\BBdo]^X_[

ti =i

ni = 0, . . . , n

W <B[�gLbNUJbN]DB(UcZ;]JBAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[pBn[�e �gBBej[�gN[YS0bjV+BBejU*(Ad [�\ ]BgN[�do[ ZB]ogLbjVfSJ$Y[ . ehV]JBJ<BS0bN]ngLU*SBU!\MU*d bjU+5�VfeN]^V%(B]odo]]B(Uf]^XdK�� SBU*Slbj] [�SB[ X_U*S*bjUJZB]AX_U*\ [�gNUfSBU Z;]^g bjeN]D(J<B] bN]]Pii

T��>��?k��! >�� doV�bN[4X_SB]^XYV�\kV*I*If]oUfej\M[YS0bN[ <BbN]od ]D$�$4V�bjV!�� ZB[ �ASB] bjV ] S (AVfgN[ Vfd do[Z;]ogLbjVfSJ$Y[g. ehV ] BJ<BS0bN]

ti+1 = ti∆pi

∆pX_UfS

∆pi = |Pi+1 − Pi|�∆p =

∑∆pi

i W <B[4g bOL <Bd bN]o\kV-�]<AV S0bN] b?oMehV%B>BBeN[4gL[�S0bjVdoVMdD<BSJI�KB[�$8$�V Z;[YdodoVkgNB([�$�$4V�bhVBi

T4�� jp�� <BSAV65�V ej]^V S0bN[ Z;[Ydod^Vlbj[�XYSB]oX�VMXdKBUfehZ�m&d [�SJI b?K Z;U�5*[!]∆pi

gNUfSAUMZB[ �ASB] bN]ZBV

∆pi = |Pi+1 −Pi|α α > 0: S BnV eNbN]^X_Ufd^V ej[ gL]9XdKB]^V \kVMX_[�S0bNej]CB([_bhV gN[

α = 1/2i

� � �Z%-��&1 � � $ �"#$��%'& � � �#$%��, #$ $&

` [�gjX_ej]C5a]^V \MUlXYUf\M[ �JXYV ehV�bNbN[�eN]D$�$4V�bjV ]oS I*[YSB[�ejVfd [ dML ] S0bN[�eNB(Ufd^V%$�] U*SB[ B(Ufdo] SAUf\M]oVfd [*i ` [Em�ASA] ej[Y\MU<BSAVMXE<AeN5�V6BAV ehV \M[YbNej]oX�V

X(t) =

(x(t)

y(t)

)

XYUfSx(t), y(t) ∈ Πn

XdKB[T] S0bN[�eNB(Ufd^Vn+1

BJ<AS*bj]��p0, . . . ,pn

iHm;X_[8Ifdo]oVf\MUr<BSAV bj[�XYSB]oX�VZ;]]BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[�]oSc\MU;Z;U!ZBV!Z;[_bj[Yej\M] SAVfeN[

t0 < . . . < tni�: S��]<B[4g bj[�XYUfSAZ;]D$Y]oUfSA]

gjV%BJBA]oVf\ U XdKB[�]od BnU*d ]oSBUf\M]oU"]oS0bN[Ye?B(Ufd^V S0bN[ � <BSB]^X_U��[�Z ��U�5�5a] U"XdKA[ XYVf\)(B]^V SnZ;U"d^VBAVfejVf\M[_bNej]D$�$�V*$Y]oUfSB[JXYVf\)(B]^V VfSAXdKB[ d^V X�<Be?5uVAi

� � � �� %&�� &��

m;XYeN]D5a]oVf\MU d^VMXE<Be?5�VX(t)

SB[�d d^V6(AVfgN[ Z;]`HFV%IfehV S>If[

X(t) =

n∑

i=0

piLi n(t) Li n(t) =

n∏

j=0j 6=i

t − tjti − tj

�D �D ��^��?��c��!�p� �� ?�� &]F

H�[ BBejU*BBej]o[_b?orXdKB[ XY] ]oS*bj[Yej[�gjgNVfSBU�Z;] �]<B[4g bhVrejV*BJBBej[�gN[YS0bjV*$Y]oUfSB[ gLU*SBUr] SASAV SJ$�] b?<;bLbjU�]<B[�d do[Z;]�Z;]^gL[8IfSBU��\ U0g bjeN]^V \MU Bn[�eLbhV S0bNU doV BBejU*BBej]o[_b?o"Z;] BAV eNbN]D$Y]oUfSA[ Z;[�d dML <ASB] bdorZ;]�]<B[4g bhVMXE<Be?5�V��

n∑

i=0

Li n(t) = 1

]oS=. V bLbN]��(gNgjV�bhVf(t) ≡ 1

eN]^g/<Ad bhV

Pn(t) =

n∑

i=0

f(ti)Li n(t) =

n∑

i=0

Li n(t) = 1

' U*S �]<B[�gLbjVsBBeNU�BBej] [Yb?ogKBUcd^VsB(U*gjgN]C(B]odo] bdocZ;],Z;[ �ASA] ej[ <BSAV X_U*\)(B]oSAV%$�] U*SB[ (AVfeN]^X_[�S*bjeN]^XYV[4Z KBU d\L ] S�5�V ej]^V SJ$4V Bn[�e�bjejV*g . U*eN\kV*$Y]oUfSB] V%0MSA] i 3 <BeNbNejU*BJB(U SBU*S 5�V do[ B([Ye��<A[�gLbjVehV%B>BBeN[4gL[�S0bjV%$�] U*SB[Hd^V BAeNU�BBeN]o[_bdoTZ;];XYUfS�5f[��]m K�<Bdod�B(Uf]^XdKJaZIfdo]

Li nB(U*gjgLU*SBUT[4gNgN[Yej[�VfSAXdKB[

SA[�I*V bN]D5a] i : SAUfd bNej[cSBUfS 5�Vfd [ doV BAeNU�BBeN]o[_bdokZ;]�5uVfeN]^V�bj] U*SZ;] \M]oSB]^g/KB]oSJI��(\kV 5�Vfd [��]<B[�d d^VZB]JBBej[�X_]^gN] U*SB[�do]oSB[�VfeN[*i�-/V do[�ehV%BJBAeN[4gL[�S*bhV%$�] U*SB[?SBU*S,� \kV ]�<;bj] do]D$�$�V bjVJSB[�d ' R� ` Bn[�e�]\MUfbN]D50];Z;[4gNXYeN] bLbj]�� gN]^V B([Ye�<Bd bj[Yej] U*eN]]BBejU*(Bdo[Y\M]aZ;]aSAV�b?<BehV S�<B\M[Yej]oX�V��gN[YSJ$4VTZ;]o\M[YS0bN]^XYVfeN[XdKA[ d\L V%I�If]D<BS0bjV Z;]�<BS S�<BU�5fUMZBV�bjUkX_Uf\B(UfeNbjV ]od�Z;U�5f[�e�ej]^XYV d^X_U*doVfeN[!d\L ]oS*bj[YehVMXE<Be?5�VBi

�&�� ! � � � �� &� % ��

HFV . Ufej\kV�Z;] " [�� bjUfS Bn[�e�ehV%BJBAeN[4gL[�S*bhV ej[ B(Ufdo]oSBUf\M] � BA]�� [�0WX_]o[YS0bN[ Z;] �]<B[YdodoV Z;]HFV%I*ejVfSJIf[ ZBV d`BJ<BS0bNUkZ;]�50]^gLbjVMZ;[Yd�S]<A\ [�eNUkZ;]FXYV d^X_U*d ]9ZAV6. V ej[�BZ;[4gNXYeN]D5a]oVf\ U*doV4(Bej[�5*[Em\M[�S*bj[��d^V(AVfgN[ Z;] " [ � bNU*S0�

w0(t) = 1

w1(t) = t − t0

w2(t) = (t − t0)(t − t1)

. . .

wi =

i−1∏

j=0

(t − tj)

P d^VMXE<Be?5�VMgL]9gjX_ej]C5*[JXYUf\M[

X(t) =

n∑

i=0

Aiwi(t)

ZBU�5f[ Ifdo]Ai

ehV%BJBBej[�gN[YS0bhV SBU d^VZ;] !,[Yej[YS>$Y[�Z;]D5a]ogN[��Z;[ �ASB] bN[ ZAVAi = F[to, . . . , ti]

�XYUfS

F[ti, . . . , ti+j ] =F[ti+1, . . . , ti+j ] − F[ti, . . . , ti+j−1]

ti+j − ti[4ZF[ti] = Pi

i H L ]o\6BAd [�\ [�S0bjV%$�] U*SB[ ZB]2�]<B[4g bjU \M[YbNU;Z;U XYUfSAgN]^g bj[�SB[�d X�V d^X_U*doVfeN[Wdo[ZB] !,[Yej[YSJ$�[�Z;]C5a]^gL[ZB([Ye�d [ gN] SJI*Ufdo[OXYUf\B(UfSB[�S*bj]a[pB(Uf]�5�V dD<;bjVfeN[Od^V . <BS>$Y]oUfSB[ # V*Zc[�gN[Y\BB]oUV bLbjejVV5*[YehgLU d\L V dDIfU*eN] bN\MUMZ;]�� U*eNSA[Ye4i

�� Z%��!&1 � � $ -"4#$�-%�& �'#�� &1_, #)�!&

RTSAV do]D$�$Y]^V \MU �]<B[�gLbNU!bN]DBnU Z;];]oS0bN[�eNB(Ufd^V%$�] U*SB[?]oSB]D$Y]^V do\ [�S0bN[OSB[YdBX�VfgNU . <BSJ$�] U*SAV do[fi�H L ]oS=mbj[Ye?BnU*doV*$Y]oUfSB[ ZB] � [�eN\M] bN[ [4g bj[YSAZB[ d\L ]oS*bj[Ye?BnU*doV*$Y]oUfSB[ B(Ufdo] SAUf\M]oVfd [ XYUfSAgN]^Z;[YehV SAZBU XYUf\M[

&�& �� 4� �D ��^��?��c��!�p� �� p� �p� ��

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4Interpolazione di Newton

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4Interpolazione Newton − Parametrizzazione uniforme

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4Interpolazione Newton − Parametrizzazione chord−length

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Interpolazione Newton − Parametrizzazione centripeta

Punti di interpolazioneParametrizzazione uniformeParametrizzazione chord−lengthParametrizzazione centripeta

q/]DI*<BehVs2ni ! ��: S0bN[�eNB(Ufd^V%$�] U*SB[TZ;] " [�� bNU*S XYUfS�Z;]D5f[�ejgN[�bj[�X_SA]oXdKB[ Z;]�BAVfejVf\ [YbNej]C$8$�V*$Y]oUfSB[

ZBV bN]FSBUfS gNUfdoUk]`BJ<AS*bj]FZ;[�d d^V. <BS>$Y]oUfSB[��n\kVkV SnXdKB[c]FZAV�bN]FeN[�doV bN]D5a]�V dodoVkZB[Yej]C5�V�bhVWZ;[�d d^V. <BS>$Y]oUfSB[��9] S \MU;Z;U ZAV ej]^X_U*gLbNe?<B]oeNSA[l]oS \MU;Z;U \ ]DIfdo]oUfej[ ]odHX_U*\6B(UfeNbjVf\M[YS0bNUni ' UfSAgN]CmZ;[�eN]^V \MU ] SB]D$Y]^V do\M[YS0bN[MZ;]HVV5*[Yej[lX_U*\ [MZBV bN]�Z;[Ydodo[ bj[YejSB[

(ti, fif′i)�i = 0, . . . , n

[MZ;]5*Ufdo[YeObNejU�5�V ej[!]od�B(Ufdo] SBU*\M] U

PN (t) ∈ ΠNbjVfd [ XdKB[

{PN (ti) = fi

P ′N (ti) = f ′

ii = 0, . . . , n

XYUfSN = 2n + 1

[ti 6= tj

B([Yei 6= j

i �T]^g bjU�XdKB[ V%(J(A]oVf\ U�X�V \)(A]oV bNU�g/BnV%$Y]oU!�Z;[ �ASB]^V \MUMdoV6(AV*gL[ Z;] � [Yej\M] bj[�

{

H(N)2i = L2

i n(t) [1 − 2L′i n(ti)(t − ti)]

H(N)2i+1 = L2

i n(t)(t − ti)i = 0, . . . , n

RTSAV do]C$8$Y]^V \MUcdo[ BBejU*BAeN]o[_bdo Z;[�I*d ]N

(N)j

i ' UfSAgN]oZB[Yej]oVf\ U ]oSB]D$Y]^V do\M[YS0bN[ ��<A[Ydod ],Z;](] SAZB]oXY[BAVfeN]��eN]^gN<Bd bhV

H(N)2i (tj) =

{0 j 6= i1 j = i' U*SAgL]^Z;[�eN]^V \MU6BnU*]DdoVMZ;[�eN]D5�V�bjV

H′(N)2i (t) = 2Li n(t)L′

i n(t) [1 − 2L′i n(ti)(t − ti)] − 2L2

i n(t)L′i n(ti)

�]<B]oSAZ;]H

′(N)2i (tj) = 0 j = 0, . . . , n

' U*SAgL]^Z;[�eN]^V \MU UfehV,�]<B[�d do[ Z;]9]oSAZ;]^X_[cZ;]^g/BnV ej] i A�]^g/<Ad bhV��

H(N)2i+1(tj) = 0 j = 0, . . . , n

� [�S0bNej[Jd^VMZ;[Yej]D5uV bjV ej]ogN<Bd bjV

H′(N)2i+1(t) = 2Li n(t)L′

i n(t)(t − ti) + L2in(t)

�D �D ��^��?��c��!�p� �� ?�� &*1

5�VfdC<;bhV SAZBUfd^V4B(Uf]�SB[Y]tj�

H′(N)2i+1(t) =

{0 j 6= i1 j = i

��Wjt��i8�\�*�� 38\��j <H

(N)j (t)

E�HJCZH j <MCZ@)NJRSIL@$C=Tm@-<MC9>P�@$C=TU< V�� "�� "�� "!$# ' UfSAgN]oZB[Yej]oVf\ U d^V XYUf\4(B] SnV%$Y]oUfSA[Jdo]oSB[�VfeN[

n∑

i=0

ciH(N)2i (t) +

n∑

i=0

diH(N)2i+1(t) ≡ 0

` U*(>(B]oVf\MU BBejU�5�V ej["XdKB[��]<B[�gLbNU VfXYX�VfZ;[ gL[ [rgLU*d U gL[ci, di = 0

i �HVfdC<BbN]^V \MU d^VXYUf\4(B] SnV%$Y]oUfSA[Jdo]oSB[�VfeN[J] S

tj�

n∑

i=0

ciH(N)2i (tj) +

n∑

i=0

diH(N)2i+1(tj) = 0

� V] S �]<B[4g bOL <Ad bj] \kV[4g/BBej[�gjgN] U*SB[MdoV BBej]o\MV"gLU*\ \kV bNUfej]^V gN]�VfSBS�<Bd d^V B([Yecb?<;bLbj]pI*d ]iVfd dML ] S=. <AUfej] ZB[Yd

jmv[4gL]o\MU!��\M[YS0bjeN[ doV gN[�X_U*SAZBV gLU*\ \kV bNUfej]^V gL]�VfSBS�<Bd d^V Bn[�e b?<;bLbj] Ifdo]

ii)A ] \kV SA[4B([YehX_] �

cj = 0Bn[�e X�<B]�] bj[YehV SAZBU gN<

jV*(J(B]^V \MU XdKB[ [�gjgN]�ZB[�5fU*SBU�[4gNgN[Yej[

b?<;bLbj]�S�<Bd do]vi ` U*(>(B]oVf\MU BBejU�5�V ej[cXdKB[ VfSAXdKB[cb?<;bLbj]�]digLU*SBUWS�<Bdod ]��9X_U0g��?X_U*SAgL]^Z;[�eNUWd^V

XYUf\4(B] SnV%$Y]oUfSA[Jdo]oSB[�VfeN[ Z;[Ydodo[cZ;[Yej]C5�V bN[�n∑

i=0

ciH′(N)2i (tj) +

n∑

i=0

diH′(N)2i+1(tj) ≡ 0

mBV%BJBB]^V \MUcXdKB[�d^VrBBej]o\MV gNUf\M\kV�bjUfej]oV � S�<Bdod^V��]Bn[�e?XE<B](ej] \kV SA[�doV XYUfSAZ;]D$Y]oUfSA[di = 0B([YeTU*IfSA]

i = 0, . . . , ni

�HtL [4g/BAeN[4gNgN] U*SB[ Z;[Yd�B(Ufdo]oSBUf\M]oU ]oS0bN[�eNB(Ufd^V S0bN[ Z;] �T[Yej\M] bj[ �cV dod U*ejV

PN (t) =

n∑

i=0

fiH′(N)2i (tj) +

n∑

i=0

f ′iH

′(N)2i+1(tj)

" [YdodoV BBejV bN]^XYV��lgjX_U*S]5*[YSB]o[YS0bj[J[4g/BAeN]o\M[Yej[ doV(AV*gL[ Z;] �T[Yej\ ] bN[cXYUfS]�BnU*d ]oSBUf\M]/Z;]HFV%I*ejVfSJIf[��B([Ye�XE<A] If[YSA[YehV do]C$8$Y]^V \MU��]<AV S0bjU V�!,[Yej\kV�bjU �ASBU*ejV4<;bj] do]C$8$�VfSAZ;U doV4(AVfgN[ Z;]" [�� bjUfS Bn[�e

ΠN�

{w20(t), . . . , w

2n(t), w2

0(t)(t − t0), . . . , w2n(t)(t − tn)}

P g/BBdo]^X_] bjV SnZ;UkdoV4. U*eN\kVMZ;] �]<B[4g bhV4(nVfgN[JUfbLbj[YSB]^V \MU

{

Bn[�e Ifdo]D] SAZB]oXY]�BAV ej]︷︸︸︷

1, (t − t0)2, (t − t0)

2(t − t1)2, . . . ,

n−1∏

i=0

(t − ti)2,

(t − t0), (t − t0)2(t − t1), (t − t0)

2(t − t1)2(t − t2), . . . , (t − tn)

n−1∏

i=0

(t − ti)2

︸︷︷︸Bn[�e Ifdo]D]oSAZ;]^X_]�Z;]^gNBAV ej]

}

: S �]<B[4g bjU \MU;Z;UMd^VkXE<Be?5�V gN]9gjX_ej]C5*[JXYUf\M[

XN (t) =

n∑

i=0

A2iw2i (t) +

n∑

i=0

A2i+1w2i (t)(t − ti)

#�2Ai ! %

1 7 �� 4� �D ��^��?��c��!�p� �� p� �p� ��

` U*(J(A]oVf\ UTbNejU�5�V ej[HdoV . Ufej\kVTZ;[8Ifdo]Aj��]<B]oSAZ;];g/<>BJBnU*SB]^V \MUJZB]*bjU*I*d ]o[Yej[ <BSBU Z;](�]<B[�gLbN]

XYU0[�0WX_]o[YS0bN]��BX_]oU/�JXYUfSAgN]oZB[Yej]oVf\ UMgNUfdoUldo[ V*gNXY]ogjgN[t0, t1, . . . , tn−1

i HFV ZB] \M[�gN]oUfSB[ Z;[�d doUgNBAV%$�] U BnVfgjgNV�Vfd doUfehV�ZBV

N = 2n+1V

2n−1� �]<B] SnZ;]�B(U*gjgN]oVf\ U�[4g/BAeN]o\M[Yej[�d\L [ �]<AV*$Y]oUfSB[

2ni ! X_U*SXN (t) = XN−2(t) + A2nw2

n(t) + A2N+1w2n(t)(t − tn)

Z;U�5*[XN−2(t)

� <BSAV XE<AeN5�V BnU*d ]oSBUf\M]^V do[ ZB] IfehVfZ;UN − 1

i ��[�Z;]^V \MU��]<B[Ydod U�XdKB[gN<AXYXY[�Z;[ gN[g5uVfdC<BbNU �]<B[�gLb8L <Bd bN]o\kV [4g/BAeN[4gNgN] U*SB[JSB[Y]

tiXYUfS

i = 0, . . . , n − 1�

XN (ti) = XN−2(ti) + A2nw2n(ti) + A2N+1w

2n(ti)(ti − tn)

� VT] d=B(Ufdo]oSBUf\M]oUXN−2(t)

�O] S0bN[�eNB(Ufd^V S0bj[tB([Ye�]CB(U bj[�gN]��\M[�S*bjeN[pIfdo];V d bNej]0gN];V SBS�<BdodoVfSBU!��]<B]oSAZ;]

XN (tj) = fj j = 0, . . . , n − 1��[4Z;]oVf\MUMV*Z;[�gjgNU ]od�X_U*\BnU*eLbhV \M[YS0bNUMZ;[�d do[cZ;[Yej]D5uV bN[��

X′N (t) = X′

N−1(t) + 2A2nwn(t)w′n(t) +

+A2n+1

[2wn(t)w′

n(t)(t − tn) + w2n(t)

]

ma[r5�VfdC<;bj]oVf\MU��]<B[4g bOL <Ad bj] \kV [�gNBBej[�gjgL]oUfSB[J]oStj# gN[Y\BBej[gBn[�e

j = 0, . . . , n − 1%

P ′N (tj) = f ′

j j = 0, . . . , n − 1

' VfdoXYUfdo]oVf\ U��]<B[YdodoUMXdKA[ V*XYX�VfZ;[gB([Ye�dML <Bd bN]o\MUWZBV�bjU!�tn�

X(tn) = XN−1(tn) + A2nw2n(tn) = fn

�]<B]oSAZ;]��BehVfXYXYU*I*d ]o[YSAZBU)B([Ye [850]^Z;[�SJ$Y]^V ej[A2n

�

A2n =fn − P2n−1(tn)

w2n(tn)

3�[Ye�do[cZ;[�eN]D5�V�bN[ ]oS�5f[�XY[

f ′n = P ′

N (tn) = P ′2n−1(tn) + 2A2nwn(tn)w′

n(tn) + A2n+1w2n(tn)

` V �]<B[�gLb8L <Bd bN]o\kVW[ �]<AV*$Y]oUfSB[ �)(AV SAVfd [leN]^XYVV5�V ej[A2n+1

BnU*]oXdKJacbN<;bNbN]/]�bj[Yej\ ]oSB]/gLU*SBUSBUfbN]��

A2n+1 =f ′

n −P′2n−1 − 2Anwn(tn)w′

n(tn)

w2n(tn)

� � � �Z%-��&1 � � $ �"#$��%'& 24�-% � � $#$%'&C1

' U*SAgL]^Z;[�eN]^V \MUlV*Z;[�gjgLUld\L ] S0bj[Ye?BnU*doV*$Y]oUfSB[JX_UfS g/BBdo]oSB[fi : d�BBejU*(Bdo[Y\kV��JVfSAV doU*I*UcV,�]<B[�d doUZ;] � [�eN\M] bN[JX_UfSngL]^Z;[YehV�bjU �ASBU*ejV!�*X_]oU/�!V*(J(B]^V \MU

n+1bN[�eNSA[

(ui,Pi,wi)[!Z;U�(J(B]^V \MU

bjeNU�5�V ej[X(u)

bjVfd [ XdKB[{

X(ui) = Pi

X′(ui) = wii = 0, . . . , n

�D �� ^��?��c��!�p� �� i�?� ��C1

1 !

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolazione di Hermite

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolazione Hermite − Parametrizzazione uniforme

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolazione Hermite − Parametrizzazione chord−length

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolazione Hermite − Parametrizzazione centripeta

Punti da interpolareParametrizzazione uniformeParametrizzazione chord−lengthParametrizzazione centripeta

q/]DI*<BehV)2ni � ��: S*bj[Ye?BnU*doV*$Y]oUfSB[ Z;] �T[Yej\M] bj[cZ;]�<BSAVMX�<BeN5�VMV%B([YeNbjV

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolazione di Hermite − Curva chiusa

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Parametrizzazione uniforme

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Parametrizzazione chord−length

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Parametrizzazione centripeta

Punti da interpolareParam. uniformeParam. chord−lengthParam. centripeta

q/]CI�<BehV42ni ,!��: S*bj[Ye?BnU*doV*$Y]oUfSB[ Z;]��T[Yej\M] bj[cZ;] <BSAVkX�<BeN5�VMXdKB]D<AgjV

1 � �� 4� �D ��^��?��c��!�p� �� p� �p� ��

' [�ejXdKB]^V \MU�V dodoUfehVX : [uo, un] → E

d [ 5*U*Ifdo]^V \MU�XdKB[��]<B[�gLbjV gL]^VC1([u0, un])

i��U�Ifdo]oVfSBU�XdKB[�d [�g/BBdo]oSB[H]oS*bj[Ye?BnU*doVfS0bN] gN]oVfSBU Z;]*I*ejV*Z;U

3�OBn[�e�XE<A]��ngjgL]^V \MU <BSrBAVfejVf\M[_bNejU

doU;XYV do[t =

u − ui

hihi = ui+1 − ui

P Z ]oSAZ;]^XdKB]oVf\MU X_U*SX(i)

d\Limv[4gL]o\MU �/Bn[8$�$�U � Z;]HgNBBdo] SB[��9XdKB[�� ehV%BJBAeN[4gL[�S*bhV%(B]odo[lSB[�d d^V

. U*eN\kV��

X(i)(u) =

3∑

k=0

b(i)k B3

k

(u − ui

hi

)

: \B(UfSB]^V \MUkV��<A[�gLb8L <Bd bj] \kVMdo[ X_UfSnZ;]C$�] U*SB]9Z;]9]oS0bN[�eNB(Ufd^V%$�] U*SB[JZB[Y]Pi�

{X(i)(ui) = Pi

X(i)(ui+1) = Pi+1⇐⇒

{

b(i)0 = Pi

b(i)3 = Pi+1

i = 0, . . . , n − 1

P gLbN[�SAZ;]^V \MU-�]<B[�gLbN[cXYUfSAZ;]D$Y]oUfSA]9V SAXdKB[ Vfd do[ Z;[Yej]C5�V bN[�

dX(i)

du=

dX(i)

dt· 1

hi

W <B]oSAZ;]{

dX(i)

du (ui) = 1hi

dX(i)

dt (0) = 3hi

∆b(i)0 = wi

dX(i)

du (ui+1) = 1hi

dX(i)

dt (1) = 3hi

∆b(i)2 = wi+1

3�[Ye�]�SBU;Z;]9]oS*bj[YejSB]Dej]ogN<Bd bN[Yedo{

b(i)1 = b

(i)0 + 3

hiwi

b(i)2 = b

(i)3 − 3

hiwi+1

� � � �� &�� & �� &� %&� �� % � ��

" [Ydod [ X_UfSngL]^Z;[YehV%$�] U*SB]9gN<Bd�BAeNU�(Bd [�\kVWZ;[YdodML ]oS0bN[Ye?B(Ufd^V%$Y]oUfSA[g. V�bLbj[ �ASAUfehVWV%(J(A]oVf\ U g/<JBJmB(U*gLbNUMZ;]DX_U*SBU*gjX_[�eN[��*U*d bjeN[JV ]�BJ<BS0bN]DZ;[�d d^V XE<Be?5�V ZBV ]oS*bj[Ye?BnU*doVfeN[��0VfSAXdKB[!]�5�Vfd U*eN]DZ;[�d d^VX�<BeN5�V+KBU;Z;U�IfehV%BJK9i/m�<JBJB(UfSB]^V \MU UfehVJZ;]BSBU*S Z;]^gNBnU*eNej[OZB]!�]<B[�gLb8L <Bd bN]o\MVJ]oS=. Ufej\kV%$�] U*SB[[ Z;]�5*Ufdo[Ye [Yd^V%((UfehV ej[+<BSAV bN[4X_SB]^XYVB([Ye V*BJBBejU*gjgL]o\kV ejdoVAi

�� gLbjV bNUc]od>BBej]o\ UcgjXdKB[�\MVgBAeNU�BnU0g bjU!�fgN]ngLbN]o\kV doV ZB[Yej]C5�V�bhV X_U*SAgL]^Z;[�eN]^V SAZBUJd^VrBAVfejVfd do[Yd^VVf] BJ<BS0bN]

Pi−1[Pi+1

iHR (J(A]oVf\ U �]<B]oSAZ;]

wi =Pi+1 −Pi−1

hi−1 − hi

3�[Ye I*d ]D[4g bjeN[�\M]9] S�5*[�X_[w0 = (P1 −P0)/h0

[wn = (Pn −Pn−1)/hn−1

i

�D �� ^��?��c��!�p� �� i�?� ��C1

1*,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spline cubica C1

Punti da interpolare

q/]DI*<BehV62Ai 2!��: S0bN[Ye?B(Ufd^V%$Y]oUfSA[ �T[Yej\ ] bN[�m&gNBBdo] SA[C1

Z;] <BSAVkX�<BeN5�VMV%B([YeNbjV

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−2

−1

0

1

2

3

4

5Hodograph

qF]DI*<AejV62AiG9(��TUaZBU*IfehV%B>K Z;[�d d^V � [Yej\M] bj[Em g/BAd ]oSB[r2Ai 2

1*2 �� 4� �D ��^��?��c��!�p� �� p� �p� ��

−5 −4 −3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

4Hodograph II

q/]DI*<BehV42Ai @�� U;Z;U�IfehV%BJK :/:OZ;[YdodoV �T[Yej\M] bj[Em g/BBdo]oSB[g2Ai 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Spline cubica C1


q/]CI�<BejV62niGF(��: S0bj[Ye?BnU*doV*$Y]oUfSB[ � [�eN\M] bN[Em gNBBd ]oSB[C1

Z;]�<ASAVkXE<Be?5�VMXdKB]C<ngNV

�D �� ^��?��c��!�p� �� i�?� ��C1

1�9

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

Hodograph

qF]DI*<AejV62Ai &��TUaZBU*IfehV%B>K Z;[�d d^V � [Yej\M] bj[Em g/BAd ]oSB[r2AifF

−10 −8 −6 −4 −2 0 2 4 6 8 10−4

−3

−2

−1

0

1

2

3

4Hodograph II

qF]DI*<AejV62Ai 1��TUaZBU*IfehV%B>K :/: Z;[�d d^V � [Yej\M] bj[Em g/BAd ]oSB[r2AifF

1�@ �� 4� �D ��^��?��c��!�p� �� p� �p� ��

�� k�k\� � �J�! ��]��O�p�� Y4��iOi8��kW <B[4g bjU��+<AS \M[YbNU;Z;UkV d bN[�eNSAV bN]D5fU6Bn[�eTXYVfdoXYUfd^V ej[ I*d ]

wii?ma]9X_U*SAgN]oZ;[�ejV

wi =dqi

du(ui) i = 1, . . . , n − 1

Z;U�5*[qi�+<ASAVkXE<Be?5�V6BnU*d ]oSBUf\M]^V do[ ��<nVfZ;ehV�bj]oX�Vl]oS0bN[Ye?B(Ufd^V S0bN[J]�BJ<BS0bj]

(ui−1,Pi−1), (ui,Pi), (ui+1,Pi+1)i ' Uf\M[?X�<BeN5�V B(Ufdo] SAUf\M]oVfd [�gL]�<;bN]od ]D$�$4V <BSnV XE<Be?5�V

Z;]�_pa8$Y]o[Ye3�(��<A] SAZB]�gL]FZ;[ �ASA]ogjX_[ ]od�BAV ehV \M[YbNejU doU;XYV do[

t =u − ui−1

hi−1 + hi

]oS \ U;Z;UkXdKB[qi(t(u)) =

{qi(0) = Pi−1

qi(1) = Pi+13�U*gjgN]oVf\ U6B([YeNbjV S0bjU [4g/BAeN]o\M[Yej[qiXYUf\M[

qi(u) = pi−1(1 − t)2 + ci2t(1 − t) + pi+1t2

` U*(J(A]oVf\ UB([Ye�� bNejU�5uVfeN[J]od BJ<BS0bNUci�JB([Ye�[4gNgNUkV dod U*ejVl] \B(UfSB]^V \MU

Pi = qi(ui) =1

(hi−1 + hi)2[h2

i Pi−1 + 2hihi−1ci + h2i−1Pi+1

]

XY] U/�ci =

(hi−1 + hi)2

2hihi−1Pi −

hi

2hi−1Pi−1 −

hi

hi−1Pi−1

RTd doUfehVld^V6. Ufej\)<AdoV4Bn[�e ]od9XYV d^X_U*d UkZ;]wi

��

wi =dqi(ui)

du=

1

hi−1 + hi

dqi

dt(ti) =

=1

hi−1 + hi

[hi

hi−1(pi − pi−1) +

hi−1

hi(pi+1 − pi)

]

ma] B>< �MSBU bhV ej[JXdKB[JdoUMgjXdKB[�\MV6q � : H`H SBU*S�� Vfd bjeNUMXdKB[ d UMgjXdKB[Y\kV Z;[Ydod [JbhV SJI*[YS0bN]9Z;]_O[�gjgN[Yd <;bj] do]C$8$�VfSAZ;U<BS BnU*d ]oSBU*\ ]oUM] S0bj[Ye?BnU*doVfS*bj[JZB]�IfehVfZ;U

1[rIfdo]

hi<ASB] . U*eN\M]vi

� �� Z%-��&1 � � $ �"#$��%'& 24�-% � � $#$%'& 2 � .0#$2 ��&C

2

H�[ XYUfSAZB]C$�] U*SB]9Z;]9]oS0bN[Ye?B(Ufd^V%$Y]oUfSA[C2

B([Ye do[cgNBBd ]oSB[ gNUfSBU��

X(ui) = Pi i = 0, . . . , ndX(i)

du (ui) = dX(i−1)

du (ui) i = 1, . . . , n − 1d2

X(i)

d2u (ui) = d2X

(i−1)

d2u (ui) i = 1, . . . , n − 1

W <B]oSAZ;]��BSB[�d9XYV*gLUBAVfeLbj]oXYUfd^V ej[!Z;[�d do[cg/BAd ]oSB[ XE<J(A]oXdKB[cgNUfSAU{

b(i)0 = Pi, b

(i)3 = Pi+1 i = 0, . . . , n − 1 (C0)

b(i)1 = Pi + hi

3 wi, b(i)2 = Pi+1 − hi

3 wi+1 i = 0, . . . , n − 1 (C1)

�D �� ^��?��c��!�p� �� i�?� �� p� �� C2

1]F

m;]oX�X_Uf\M[ SBU*S XYUfSBU0gNXYU4I*d ]wi

]o\BnU*SJIfUMd^VkX_UfSnZ;]C$�] U*SB[C2

d2X(i)

du(ui) =

d2X(i−1)

du(ui) i = 1, . . . , n − 1

XY] U/��Ag . e?<;bNbjV SnZ;UMdoV4. Ufej\kVMZ;]`_Za�$�] [�e �

1

h2i

d2X(i)

dt(0) =

1

h2i−1

d2X(i−1)

dt(1)

P g/BBdo]^X_] bjV S0bjUkdoVMZ;[�eN]D5�V�bjVMgN[�XYUfSAZBV gN]�KAV

1

h2i

(b(i)2 − 2b

(i)1 + b

(i)0 ) =

1

h2i−1

(b(i−1)3 − 2b

(i−1)2 + b

(i−1)1 )

m;U*gLbN] bN<B[�SAZ;Uk]oS��]<B[4g bOL <Ad bj] \kV d [cXYUfSAZB]C$�] U*SB] Bn[�e�] d�ejV*XYX_U*ejZBU V%(>(B]oVf\MU1h2

i

[(Pi+1 − hi

3 wi+1) − 2(Pi + hi

3 wi) + Pi

]=

= 1h2

i−1

[

Pi + −2(Pi − hi−1

3 wi) + (Pi−1 + hi−1

3 wi−1)]

� Ufd bN]DBBd ]^XYVfSAZ;UM[YS0bjejVf\)(B]D]9\M[�\)(Bej] Bn[�e3hi−1hi

U bNbN[�SJIfU

hiwi−1 + 2(hi−1 + hi)wi + hi−1wi+1 =

= 3

Pi+1 −Pi

hihi−1 +

Pi −Pi−1

hi−1hi

︸︷︷︸

TNi

[,��<A[�gLbNU B([Yei = 1, . . . , n − 1

i R (J(A]oVf\ U <BS gL]^gLbN[Y\kVWZB]n − 1

[ �]<AV%$�] U*SB]9]oSn + 1]oSAXYU*IfSA] bj[�%B([YeHX�<B]B\kV SAX�VTdML <BSB]^X_] b?oBiHma]BV*I*I*]C<BS>IfUfSAU!V dodoUfehVTZ=<B[ XYUfSAZB]C$�] U*SB];V%<AgN] do]^V ej] [

V*Ifdo]*[4g bjeN[�\M]f[HbhV do]0XYUfSAZB]C$�] U*SB]��w0

[wn

Z;[_bj[Yej\M] SAVfSBU ]odfbj]CB(U!ZB]agNBBdo] SA[fi/m=<JBJB(UfSB]^V \MUZB]�gNXY[�I*d ]o[Yej[JVkX�VfgNUlbhV do]9X_U*SAZ;]D$Y]oUfSB]��;] dFgL]^g bj[Y\kVMXdKB[ U bNbN[YS>IfU��

2(h0 + h1) h0 0 . . . . . . 0h2 2(h1 + h2) h1 0 . . . 0

0

i i i i i i i i iiii i i i i i i i i i

hn−2

0 . . . . . . 0 hn−1 2(hn−2 + hn−1)

w0

w1iiiiii

wn

=

w0

TN1iiiTNn−1

wn

HFV\kV�bjeN]^X_[�ZB[Y] X_Ua[E0WX_]o[YS0bj]��WbjeN]^Z;]^V%I*UfSAVfd [M[�Z V Z;]^V%IfU*SAV do[WZ;U*\ ]oSAVfS*bj[� �]<B] SnZ;]�d^VgNUfdD<J$�] U*SB[?[4gL]^g bj[�[4Z �Z<BSB]^XYVAi/RTSAV do]C$8$Y]^V \MU!V*Z;[�gjgNUT]�5�V ej]fbj]CBB]BZB]agNBBdo] SA[OXdKB[ gN]]B(U*gjgNUfSBUI*[YSB[�ejVfeN[ gjX_[�I*d ]o[YSnZ;U

w0[wn

i�H�[�g/BBdo]oSB[ XE<J(A]oXdKB[ SAV bN<BehV do]BVfZ [�gN[Y\BB]oU+KAV SBSAU XYUf\M[<Ad bj[Yej] U*eN]9XYUfSAZ;]D$Y]oUfSA]

{d2

X(0)

d2u (u0) = 0d2

X(n−1)

d2u (un) = 0⇐⇒

{d2

X(0)

d2t (0) = 0d2

X(n−1)

d2t (1) = 0

W <A[�gLbNUM[ �]<B]D5�V do[ V{

∆2b(0)0 = 0

∆2b(n−1)1 = 0

⇐⇒{

(P1 − h0

3 w1) − 2(P0 + h0

3 w0) + P0 = 0

Pn − 2(Pn − hn−1

3 wn) + (Pn−1 + hn−1

3 wn−1) = 0

1�& �� 4� �D ��^��?��c��!�p� �� p� �p� ��

: S(�nSB[rBnU0gNgNUMgNXYeN]D5f[�eN[��{

2h0w0 + h0w1 = 3(P1 −P0)hn−1wn−1 + 2hn−1wn = 3(Pn −Pn−1)

: S . Ufej\kVl\kV bNej]oXY]oVfd [ V%(>(B]oVf\MUMVfd doUfehV

2h0 h0

h1 2(h1 − h0) h0i i i i i i i i ihn−1 2hn−1

W <B[4g bOL <Ad bj] \kV � <ASAVc\kV�bjeN]^X_[ VlZ;]oV*IfU*SAV do[ Z;Uf\M]oSAV S0bj[ Bn[�e�X�<B]nd^VcgNUfdD<J$Y]oUfSA[�[�gN]ogLbN[ [4Z�g<BSB]^XYVAi

� S V d bNejUWbj]CB(U ZB]�XYUfSAZB]C$�] U*SB[lV%<ngL]od ]^V ej]^V # XdKB[4Bn[�e [�gN[Y\BB]oU <AgjV � R - HFR _ % � d^V�LSBU bTV�*aSBUfb � �?]oS [�gjgNVkgN]9]o\6B(UfSA[lXdKB[

u1[�Z

un−1SAUfS gN]oVfSBUBB]��SAUaZB]FZ;[�d d^Vkg/BAd ]oSB[�

\kV gNUfdoUrBJ<BS0bN](Z;]B]oS*bj[Ye?BnU*doV*$Y]oUfSB[ # ] S BAejV bN]^XYV gL](Z;]o\ ]oS�<B]^gNXY[ Z;]2d^V Z;]o\ [�SAgN] U*SB[TZ;[�d d^V

gNBAV%$�] U/% i W <B[�d doUXdKB[k] S BBehV�bj]oX�V 50]o[YSA[6B(Uf] . V�bLbjU �kX_UfSngL]^Z;[YehV ej[��<A[�gLbN]?XYUf\M[MSBU;Z;]��\kV ] \B(UfSB[�SAZ;UWgN<JIfdo]9gLbN[�gjgN]DejV*XYX_U*ejZBU Z;]Dbj]CB(U

C3 i��[�ZB]oVf\ U VfZ;[4gNgNUlXYUf\M[TX_ej[�VfeN[ X�<Be?5f[TgNBBdo] SB[JXdKB]D<AgL[JX_U*SWejV*XYXYUfehZ;U ] SP0 = Pn

Z;]bj]CB(U

C2i�H�[TX_U*SAZ;]D$Y]oUfSB]WBn[�eHbhV do[ bN]DB(UlZ;](eN[8IfUfd^V ej] b?o gLU*SBU U�5�5a]oVf\ [�S0bN[�dML <JI�<AV%I*d ]^V S>$�V

Z;[�d d^V4. <BSJ$�] U*SB[ [ Z;[Ydod [cZB[Yej]C5�V�bhV6BBeN]o\kV SB[Yd9ehVfX�X_U*ejZ;U [

d2X(0)

d2u(u0) =

d2X(n−1)

d2u(un)

3�VfgjgjV SAZ;UMV ] BnV ehV \M[_bjeN],doUaX�V do]��]<B[4g bOL <Ad bj] \kVM[ �]<B]D5�V do[ V

1

h20

d2X(0)

d2t(0) =

1

h2n−1

d2X(n−1)

d2t(1)

qnVfX_[�SAZ;UMdo[ gNU*gLbN] bN<>$Y]oUfSB]DU�BJBnU*eLb?<BSB[��

h2n−1

[(P1 − h0

3 w1) − 2(P0 + h0

3 w0) + P0

]=

= h20

[

Pn − 2(Pn − hn−1

3 wn) + (Pn−1 + hn−1

3 wn−1)]

ma[8BAV ehV SnZ;U U*BJB(UfeNbN<ASAV \M[YS0bj[�]pjZBV ]

wj[T\ U*d bj]CBAd ]^XYVfSAZ;UsB(Uf]WBn[�e

h0hn−1U bLbj[YSJI*U

2hn−1w0 + hn−1w1 + h0wn−1 + 2h0wn =

= 3[

hn−1

h0(P1 −P0) + h0

Pn−Pn−1

hn−1

]

A�]^X_U*ejZAV SAZ;UMXdKB[ V%(>(B]oVf\MU6B(U*gLbNUkXYUf\M[ X_UfSnZ;]C$�] U*SB[w0 = wn

gL] KAV

h0wn−1 + 2(h0 + hn−1)wnhn−1w1 =

= 3[

hn−1

h0(P1 −P0) + h0

Pn−Pn−1

hn−1

]

: S . Ufej\MV \kV�bjeN]^X_]^V do[!V*(J(B]^V \MU��]<B]oSAZ;] <AS gN]ogLbN[�\MVn× n

]oS . Ufej\MVcbNej]oZB]oV*IfUfSnV do[!X_U*SdML V*I*I*]C<BS0bhV"Z;]�Z=<B[ [�d [�\ [�S0bN] SB[YdodML <Bd bN]o\kVrX_UfdoUfSASAV"Z;[�d d^V BBeN]o\kV"ej]DI*V [ SB[�d d^V BBeN]o\kVXYUfdoUfSBSAV Z;[�d dML <Ad bj] \kVMej]CI0VBi

�D ��a � ��o� �� ?� �� 1*1

−2 −1 0 1 2 3−3

−2

−1

0

1

2

3

4

−4 −3 −2 −1 0 1 2 3−8

−6

−4

−2

0

2

4

6

0 200 400 600 800 10000

50

100

150

200

250

300

350

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spline cubica C2


Hodograph

Hodograph II Curvatura

q/]DI*<BehV)2ni !O7 ��: S0bN[�eNB(Ufd^V%$�] U*SB[ �T[Yej\M] bj[Em g/BBdo]oSB[C2

Z;]�<BSAVkX�<Be?5uV V%B([YeNbjV

�� ^�31� �#$% &�%�&1 ��

A ]oXYUfehZ;]oVf\MUMXdKA[JSA[Yd9XYV*gLU6. <BS>$Y]oUfSAVfd [g5�V do[ ] d�gN[�I�<B[YS0bj[!bj[YU*eN[�\MV!�

��Wjt��i8�\�*�� 38 S��!@�E b j <MCZ@gG$hDK�<BG)fq@nCZNJT]hiReNJj <fqNPCDCZH hiCZk <MI b HJRST NJCDT @ b R[H b RS<B@�T �� NJT NQhiCZN �)hiC O�<BHJCZ@y ∈ C2([a, b])

<MC=Tm@�R b HJj NPCDT @�j @�G)H b b <B@ (xi, fi)G)HJC

i = 0, . . . , n �ES<�fDN∫ xn

x0

�:(x)2dx ≤

∫ xn

x0

�� (x)2dx

>�HJ`J@��hD@SEST]<?<MC=Tm@mr ReNJj <> @�jMj @ >�@$RS<M`JNJT @QE�@)G)HJCZ> @ b R[@$CZ> HJCZH'<Mj�C9HJIL@L>J< ] SnZ;]oXY[ Z;]�m;VfejZ@-ReN b b R[@SE�@�CDT NJC9H hiC9NLI�<FEShiReN�> @�j=rPR[N > H'>J<HaE�G$<MjMj N�O�<BHJCZ@L> @�jMj N��)hiC O�<BHJCZ@ V�� "�� "�� "!$#$%R (J(B]^V \MUWXdKB[

��(x) =

�m(x) + (

��(x) −

�m(x))

P d [85uVfSAZ;UWVfd ��<nVfZ;ehV�bjU [�S*bjejVf\)(B]D]9bN[Yej\M] SA]D[rBAVfgjgNVfSAZ;UMV%I*d ]D]oS0bN[�I*ejVfd ]

∫ xn

x0

��(x)2dx =

∫ xn

x0

�m(x)2dx+

∫ xn

x0

( ��(x) −

�m(x))2

dx+2

∫ xn

x0

�m(x)( ��

(x) −�m(x))dx

!O7*7 �� 4� �D ��^��?��c��!�p� �� p� �p� ��

d^V"Z;]^gN<JI*<AV*Ifdo]oVfSJ$�V"XdKB[ 5*U*Ifdo]^V \MU"Z;]o\MU*gLbNehV ej[WeN]^gN<Bd bhV 5f[�ejV gL[ d\L <Bd bN]o\MU ]oS0bN[�I*ejVfd [ �S�<Bdod U��BX_]oU/� gN[�G∫ xn

x0

�m(x)( ��

(x) −�m(x))dx =

�m(x)( �

(x) − � (x))∣∣∣

xn

x0

−∫ xn

x0

m ��L(x)( �

(x) − � (x))dx =

Z;U�5*[ �m(x0) =

�m(xn) = 0

B([YelZB[ �ASB]D$Y]oUfSA[WZ;]�gNBBdo] SB[kSnV�bN<AejVfd [k[ �]<B]oSAZ;]�] d�bj[Yej\M] SB[SBU*S] S0bj[�IfehV�bjUWgL]�V SAS]<Ad d^V��n] SBU*d bjeN[Mm �]L

(x)SBU*S � XYUfS0bN]oS�<AVM\kV�X_U0g bhV S0bN[cVWbNehV�bNbN] # doV

Z;[�eN]D5�V�bhVlbj[Ye?$�V Z;]�<BSnV6. <BSJ$Y]oUfSA[ X�<J(B]^XYV!�!�]<B] SnZ;]��r<BSAVMX_U0g bhV S0bN[ % i W <B]oSAZ;]

= −n−1∑

i=0

∫ xi+1

xi

m �]L(x)( �

(x) − � (x))dx =

= −n−1∑

i=0

m ��Li(x)

∫ xi+1

xi

( �(x) − � (x)

)dx =

= −n−1∑

i=0

m ��Li(x) (

�(x) − m

(x))∣∣∣

xi+1

xi

= 0

ZBVfdn\MU*\ [�S0bNUlXdKB[y[�Z

SgNUfSBUl] S0bN[�eNB(Ufd^V S0bj]ASB[�]nSBU;Z;],[JZ=<BS!�]<B[ <JI*<AVfd ]nSB[�I*d ]([�gLbNej[Em

\M]vi�-/V do[ bN[�Ufej[Y\kV��AgN[cV SnV do]C$8$Y]^V \MUMdoVkX�<BeN5�VkXYUf\BnU*SB[YS0bj[sBn[�eTX_U*\BnU*SB[YS0bN[��>5�V do[cV SnXdKB[

SB[�dFX�VfgNU6BnV ehV \M[_bjeN]^X_Uni -`<;bNbjVV5a]^V��]<B[�d doUWXdKB[cXY]9] S0bj[Yej[�gjgNV SBU*S � ] d/X_Uf\B(UfeNbjVf\ [�S0bNUZ;[�d do[MgL]oSJIfU*d [MX_U*\BnU*SB[YS0bN]��D\kV��]<B[�d doU Z;[YdodoV XE<AeN5�V ] S If[�SB[YehV do[�,]oSAZ;]DBn[�SAZ;[YS0bj[Y\M[YS=mbj[WZBV dodoV BnV eNbN]^X_Ufd^V ej[6BAV ehV \M[_bjeN]D$�$4V%$�] U*SB[4<BbN]od ]D$�$4V�bjVAi ` [ �ASA]oVf\ U Vfd doUfehV <BS�L V SAVfd U�I*V\M]^g/<BehV Z;[�d dML ] SAZB]oXY[ Z;]�m;VfejZ�SA[Yd XYVfgNU BAVfejVf\M[_bNej]^X_U!��XdKB[ BBeN[�SAZ;[ ] d SBU*\ [ Z;] E)T]ReNP<MC@�C9@�RUr � [�Z0� ZBV�bhV ZAV

∫ un

u0

κ2(u)du

-�<;bLbhVV5a]oV!� 5�V dD<;bjVfeN[��]<B[�gLbjV��]<AV S0bN] b?o1� V SAVfd ] bN]^XYVf\ [�S0bN[ X_Uf\BBdo]^XYV�bjU #�Bn[�ejXdK��X*L � ] dBBejU;Z;U bNbNUs5*[_bNbNUfej]^V do[3% �*] SAUfd bNej[�*gN[ ej]oZJ<AX_]^V \MUc]od(X�VfgNUrBAV ehV \M[YbNej]oXYU VfdnX�VfgNUr. <BSJ$�] U*SAV do[XYUfS

X(u) =

(u

y(u)

)

UfbLbN[�SB]^V \MU∫ un

u0

κ2(u)du =

∫ un

u0

y(u)′′2

(1 + y′(u))3du

A�]^gNUfdD5f[Yej[ �]<B[�gLb8L <Bd bN]o\kV . Ufej\)<Bd^V �?VfSAV do] bN]^XYV \M[�S*bj[?X_U*\BBd ]^XYV bNU # X_Uf\BAVfeN[?doV ZB[Yej]C5�V�bhVBBej]o\MVkVfd�Z;[�SBUf\M]oSAV�bjUfej[3% i ma[�I�<B[lXdKB[ SAUfS��lgN[Y\BBej[r5f[YejUkXdKB[cd^Vkg/BAd ]oSB[lX�<J(B]^XYV��cd^VBB]��gL\MUaU b?K9i

� ��ν− � � $#$%'&

H�[�gNBBd ]oSB[OXdKB[?[4gNVf\ ]oSB]^V \MU!V*Z;[�gjgNU KAV SBSAUTd^VTXYVfejV bLbj[Yej]ogLbN]^XYV�Z;]a[4gNgN[Yej[?XE<>(B]oXdKA[��]oSBUfd bNej[KAVfSBSBUcXYUfS0bN]oS�<B] bdo

C1 ∩G2gN<MbN<;bNbNUcdML ] S0bN[�eN5�Vfd doUAi W <B[�gLbNUcgN]DIfSB]��nXYV XdKB[TdoVcX�<Be?5uV bN<BehV

� X_U*S0bN]oS]<nV��;\M[YS0bNej[ d [cZB[Yej]C5�V�bj[ gL[4X_UfSnZ;[ SBUfS gLU*SBUkX_U*S*bj] S�<B[*iK�� cU:%8�`�9%.\c�2�NuY�8�95Y�.\9T:%1*k�Q�1#N�t

Z

f(x)g′(x) dx = f(x)g(x) −

Z

g(x)f ′(x) dx

�D �q ν− � �4� �� !O7J!

H�[ν− g/BBdo]oSB[cgLU*SBUMZ;[ �ASA] bj[lZBV

X(u) = Xi(u) con u ∈ [ui, ui+1] i = 0, . . . , n − 1

: d`If[YSA[Yej]oXYU �/Bn[8$�$�U � ZB]�g/BAd ]oSB[ � [�gNBBej[�gjgLU SB[YdodoV6. U*eN\kV Z;]`_Za�$�] [�e �

Xi(u) =

3∑

i=0

bk iB3k

(u − ui

hi

)

[4Z ]�BJ<BS0bN]FZ;]�X_U*S0bNejUfdod U Z;] �]<B[�gLb8L <Bd bN]o\kVM[�gNBBeN[4gNgN]oUfSB[ gLU*SBU!�

b0 i = Pi

b1 i = Pi + hi

3 wi

b2 i = Pi+1 − hi

3 wi+1

b3 i = Pi+1

��[4Z;]^V \MUWX_U*\ [ Z;[ �ASB]oeN[G2

gN<B]9SBU;Z;]��HX_U*SAgN]oZ;[�eN]^V \MU6<BSAV ej]CBAVfejVf\M[_bNej]D$�$�V*$Y]oUfSB[!V \6m\M]^gNgN]D(B] do[

u = u(v)XYUfS

v ∈ [vo, vn]i W <B[�gLbNU gL]DIfSA].�nX�V"XdKB[ ]oS-U�IfSB]ZBn[8$�$�U Z;[�d d^V

gNBBdo] SA[ V%(>(B]oVf\MUui = u(vi)

�i = 0, . . . , n

i"m;V%B>BB]oVf\MU"XdKB[kbhV do[ BAVfejVf\ [YbNej]C$8$�V*$Y]oU%mSA[4KAV X_U*S*bj] S�<B] b?o

C1�,\kVW]oS If[�SB[YehV do[cd^V�Z;[�eN]D5�V�bhVWgL[4X_U*SAZBVkSBU*S � ZB[ �ASB] bjV!��<A] SAZB]

XYUfSAgN]^Z;[Yej]oVf\MU!�{

u+′′

i = d2ud2v (v+

i )

u−′′

i = d2ud2v (v−i )

A ]CBAVfejVf\M[_bNej]D$�$Y]^V \MUMV dod U*ejV d^V X�<Be?5uV!�

dXi

dv=

dXi

du

du

dvB([Ye!XE<B]9d^VMZ;[Yej]D5uV bjVMgN[�X_U*SAZBV eN]^gN<Bd bhV

d2Xi

d2v=

d2Xi

d2u

(du

dv

)2

+dXi

du

d2u

d2v

' VfdoXYUfdo]oVf\MU��]<B[4g bOL <Bd bN]o\kVMSB[Y]vi�

d2Xi

d2v(vi) =

d2Xi

d2u(vi)u

′(vi)2 +

wi︷︸︸︷

dXi

du(vi) u+′′

i (vi)

d2Xi−1

d2v(vi) =

d2Xi−1

d2u(vi)u

′(vi)2 +

dXi−1

du(vi)

︸︷︷︸

wi

u−′′

i (vi)

� I*<AV*Ifdo]oVfSAZ;U d [rBAeN[4X_[�ZB[YS0bN]D[4g/BBej[�gjgN] U*SB],U bLbj[YSJI*U(

d2Xi

d2u(vi) −

d2Xi−1

d2u

)

u′(vi)2 = wi(u

−′′

i − u+′′

i )

` [ �nSB]oVf\MUνi = (u−′′

i − u+′′

i )/u′(vi)2�AV dod U*ejV V%(>(B]oVf\MU XdKB[

d2Xi

d2u(vi) −

d2Xi−1

d2u= νiwi

!O7 � �� 4� �D ��^��?��c��!�p� �� p� �p� ��

W <B[4g bjU gN]CI*SB]��nXYV XdKB[ ]od�BBej] \MU�\M[�\)(BejU KnVWd^V gLbN[4gNgjV�Z;]oej[�$Y]oUfSA[ Z;[�d d^V ZB[Yej]C5�V�bhV BBej]Cm\kVBi H�[ Z;] !,[Yej[YS>$Y[ Z;[YdFgNVfd bjU

νi�BZ;[YbLbj[ b NPReNJIL@$T]RS<4>P< Tm@�CqES<BHPC9@ B(U*gjgNUfSBU [�gjgL[�eN[ gjX_[Yd bN]VfeN(B] bNehV ej]^V \M[YS0bN[�[Jdo],gN<JBJB(UfSA]oVf\ U4BnU0gL] bN]D5a] i P gNBBdo]oXY] bj]oVf\MU��]<B[4g bOL <Bd bN]o\kVl[4g/BAeN[4gNgN] U*SB[

B([Ye�UfbLbN[�eN[�SB[ ] dFgL]^g bj[Y\kV \kV�bNej]^X_]^V do[�

1

h2i

d2Xi

d2t(0) − 1

h2i−1

d2Xi−1

d2t(1) = νiwi

: S bN[�eN\M]oSB]9Z;]�B><BS0bN]�Z;]FX_UfS0bjeNU*d doU V*(J(B]^V \MU [ �]<B]D5uVfd [�S0bN[Y\M[�S*bj[�

6

h2i

∆2b0 i −6

h2i−1

∆2b1 i−1 = νiwi

ma[cgN5a] dD<JBJBA]oVf\ U6Ifdo]9U�Bn[�ejV bNUfej]∆ej]ogN<Bd bjV ] S��ASB[

hiwi+

[

2(hi−1 + hi) +1

2hi−1hiνi

]

wi+hi−1wi+1 = 3

[Pi+1 −Pi

hihi−1 +

Pi −Pi−1

hi−1hi

]

XYUfSi = 1, . . . , n−1

i � [�5a]oZB[YS0bN[ XdKB[ V dodoV BBej[�X_[4Z;[YS0bj[?\kV SAX�V SBU V SnX_UfehV2X_UfSnZ;]C$�] U*SB]

B([Ye�dML <ASB]oXY] bdoBi -/V do]�X_U*SAZ;]D$Y]oUfSB] B(U*gjgNUfSBU [�gjgL[�eN[ Z;[YbN[Yej\M] SnV�bN[ ]oS Z=<B[ \MU;Z;]��

• gL[OgL]]5�<BUfd0X_ej[�V ej[ <BSnV XE<Be?5�VTV*Bn[�eLbhV gL]aZ;[ �ASB]^gNXYUfSBUw0

[wn

bjVfd ]aXdKB[?do[�ZB[Yej]C5�V�bj[gL[4X_UfSnZ;[ SB[�I*d ]9[4g bjeN[�\M]9gL]^V SBU S�<Bdod [��JBn[�e XE<B]FVV50ej[Y\MU d [ XYUfSAZ;]D$Y]oUfSA]

{2h0w0 + h0w1 = 3(P1 −P0)hn−1wn−1 + 2hn−1wn = 3(Pn −Pn−1)

• SB[YdcX�VfgNU-]oS X�<B] ]oS]5*[�XY[ d^V-X�<Be?5uV ZBV-ejV*BJBBej[�gN[YS0bjVfeN[ gL]^V XdKB]C<ngNV!�!Z;U�(J(B]^V \MU] \B(UfejeN[

{

w0 = wn

d2X0

d2u (u0) − d2Xn−1

d2u (un) = νnwn

qAVfX�X_]^V \MU�V*Z;[�gjgLU V d^XE<BSB[ X_U*SAgL]^Z;[�ejV*$Y]oUfSB]�VfSAV do] bN]^XdKB[ gN<Bd�gL]^g bj[Y\kV�ZAVkeN]^gNUfdD5f[Yej[Bn[�ecZ;]ogN[�I*SAV ej[ldo[

νm&gNBBdo] SA[fi ma[kVfSAV do]C$8$Y]^V \MU!�(]oS BAVfeLbj]oXYUfd^V ej[�(] dHX�VfgNU�]oS X�<B]�gL]

5fU*I*d ]oUfSAU XYeN[4V ej[ Z;[Ydodo[TXE<Be?5f[ XdKA]C<AgN[�0]odngN]ogLbN[�\MVsKAV d^Vg. Ufej\kV(A+Dν)w = TnX_UfS

A =

2(h0 + h1) h0 0 . . . 0 . . . h1

h2

i i i i i i0

0

i i i i i i i i i iiiiii i i i i i i i i i

00 hn−1 2(hn−2 + hn−1) hn−2

hn−1 0 . . . 0 h0 2(h0 + hn−1)

[Dν = 1

2diag(h0h1ν1, . . . , hn−2hn−1νn−1, hn−1hoνn)i ' UfSngL]^Z;[YejU ]

νiZ > 0] S \ U;Z;U XdKB[Td^Vc\kV�bjeN]^X_[(A+Dν)

gN]oVlVlZ;]^V%I*UfSAVfd [ Z;U*\M] SAVfS0bN[T[4Zk]oS]5*[YeNbN]D(B]od [*iA�]ogjX_ej]D50]^V \MU6BnU*]9]od�gL]^gLbN[Y\kV SB[�d d^V4. Ufej\MV

(AD−1ν

︸︷︷︸

tν

+I)

wν︷︸︸︷

Dνw = Tn

�D �� ^��?��c��!�p� �� ?�� !O7 ,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q/]DI*<BehV42Ai !*! ��: S0bN[�eNB(Ufd^V%$�] U*SB[νm&gNBBdo] SA[�BXE<AeN5�VMV%B([YeNbjV

X_]oU/�(I + tν)wνTn

iHma[!ej]^g/<Bd bjV ‖tν‖ < 1#�(AVfgLbjV XdKB[JX_]DgL]^Vs<AS

νiV%(>(AVfgLbjVfSJ$�V

IfehV SnZ;[3%OV dodoUfehVcej]ogN<Bd bjV

‖(I + tν)−1‖ <1

1 − ‖tν‖W <B]oSAZ;]‖wν‖ ≤ 1

1 − ‖tν‖‖Tn‖

�]<B]oSAZ;]*B([Yeνmax → +∞ do[HX_Uf\B(UfSB[�S0bN]fZ;]

wνgLU*SBU d ]o\M] bhV�bj[H[ �]<B[�gLbNU gN]CI*SB]��nXYV

XdKB[�d^VcXE<AeN5�VJbN[�SAZ;[TV*ZM[�gjgL[�eN[�gN[Y\BBej[ BB]�� 50]^X_]oSAVcVfd d^V gNBn[8$�$�V bjV XdKB[ XYUfSJI*]C<ASJIf[]Pii

�� Z%��!&1 � � $ -"4#$�-%�& 2��% . � � $#$%'&

' U*SAgN]oZ;[�eN]^V \MU V*Z;[�gjgLU�dML ] S0bN[�eNB(Ufd^V%$�] U*SB[HZ;]aZBV�bj] bjejVf\ ] bN[OXE<Be?5f[ _pm g/BAd ]oSB[��XdKB[ . ejV b?<;bLbj[do[�X�<Be?5f[HgNBBd ]oSB[OgLU*SBU�]<B[Ydod [OXdKB[Hej]oXdKA] [4Z;UfSBU�\M[�SBU �]<AV S0bj] bdo Z;]f\M[�\ U*eN]^VBi�H�V I*[YSB[�eN]^XYVX�<Be?5uV_pm g/BBdo]oSB[ �cZ;[ �ASA] bhVkZBV

X(t) =n∑

i=0

diNi k(t), t ∈ [tk−1, tn+1]

XYUfSQ5f[YbLbNU*eN[ Z;[�]�SBU;Z;]�[�gLbN[4gLUT =

{t0, . . . , tn+k

} i�m=<JBJB(UfSB]^V \MUrV dod U*ejV Z;]TVV5f[�eN[n + 1

XYU*BJBA] [(ti,Pi), i = 0, . . . , n

bhV do]9XdKB[tk−1 = a ≤ t0 < t1 < . . . < tn ≤ b = tn+1

!O7 2 �� 4� �D ��^��?��c��!�p� �� p� �p� ��

−3 −2 −1 0 1 2 3 4−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

q/]CI�<BejV62ni ! � � ` [�eN]D5�V�bjVMZ;[�d d^Vνm g/BAd ]oSB[g2Ai !*!

−8 −6 −4 −2 0 2 4 6 8 10−5

−4

−3

−2

−1

0

1

2

3

q/]CI�<BehV)2ni ! ,�� ` [�eN]D5�V�bhV gN[�XYUfSAZAV Z;[�d d^Vνm g/BBdo]oSB[r2Ai !*!

�D �� ^��?��c��!�p� �� ?�� !O7 9

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

q/]CI�<BehV42ni ! 2!� ' <Be?5uV bN<BehVMZ;[Ydod^Vνm&gNBBdo] SB[g2ni !�!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q/]DI*<BehV62Ai ! 9(��: S0bN[�eNB(Ufd^V%$�] U*SB[νm&gNBBdo] SB[��AXE<AeN5�VMXdKB]D<AgNV

!O7 @ �� 4� �D ��^��?��c��!�p� �� p� �p� ��

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

5

q/]CI�<BejV62ni ! @�� ` [�eN]D5�V�bjVMZ;[�d d^Vνm g/BAd ]oSB[g2Ai ! 9

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

25

q/]CI�<BehV)2ni ! F/� ` [�eN]D5�V�bhV gN[�XYUfSAZAV Z;[�d d^Vνm g/BBdo]oSB[r2Ai ! 9

�D �� ^��?��c��!�p� �� ?�� !O7 F

0 200 400 600 800 1000 12000

50

100

150

200

250

300

q/]CI�<BehV42ni ! &�� ' <Be?5uV bN<BehVMZ;[Ydod^Vνm&gNBBdo] SB[g2ni ! 9

0 0.2 0.4 0.6 0.8 10

0.5

1

0 0.2 0.4 0.6 0.8 10

0.5

1

0 0.2 0.4 0.6 0.8 10

0.5

1

0 0.2 0.4 0.6 0.8 10

0.5

1

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

ν = 1

ν = 0.1

ν = 10

ν = 100 ν = 1000

qF]DI*<AejV62Ai ! 1!� � UaZB].�nX�VkZ;[Y] BnV ehV \M[_bjeN]9Z;],bj[YSAgN]oUfSB[cZ;] <ASAVνm g/BBdo]oSB[

!O7 & �� 4� �D ��^��?��c��!�p� �� p� �p� ��

` U*(J(A]oVf\ U bjeNU�5�V ej[X(t)

bhV do[ XdKA[

X(ti) = Pi i = 0, . . . , n

W <B]oSAZ;]

X(ti) =n∑

j=0

djNjk(t) = Pi i = 0, . . . , n

` U�5aeN[�\ U6B([YehX_] � ej]^gLU*dC5*[Yej[T]od�gL]^gLbN[Y\kV do] SB[4V ej[

N0 k(t0) . . . Nn k(t0)iii iiiN0 k(tn) . . . Nn k(tn)

d0iiidn

=

P0iiiPn

: d gL]^gLbN[Y\kV BAeN[4X_[�ZB[YS0bN[ KnV <BSAV g bjeN<BbLbN<AejV gNBAVfejgjV�� ]<AVfgN]�V (nV SAZBV!� BnU*]oXdK!��g/< U*IfSA]ej]CI0V XY]�gNUfSBU

k[Ydo[Y\M[YS0bj]�X_U*SAgL[4XE<;bj]C5a]DSBU*S S�<Bdodo]��n] SBU*d bjeN[ ]

tigNUfSAUMZB]ogLbN]oS0bN]��!�]<B] SnZ;]�d^V

\kV�bjeN]^X_[ �JSBUfS gL]oSJI*Ufd^V ej[fi�-/V do[ X_UfSngL]^Z;[YehV%$�] U*SB[ B([Ye��65uVfd [JSB[�d dML ]DB(U bN[4gL]9XdKA[!]tigN]oVfSBU

<BSA] . U*eN\M]��FV d bNej]o\ [�S0bN]�d^V \kV bNej]oXY[MXdKA[WgL] KAV0�WZ;]HbN]DBnU gNBAV ehgNUAi��B(U*gjgN]C(B]odo[6BAeNU�5�V ej[B([Ye��kXdKB[ SB[Ydod\L ]DBnUfbN[�gN]�# Z;]��] bNSB[ ��m m;XdKBUa[�S](([Ye?I %

ti < ti < ti+1 i = 0, . . . , n

d^Vk\kV�bjeN]^X_[ ej]ogN<Bd bjV<JI�<AV do\M[YS0bN[lSBUfS gN] S>IfUfd^V ej[fip: S If[�SB[YehV do[ V%(J(B]^V \MU�ZB[Y]`B><BS0bN]PiZBV ]oS0bN[�eNB(Ufd^V ej[� gN]BbjeNU�5�V SAU!]

ti[ g/<nXYX_[4gNgN]D5uVf\M[YS0bN[�gN](�ngjgNVfSBU ]

ti�fbNehV \M] bN[�d^V+. Ufej\)<Bd^V

ti+k =1

k + 1

∑

j = 0kti+j i = 0, . . . , n − k

" 5�5a]oVf\M[YS0bN[��gL[kgN] BBej[YSAZB[tk−1 = t0, . . . , tn−1 = tn

�]<B[Ydod UXdKB[kgN]HU bNbN]o[YSB[-�<BSnVX�<BeN5�VMgNBn[8$�$�V bjVAi

�D �� ^��?��c��!�p� �� ?�� !O7 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

q/]DI*<BehV 2Ai �%7 �s_OV*(J((U SnV�bjVfd [�� d [WX�<Be?5f[M] S�ejU*gjgNU gNUfSBUνm gNBBd ]oSB[�� ]<B[Ydodo]H] S (BdD<�gNUfSBU

X�<Be?5f[ Z;]`_��$�] [�e

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

q/]DI*<BehV)2ni �J! �Hm;V*IfUf\kV Z;] <BSAVkX�VV5�V dod U eN[4V do]C$8$�V bNU ] S0bN[�ejVf\M[YS0bN[ X_U*Sνm&gNBBdo] SB[

!�!87 �� 4� �D ��^��?��c��!�p� �� p� �p� ��

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

q/]DI*<BehVs2ni � � ��: dDd U�IfU R BJBBdo[�;ej[�Vfd ]D$�$4V�bNU X_U*Sνm g/BAd ]oSB[�# (BdD<�% �W_ �8$Y]o[Ye # 5*[YehZ;[3% ��_��8$Y]o[YeNm

gNBBdo] SB[C0

# eNU0gNgNU %

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

q/]DI*<BehVg2Ai � ,!�`-`< ��0eN[4V do]C$8$�V�bjU X_UfSνm&gNBBdo] SB[,# (BdD<�% �=_��$�] [�e # ejU*gjgNU % ��_ �8$Y]o[YeNm&gNBBdo] SA[

C0#�5f[YehZ;[ %

�� F��M��,��A��X�� * �T$��w�*� �� u�� \�T�� )!"��T ��u��*��\�\��4�w+� �� U� �� u��T� ��uu��)��l �� u�� %$��#�� 4��T�� X��T�*��T�� u �%�#�;��w�*��T��\�T�*� �� V�� # �U��?��J�T�� w�4 �� T�\�%��d��l��,�� %$�� "�� w��u�\��u� �� e�,�u��,+

h�g � .\7�7�.\`�8

5� C>f@ � �r�� D �)@ � �� B �C> d<s�

HtL V*BJBBejU*gjgL]o\kV%$�] U*SB[cZ;]�. <ASJ$Y]oUfSB] �)<AS"VfgNBn[YbLbjUW] \B(UfeNbjVfS*bj[ Z;[�d ' R� ` �]<AVfgN] �]<AVfS=mbjU d\L ] S0bj[Ye?BnU*doV*$Y]oUfSB[*i ` [Yd�ej[�gLbNU!� <AS \MU;Z;U Bn[�e V%BJBBejU*gjgN] \kV ej[)<ASAV ZAV�bjV . <BS>$Y]oUfSB[��V*BJBJ<BS0bjU �]<B[Ydod UWZB]9bNejU�5�V ej[g<BSAVWX�<BeN5�VM]oS0bN[Ye?B(Ufd^V S0bN[ ]FZAV�bN]/X_U*SBU*gjX_]D<;bN]vi�m;X_U�BnUWZ;[�d d^VgN[�I�<B[YS0bj[ bjejV bLbhV%$Y]oUfSA[ gjV edo��]<B[�d doU ZB]�[�gjV \M] ehV ej[)<ASAV 50]^V V d bN[YejSAV bN]D5uV V dodML ]oS0bN[Ye?B(Ufd^V�m$�] U*SB[� ] S�5*[�X_[WZB]?]oS*bj[Ye?BnU*doVfeN[M]OZBV�bj]?X_[�ejXdKB[�eN[�\MU ZB]?\M] SA] \M]D$�$�VfeN[kd^VZ;] !([�eN[�SJ$�V . ehVbhV do]nZBV bN]B[�d^V+. <BSJ$�] U*SB[ V%B>BBeNU0gNgN]o\MV bjVAi�-/V do[ gLU*dC<J$�] U*SB[ KAV Z=<B[ . U*SAZBVf\ [�S0bjV do]B\MU bj]C5a]��

• gNBn[4gNgNU ]�ZBV bN]HgNUfSBU gLU�I*I*[_bLbj]/V*Z"[�eNejUfej]FZB]�\ ]^gN<BejV*$Y]oUfSB[��<A] SAZB] � \M[�I*d ]oU SBUfS]oS*bj[Ye?BnU*doVfeNdo],[�gjV�bNbjV \M[�S*bj[�;\kVkZ;]9V*BJBBejU*gjgL]o\kV ejd ]'�ngjgNV bjV4<BSAVkXY[YeNbjVlbNUfdodo[YehV SJ$4V

• ]oS�\MUfd bN[ V%B>BBd ]^XYV*$Y]oUfSB](d^V��]<AV S0bj] bdo Z;[�]DZBV�bj]'�J\MUfd bNU6IfehV SAZB[ �]<B]oSAZ;]�<;bN]odo]C$8$�V ej[<BSAVkgjXdKB[�\MV ]oS0bN[Ye?B(Ufd^V�bNU*eN]oU)B>< �keN]^gN<Bd bhV ej[ ] SB[�0WX_]o[YS0bN[cZBVfd BJ<BS0bNUWZB]�5a]ogLbjVkZ;[�dX_U0g bjU XYUf\BJ<;bhV%$�] U*SAV do[,# gL] B([YSAgN] B(Uf]9V X_U*gjV X_Uf\B(UfeNbN[�eN[8(J(n[!d\L V%I�If]D<BS0bjV Z;]�<BSS�<BU�5fUMZBV bNU � %

RTd d^V dC<AXY[ Z;] �]<B[4g bj]`. V bLbN]��,] d�SBU0g bjeNU gNXYU*B(U�gNVfe?o�Z;[�gjX_ej]C5*[Yej[r<BS"]oSAgN] [�\M[ Z;]�ZBV bN]/[Y\6mBA] ej]oXY]��,Z;]^g bjeN]D(J<B] bN]�gN[�X_U*SAZ;U <BSnV. <BSJ$�] U*SB[lgjX_UfSAU*gjX_]D<;bjV

f�(X_UfS <AS�L Vfd bjejV . <BSJ$Y]oUfSA[

f∗XdKA[ \ ]oSB]o\M]C$8$Y]9d^VMZ;]ogLbjVfSJ$�VMXYUfS ]�ZBV�bj] i

�L�� *+ �3� � ��%'"#$��%' $&

m=<JBJB(UfSB]^V \MU"Z;] VV5f[�eN[ <ASAV . <BSJ$�] U*SB[fX_U*S0bN]oS]<nV] S <AS�] S0bj[Ye?5uVfd doU

[a, b]i"m;]oV bhV do[

. <ASJ$Y]oUfSB[�SBU bhVg/< {x1, . . . , xN} �/V*(J(B]^V \MUX_]oU/� <BS�] SngL]o[Y\M[�Z;]OX_U�BJBB]o[ (xi, f(xi))�

i = 1, . . . , Ni��U�Ifdo]oVf\MU bjeNU�5�VfeN[ <BSAV6. <BSJ$�] U*SB[

f∗ V*BJBAV eNbN[�SB[YS0bj[ Vfd doUMg/BAV*$Y]oU

V =< φ0(x), φ1(x), . . . , φn(x) >⊆ C([a, b])

!�!*!

!�! � ��o� ��4� �7 �� 4�� g�p� �� ^��p��

P0

P1

P2

P3

P4

P5

P6

q/]CI�<BejV 9Bi ! ��R BJBBejU*gjgL]o\kV%$�] U*SB[JZB]�. <BSJ$Y]oUfSA]

-/V do[ XdKB[�G|f − f∗| ≤ |f − g| ∀g ∈ V

m�5a]odC<>BJBAV SnZ;U-��<A[�gLbjV [ �]<AV*$Y]oUfSB[ U bNbN[�SB]oVf\MUN∑

i=1

|f(xi) − f∗(xi)|2 ≤N∑

i=1

|f(xi) − g(xi)|2#\9Bi ! %

Z;U�5*[g(x) =

∑ni=0 aiφi(x)

i4: d BBejU*(Bdo[Y\kV XdKB[kZ;U�5aeN[�\ U�eN]^gLU*dC5*[Yej[�DZ;[YbLbNU N b b R[HaE)ES< �ILN�O�<BHJCZ@QNJ<_I^<MCD<MCD< ��hDN >PReNJT]< gjV edo��

mina0,...,an

N∑

i=1

f(xi) −n∑

j=0

ajφj(xi)

2

m;XYeN]D5a]oVf\MU��]<B[4g bjU6BAeNU�(Bd [�\kV ]oS . Ufej\MV \kV�bjeN]^X_]^V do[��B(UfSB]^V \MU

f = (f(x1) · · · f(xn))T ∈ R

N a = (a0 · · · an)T ∈ R

n+1

A =

φ0(x1) . . . φn(x1)iii iiiφ0(xN ) . . . φn(xN )

N×n+13�[Ye X�<B]9]od�BBejU*(Bdo[Y\kV Z;]D5f[YS0bhVmin

a||f − Aa||22

︸︷︷︸

d(a)

` [Yej]C5a]^V \MU d^V+9Bi ! eN]^g/B([_bNbNU VfZak[�Z UfbLbN[�SB]^V \MU ∂d

∂ak

��Bn[�ek = 0, . . . , n

i P gNBBd ]^X_] bjVfSAZ;U]�X�V d^X_U*d ]DgN]9U bNbN]o[YSB[

−2N∑

i=1

f(xi) −n∑

j=0

ajφj(xi)

φk(xi) = 0

K�� :�1#6#1#m�m�1O.\]AN[1#c��nS�8�W%:�.'2�8�Z�c�1#m�1#N�c�8j1#6HQ�N�c�Q�8�:T:�Nf2�1�� k�Q�1#Nnt

|f | =N

X

i=1

f2(xi)

� �� p��p� �� c�o� � !�! ,

XdKA[cgL] BJ< �WV SAXdKB[ gjX_ej]C5*[Yej[JXYUf\M[

n∑

j=0

aj

(N∑

i=1

φj(xi)φk(xi)

)

=

N∑

i=1

f(xi)φk(xi) k = 0, . . . , n

mBX_ej]C5a]^V \MU�S�<BU�5�V \M[�S*bj[,�]<B[4g bOL <Bd bN]o\kV�[4g/BAeN[4gNgN] U*SB[c]oS . U*eN\kVW\kV bNej]oXY]oVfd [��n]oSAZ;]^XYVfSAZ;UXYUfS

[φj , φk] =

N∑

i=1

φj(xi)φk(xi)

RTd doUfehV

b =

[f, φ0]iii[f, φn]

M =

[φ0, φ0] . . . [φ0, φn]iii iii[φn, φ0] . . . [φn, φn]

: d;gN]ogLbN[�\kV!XdKB[OZ;U*(>(B]oVf\MUTej]ogNUfdD5f[�eN[��4Z;[YbLbjU ES<FESTm@�ILN�> @�jMj @p@ ��hDN�O�<BHPCD< C9HPRSIQNPj < �Ma =

bi H�V\kV�bjeN]^X_[

MKAV Z;]o\M[YSAgN] U*SB]

(n + 1) × (n + 1)[4Z$� (nV SAVfd [ 5*[Yej].�(XYV ej[WXdKB[

M = AT A��X_]oU ��gL[

AKnV�ejVfSJIfU \kVfgjgN] \MU V dodoUfehV

M�?gN]o\ \M[YbNej]oX�V�[?Z;[ �ASB] bjV B(U*gN] bj]C5�VAi

qnVfX�X_]^V \MU V*Z;[�gjgNU V d^XE<ASB[ XYUfSAgN]^Z;[YehV%$�] U*SB]OgN<Bdod UrgNBAV%$�] UVi ma[V%(J(B]^V \MU gNXY[Yd bNU

V = Πn� Vfd doUfehV (AV*g bhV XdKB[?do[�V*gNXY]ogjgN[?gL]^V SAUTZ;]^g bj] S0bN[pBn[�e/VV5*[Yej[

AXYUfScehV SJI*U�\kVfgjgL]o\MUJI

�]<B[4g bjU5uVfd [+B([Ye�d^V4(nVfgN[JX�V SBU*SB]oX�V��B([Ye ] BnU*d ]oSBU*\ ]9ZB]�_�[�eNSng bj[Y]oS [4ZW]oS I*[YSB[�ejVfd [ Bn[�eU�IfSB]>(AVfgN[ Z;]

ΠniHma[ <;bj] do]C$8$YU ]oS�5f[�XY[�doVg(nVfgN[ Z;[Ydod [ _tm gNBBd ]oSB[ {N0 k(x), . . . , Nn k(x)}XYUfS 5f[YbLbNU*eN[ [4g bj[�gNU Z;[�]DSBU;Z;]

T = {x0, . . . , xn+k}�Bd^V \kV bNej]oXY[JZ;]FX_UfdodoUaX�V%$�] U*SB[ �cZ;[�d

bj]CB(U

N0 k(x1) . . . Nn k(x1)iii iiiN0 k(xN ) . . . Nn k(xN )

R 0kSAXdK�� bhV do[�\kV�bNej]^X_[TV*(J(B]^V ejVfSJIfU \kVfgjgL]o\MUcZ;[�5*[ [�gN]^g bj[Yej[ <BSAVcgNU bNbNU*gN<AX�X_[�gjgN] U*SB[�Z;]n + 1

bj[Yej\M] SB]9bjV do]9ZBVkgNU;ZBZ;]^g . VfeN[Jd [cXYUfSAZB]C$�] U*SB]9Z;] ��] bNSB[ ��m m�KBU*\)(([Ye?IAiH�[ ZB] \M[YSngL]oUfSB]�Z;[�d d^V \MV bNej]oXY[

A# d^V \MV bNej]oXY[ ZB] X_UfdodoUaX�V%$�] U*SB[3%gB(U*gjgNUfSBU [�gjgL[�eN[

\MU*d bjU)IfehV SnZ;]�� ]<B] SnZ;]n]od,gL]^g bj[Y\kVMa = b

� ]oS If[�SB[YehV do[ <BS BBeNU�(Bdo[Y\kV \kV d^X_UfSnZ;]C$�] U*mSnV�bNUniHma]�<;bN]od ]D$�$4VlVfd doUfehV d^Vr. V�bNbNU*eN]D$�$4V%$Y]oUfSA[

QR# Z;U�5f[

Q� <ASAVl\kV bNej]oXY[ UfeNbNU�IfUfSnV do[�

d^VfgjX_]^V ] SnV d bN[YehV�bhVld^V SBUfej\kV2% � B(U*gLbNU

A = QReN]^gN<Bd bhV

||f − Aa||22 = ||QT f − Ra||22 = ||f − Ra||22ZBU�5f[

f = (f1 f2)�;X_U*S

f1 ∈ Rn+1

[R =

(R1

0

) X_U*SR1 ∈ R

(n+1)×(n+1)i H L [ �]<AV*$Y]oUfSB[

ZB]9\M] SB]o\MUWV dod U*ejV Z;]D5f[�S*bhV

mina

∥∥∥∥

f1 − R1a

f2

∥∥∥∥

2

2

= mina

(

‖f1 − R1a‖22

)

+ ‖f2‖22

" 5�5a]^V \M[YS0bN[��8B([Ye/X_U0g bjeN<J$�] U*SB[�AKAV ehV SJI*U \MV*gNgN]o\ U gN[�[?gNUfdoU gL[

R1�HSAUfScgN] SJI*Ufd^V ej[fi

3�U*]oXdK�� g/<Ad?bj[Yej\ ]oSB[ ‖f2‖22

SBUfS BnU0gNgN]oVf\MU . V e S]<Ad d^V�� ]<B[YdodoU"XdKB[�ej]o\MVfSB[�ZBV . V ej[ �ej]^gLU*dC5*[Yej[!]od�gN]ogLbN[�\MV

R1a = f1i

!�! 2 ��o� ��4� �7 �� 4�� g�p� �� ^��p��

�Q�)( *+ �!� �0 �1� -, &�31�#$2�

" [Yd�X�VfgNU6BnV ehV \M[_bjeN]^X_UkV%(>(B]oVf\MU6<AS <AS] SAgN]o[Y\M[cZ;]`B><BS0bN]Pi ∈ E

d�

i = 1, . . . , N[

5*U*Ifdo]^V \MUcbjeNU�5�V ej[JdoVMXE<AeN5�V

X(u) =n∑

i=0

aiφi(u)

XdKB[ V*BJBAV eNbN]o[YSA[JVfd doUkg/BAV*$Y]oUV =< φ0(u), . . . , φn(u) >

[�Z V%BJBAeNU0gNgN] \kVld\L ] SngL]o[Y\M[cZ;]BJ<AS*bj]0ZAV�bNUniFRT]�BJ<AS*bj];ZBV�bj]�?SA[�X_[4gNgjV ej]oU VfgjgLU;X_]^V ej[ <BSsBnV ehV \M[_bjeNU

ui[?bjV do[�gjX_[�d bhV ��gN]Cm

I*SB].�(XYV�bj]C5�V B([YeF] d]BBejU*(Bdo[Y\kV�Z;[�d dML V*BJBBejU*gjgL]o\kV%$�] U*SB[fi H�V . <BSJ$Y]oUfSA[?XdKB[H\M]oSB] \M]D$�$�[Yej[Y\MUgjV edo��

mina

N∑

i=1

‖Pi −X(ui)‖22

#\9Bi � %

�HV SBUfbjV�bjU XdKB[ ]oS ej[�Vfd bdolVfSAZ;ej[�(J(([ \ ]oSB]o\M]C$8$�V�bhVd(P,X) = minu∈[a,b] d(P,X(u))

�\kV��]<B[�gLbNU��k[�5a]^Z;[YS0bj[Y\M[YS0bN[k]o\BnU0gNgN]D(B] do[Bn[�e��]<B[�gLbN]oUfSA]HZ;]?X_U*\BBd [4gNgN] b?o X_U*\6B><;bjV%m$�] U*SAV do[fi�H�V �]<AVfS*bj] bdo

d(P,X(ui))BBej[YSnZ;[ ]odASBU*\ [!Z;] >P<FE)T NJCiO\NL@$hDG�j @)N �0[4Z�� BBej[�gjV

dD<BSJI*U doVrB([Ye?B([YSAZ;]^X_U*doVfeN[�Vfd d^VJbjVfSJIf[�S*bj[TZ;[�d d^V XE<Be?5�V [ BAV*gNgjV S0bN[ B([Yeuii�HtL [3�]<AV%$�] U*SB[

9Bi � BJ< �M[4gNgN[Yej[JgjX_ej] bLbjVMXYUf\M[�N∑

i=1

d∑

j=1

(pij − Xj(ui))2

=

=

d∑

j=1

N∑

i=1

(pij − Xj(ui))2

=

=

d∑

j=1

N∑

i=1

(

pij −n∑

k=0

akjφk(ui)

)2

: S��]<B[4g bjUW\MU;Z;U V*(J(B]^V \MUWej]oZ;UfbLbjUW] d�BBejU*(Bdo[Y\kV BnV ehV \M[_bjeN]^X_UkVdBBejU*(Bdo[Y\M]�. <ASJ$Y]oU%m

SAVfd ]vi ` U�50ej[Y\MU��]<B] SnZ;]9ej]ogNUfdD5f[�eN[dgL]^gLbN[Y\M]Ddo] SA[�V ej]9Z;[�dDbN]DBnU

Maj = bji

� � �� $� ��

��U�Ifdo]oVf\ UrbjeNU�5�VfeN[��OXYUf\M[ VfX�X_[YSASAV�bjU BBeN[4X_[4Z;[YS0bN[�\M[YS0bN[��O]od+BJ<BS0bNUu∗

i

Z;U�5*[(Pi −

X(u∗i ))X

′(u∗i ) = 0

i � bj] do]C$8$Y]^V \MU doV bj[�X_SA]oX�VZ;[_bNbjV b NPReNJIL@�Tm@�R G)HJRSRe@)G�TU<BHPC �lgN5a] dD<JB=mBB]^V \MUM]oS gN[Yej]o[JZ;]�-FV �adoUfe

Pi −X(u∗i ) ≈ (Pi −X(ui)) −X′(ui)(ui − u∗

i )m�<>BJBnU0g bhV BnU*]HejV*If]oUfSB[85fU*d \M[YS0bj[)BB]^XYXYUfd^V�d^V Z;]ogLbjVfSJ$�V . ehVu∗

i

[ui�/V dod U*ejV

X′(ui) ≈X′(u∗

i )�!�]<B]oSAZ;]

0 ≈ (Pi −X(ui))X′(ui) − (u∗

i − ui)|X′(ui)|2` VMXE<B]9ej]^XYVV5fU

u∗i = ui +

(Pi −X(ui))X′(ui)

|X′(ui)|2-/V do[BBeNU;XY[�Z;]o\M[YS0bNU BJ< �[4gNgN[Yej[MV%BJBBdo]^XYV�bjU BB]�� 5*Ufd bN[�`<;bN]od ]D$�$4V SAZBU X_Uf\M[ui] BJ<BS0bN]

Z;[�d d^V XE<Be?5�VkbNejU�5�V�bj] i " 5�5a]oVf\ [�S0bN[ gN] �ngjgNV <BSAV�bNUfdodo[Yej[YSJ$4V�gN<Bd dML [�gNBBeN[4gNgN]oUfSB[sBAeN[4X_[YSJmZ;[�S0bN[��SAUfS �OSB[�XY[�gjgNVfeN]oU V ejeN[4g bhV ehgL]��]<AVfS0bNU ]od0gN[�XYUfSAZBU \M[Y\)(AeNUTZB[YdodoV!gLU*\ \kV �OS�<Bdod Uni

� �D �� c�� p�� !�! 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6Punti da approssimarePoligono di controlloCurva di Bezier

q/]DI*<BehV69Bi � � � ] SA] \M] �]<AV*Z;ehV�bN]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Punti da approssimarePoligono di controlloCurva di Bezier

q/]DI*<BehV69Bi ,!� � ] SA] \M] �]<AV*Z;ehV�bN]

!�! @ ��o� ��4� �7 �� 4�� g�p� �� ^��p��

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

q/]DI*<BehV9;i 2!��3�V ehV \M[_bj[Ye�X_U*eNej[�X_bN]oUfS1# ZBV�bj]�VfZ V SnZBV \M[YS0bjU do]oSB[�VfeN[ %

−6 −4 −2 0 2 4 6−100

−50

0

50

100

150

DatiMinimi quadratiParameter correction IParameter correction IIParameter correction III

q/]DI*<BehV9;iG9(��3�V ehV \M[_bj[Ye�X_U*eNej[�X_bN]oUfS1# ZBV�bj]�ZBVMV SnZBV \M[YS0bjUMX�<J(B]^X_U/%

� �U��A�%$��5��w�*��%� �A��U� �U��%$��#�[�� z��" �� %$�� $��;�� u��%$��[�� U�� # ��'�%$�� $� �#)�*�� u�� 4�u��%��? ��4�� *��T��# ��[�� # ��[ ��i��*�,+

�eg�-�8�W%Qw.\9;:%8�W

6" BC>�# @�� +M<

H�[TXE<AeN5*[OU�!,gN[_bHU # XE<Be?5f[ BAVfejVfd do[Ydo[%�gNUfSBUg<BSMbj]CB(UcZ;]nX�<Be?5f[�XdKB[ KAVfSBSBUg<BSAVcZB]ogLbjVfSJ$�VXYU*gLbjVfS*bj[kZBV <ASAV XE<Be?5�V ]oSB]D$Y]^V do[kZBV�bhVBi �TV SASBU <BSAV SBU bj[�5*Ufdo[ ]o\BnU*eLbhV SJ$4V V do]C5*[Ydod UV*BJBBdo]oX�V�bj]C5*U!�BVfZ [�gN[Y\BB]oUMSB[Ydod^V4I*[�gLbN]oUfSB[ Z;[�]D\MU�50]o\M[YS0bN]9ZB]�(BehVfXYXY]D\M[�XYX�V SB]^X_] B([Ye�d^Vd^VV5fU*ejV*$Y]oUfSB[OZ;];\kV�bj[Yej]oVfd ]�� SB[�d d^V Z;[ �ASA]C$�] U*SB[ Z;];ej[�If]oUfSA]BZ;]abNU*d do[YehV SJ$4V��SB[�d d^V X_ej[�V*$Y]oUfSB[ZB]�gNXdKA[Ydo[_bNej]9Z;]9\MU;Z;[�d do]�If[�Uf\M[_bjeN]^X_]vi

� �� Z%��31��'"4#��%�&

5 �76�� 38�� NJT N hiC9N G�hiRS`JN b <BNJCZN Re@mr7HJj NPRe@X(t) : [a, b] → E

2 � @)> hiCC=hiIQ@�ReH

α ∈ R � ES<?>�@BA?CD<FE�G)@�G�hiRS`aN�H �E�@�T

Xα = X(t) + αN(t)

>�HJ`J@N(t)

�-<Mj3`J@$R)E�HPRe@�CZHJRSILNJj @^NJjMj N'G�hiRS`JN VR�Z [4gL[�\6BA] U��BgL[cXYUfSAgN]oZB[Yej]oVf\ U ] dFX_[YehXdKB]oU!�a[�gjgNU6KnV [3�]<AV%$�] U*SB[

X(t) =

(cos tsin t

)

` <BS!�]<B[l] d 5*[_bLbjUfej[JbjV S>If[YS0bj[J[ SBU*eN\kVfd [ gNUfSBU eN]^gNBn[YbLbN]D5�V \M[YS0bj[

t(t) =

(− sin tcos t

)

N(t) =

(cos tsin t

)

HFVkXE<Be?5�VlU�!,gN[_b�gB([YeNbjVfS*bjUkZBV�bhV ZAV

Xα(t) = (1 + α)

(cos tsin t

)

W <A[�gLbjV �rgN[Y\BBej[X<BSAV X_]oehX_UfSJ. [Yej[YSJ$4V��\kV-X�V \4(B]oV ] d ehV%I�If]oUAi : SAUfd bNej["]od+5*[YehgLU*eN[bhV SJI*[YS0bN[��]oZ;[�S0bN]^X_U V$�]<B[�d doU�Z;[Ydod^V�XE<Be?5�VrUfej]CI*] SAVfeN]^VBi : S If[YSA[YehV do[� VV5f[YSnZ;U <BSAV

!�! F

!�! & �� 7 �p�!�� +� �� c�

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Curva originaleCurva offset (α = 0.2)Curva offset (α = −0.4)

q/]CI�<BehV4@Ai ! � "�!DgN[_bTZ;[�d d^VMX_]oejXYUfS=. [�eN[�SJ$�V

X�<BeN5�VX(t) = (x(t), y(t))T

�B]od�5*[_bLbjUfej[!SAUfej\MVfd [ ej]ogN<Bd bjV

N(t) =1

√

x2 + y2

(−yx

)

W <B]oSAZ;]��AgN[ XYV d^X_U*d U d^V ZB[Yej]C5�V�bhVMZ;]Xα

dXα

dt=

dX

dt+ α

dN

dt

HtL [4g/BBej[�gjgN] U*SB[JZB]dN/dt

�

dN

dt=

1

|X(t)|3(

x(xy − xy)y(xy − xy)

)

A�]^X_U*ejZAV SAZ;U d^V ZB[ �ASB]D$Y]oUfSA[cZ;]9XE<Be?5�V�b?<BejV��]<B[�d doUkXdKB[ gL]DUfbLbN]o[YSA[ �

dN

dt= −κ(t)

(xy

)

RTd doUfehVld^VkZ;[�eN]D5�V�bjV Z;[�d d^VkXE<Be?5�V U !DgL[Yb ej]^g/<Bd bjV

Xα(t) = X(t) − ακ(t)X(t) = (1 − ακ(t))X(t)

: d�5*[YehgLU*eN[TbjVfSJIf[�S*bj[ V dodoV XE<Be?5�V U !,[�gN[_b ej]ogN<Bd bjV

tα(t) =1 − ακ(t)

|1 − ακ(t)|t(t)

�]<B]oSAZ;]9SAUfS0�cZ;[_bNbNUkXdKB[cgN[Jd^VMXE<Be?5�V�� eN[8IfUfd^V ej[!VfSAXdKB[ d^V X�<Be?5uV U�!,gN[_b gN]oV ej[�IfU*doVfeN[*i

� �D �p�� +� �� c� � �� ^�� 3�� o�o� � !�! 1

� �[( * ��1 ��& �� !&� & � ��%��!# � �31� ��1 ��#$%' �1�#

m=<JBJB(UfSB]^V \MU!ZB]*VV5f[�eN[ <ASAV XE<AeN5�VX(t)

[4Zg<BSsBJ<BS0bjUtC ∈ (a, b)

bhV do[?XdKB[κ(tc) = 1/α[

κ′(tC) 6= 0iHR dodoUfehV

Xα(tC)�g<BSAVkX�<Ag/BA]oZ;[ U*ejZ;]oSAVfeN]^V��a]oS=. V�bNbN]

Xα(t) = (1 − ακ(t))X(t) − ακ′(t)X(t)

' VfdoXYUfd^V SAZBUlbhV do[ ] StCU bLbj[YSB]^V \MU

Xα(tC) = −α

6=0 per Hp︷︸︸︷

κ′(tC) X(tC)︸︷︷︸

6=0 per la regolarita

6= 0

� bN]od ]D$�$4V SAZBUkd UkgN50]odD<JBJB(UWZ;]�-FV �adoUfe ej]ogN<Bd bjV

Xα(t) ≈ Xα(tC)︸︷︷︸

0

+Xα(tC)(t − tC)

XY] U/�

tα(t) ≈ Xα(tC)

|Xα(tC)|t − tC|t − tC |

]od�gN[�I*SBUkXYV \4(B]^V)BBejU*BAeN]oU ]oStC

��<A] SAZB] KBU <BSAVMXE<ng/BB]^Z;[ U*ejZ;]oSAVfeN]^VBi' U*SAgN]oZ;[�eN]^V \MUMVfZ;[4gNgNU ] d�X�VfgNU ]oSXE<B]

κ(tC) = 1α

κ′(tC) = 0κ′′(tC) 6= 0

ZB] \MU*gLbNej]^V \MUJXdKA[tC� <BS4B><BS0bNU g bjejVfUfehZ;]oSAV ej] Uni�m;V%B>BB]oVf\MU!��<;bj] do]C$8$�VfSAZ;U+Ifdo]Bg/5a]odC<JB>BB]

BAeN[4X_[�ZB[YS0bN]��AXdKB[Xα(tC) = 0

�B\M[YbNej[...

Xα(tC) = −ακ′′(tC)X(t)

W <A[�gLbNUkgL]DIfSA].�nX�VMXdKB[ doVMZ;[�eN]D5�V�bjV6BBej]o\MV ej]ogN<Bd bjV

Xα(t) ≈ 1

2

...

Xα(tC)(t − tC)2

[Z=<AS!�]<B[ SBUfS XYVf\)(B]^V ZB] gL[8IfSBUni W <B[4g bjU gN]CI*SB]��nXYV XdKB[ X%L � <AS�L ]oejeN[8IfUfd^V ej] b?o Z;[�d d^VBnV ehV \M[_bjeN]D$�$4V%$Y]oUfSA[�0SBUfSZ;[�d d^VMXE<Be?5�VBi

3�[�e4I*[YSB[�ejVfd ]D$�$4V ej[ ��<nV S0bNU ] S0bNejU;Z;U bNbNU��ASBUfehV��SAU bN]^V \MU XdKB[�dML [4gNgN[Yej[ X�<AgNBB]oZB[�UB><BS0bNUWgLbNehV UfehZ;]oSAV ej]oUlZ;]DBn[�SAZ;[lZBVfd do[gBnUfbN[�SJ$Y[cZ;[�d doUkg/5a]odC<>BJBnUkZ;]�-FV �adoUfe3�0]oS=. V bLbN]

κ(m+1)(tC) 6= 0 →{

m pari ⇒ cuspidem dispari ⇒ punto straordinario

3�U*]{

κ(t) < 1α ⇒ tα(t) = t(t)

κ(t) > 1α ⇒ tα(t) = −t(t)

! �%7 �� 7 �p�!�� +� �� c�

−2 −1 0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

−1 −0.5 0 0.5 11

1.5

2

2.5

3

0 20 40 60 80 1000

5

10

15

20ddX

dXX

κ

q/]DI*<BehV@Bi � ��3�V ehV%((Ufd^V��KBU;Z;U*I*ejV*BJK�[ X�<BeN5�V bN<BehV

� � � � �!&� �'# �'%� �0 �1� �. � �

P gjV \M]oSB]^V \MU�d^V XE<AeN5�V�U�!,gN[_b ZB]+<ASAV BAVfejV*(nU*doVAi H L [ �]<AV%$�] U*SB[ BAVfejVf\M[_bNej]^XYV�Z;[�d d^VBAVfejV*(nU*doV,��

X(t) =

(tt2

)

W <B]oSAZ;]�Z;[�eN]D5�V�bhV BBej] \kV!�,ZB[Yej]C5�V�bhV�gN[�XYUfSAZBV!�,SBUfej\kV do[l[MXE<Be?5�V�bN<AejV gLU*SBU�ej]ogNB([_bLbj]C5�V%m\M[YS0bj[�

X(t) =

(12t

)

X(t) =

(02

)

N(t) =1√

1 + 4t2

(−2t1

)

κ(t) =2

(1 + 4t2)3/2

A�]^gN<Bd bhVM] S BAVfeLbj]oXYUfd^V ej[JXdKA[J]od�\kV*gNgN] \MUMZ;[Ydod^VkXE<Be?5�V�bN<AejV��2��]<B]oSAZ;]/gL[gBBej[YSnZ;U

α <1/2

] d�gN[�I*SBUkZ;[Yd`5f[YehgNUfej[TbhV SJI*[YS0bN[ �cgN[Y\BBej[cX_UfSnX_UfehZ;[*i?ma[ ] S�5*[�X_[gBBej[YSnZ;Uα ≥ 1/2V*(J(B]^V \MUkXdKB[

κ(t) = 1/αKAVMZ=<A[cgLU*dC<J$�] U*SB]�� Bn[�e

α = 1/2gN]�KAV<BSAVkZBU*BJBB]^V ejV*Z;]^X_[

]oSt = 0

� �]<B]oSAZ;]HV*(J(B]^V \MU <AS BJ<AS*bjUg bjejVfUfehZ;] SnV ej] Uni ma[M]oS�5f[�XY[α > 1/2

V%(J(B]^V \MUI*ejV��nXYVf\ [�S0bN[ Z=<B[ XE<AgNBB]^Z;[ ]oS

t1, t2#�BJ<BS0bN]�XYUfS doVM\M[4Z;[�gN] \kVMXE<AeN5�V�b?<BehV�% i

� �q ;� �� c� ��^� �c� � �o�� 3� ! �=!

−2 −1 0 1 20.5

1

1.5

2

2.5

3

3.5

4

4.5

−10 −8 −6 −4 −2 0 2−4

−3

−2

−1

0

1

2

3

4

−100 −50 0 50 100−400

−350

−300

−250

−200

−150

−100

−50

0

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

−8

καddXα

dXα

Xα

q/]DI*<BehV4@Ai ,!� "�!,gN[_b BAVfejV*(nU*doV!��KAUaZBU*IfehV%B>K�[cXE<AeN5�V�b?<BehV XYUfSα = 0.5

−1 −0.5 0 0.5 11

1.5

2

2.5

3

3.5

4

4.5

−25 −20 −15 −10 −5 0 5−10

−5

0

5

10

−300 −200 −100 0 100 200 300−1000

−800

−600

−400

−200

0

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5x 10

−6

καddXα

dXα

Xα

q/]DI*<BehV4@Ai 2�� "�!,gN[_b BAVfejV*(nU*doV!��KAUaZBU*IfehV%B>K�[cXE<AeN5�V�b?<BehV XYUfSα = 1.2

! � � �� 7 �p�!�� +� �� c�

−3 −2 −1 0 1 2 3−1

0

1

2

3

4

1 2 3 4 5−5

0

5

−40 −20 0 20 400

20

40

60

80

100

120

140

160

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3x 10

−15

καddXα

dXα

Xα

q/]DI*<BehV6@BiG9(� "�!,gN[_b BnV ehV%((Ufd^V��KBU;Z;U*I*ejV*BJK�[ X�<BeN5�V bN<BehV XYUfSα = −0.2

$e��F@�� > ��

� !�� 3�VfUfdoU � V ehX_[Ydodo] SB]�� ' V ejdoUkm=(nU*ejZ;U*SB[� NJj G[HPj H ��H9]DI*<BU*eN] P Z;] bjUfej[ ! 1*1 �

� �� -�U�(B]oV*g "![_bj]�**[Ye3�� <J(([YeNb43�V eNbNd��: eN[�SB[ � �aSAV!� P d ]^gNV*(n[YbNK-mBXdKBd [8Ifd�� X�C9N �]I^<BG)NTmNPCDT H�� K$Re@�`J@-<MC=TUR[H�>Ph�O$<BHPC9@^N � ��

ε� �*7*7*7

� , � �HV do[Yej]oVfSBU ' Uf\M]oSAX_]oUfdo]�� CZNJj <FES<��hiIL@$RS<BG)N \M[_bNU;Z;]��;\MU;Z;[Ydod ]��;V%BJBAd ]^XYV*$Y]oUfSB]vi P Z9i� X �JehV � m �T] dod ! 1�1�9

� 2 � �J[YehV d^Z qAV ej]oS �� hiRS`J@SEpNPC9>�:9hiR �$N G)@SE��$HPR HPI b hiTm@�R � �@[> ��@)HJIL@�TURS<BG � @SES< r C �N b R[NJT]<BG[NPj9r hi @ inR�X�VfZ;[�\ ]^X+3Hej[�gjg �=: SAXfi ! 1�&*&� 9 �� U0gL[�. �TU*gjXdKB[ * � ` ] [YbN[�e HFVfgjgL[�e �� HJI b hiT @$R � @)> ��@)HPIQ@�TURS<BG � @$E)< r C i R v

3/[YbN[�ejg �JHDbhZDi ! 1�1*,

! � ,

2004 2005 - nepero.net · AeN[4gL[ SbhV %$ ] USB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

Documents

Transcript of 2004 2005 - nepero.net · AeN[4gL[ SbhV %$ ] USB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

2004 2005 - nepero.net · AeN[4gL[ S*bhV %$ ] U*SB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

Documents

Transcript of 2004 2005 - nepero.net · AeN[4gL[ S*bhV %$ ] U*SB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

2004 2005 - nepero.net · AeN[4gL[ SbhV %$ ] USB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...

Transcript of 2004 2005 - nepero.net · AeN[4gL[ SbhV %$ ] USB[JZB] X E< Be?5 f[X_U*SB]^X dK B[iJi e2 1 h T4...