Free-form design using axial curve-pairs

K.C. HuiCUHK

Computer-Aided Design 34(2002)583-595

OUTLINE

1.About Author

2.Overall View of The paper

3.Previous Work

4.Axial Curve-pairs

5.Implementation and Results

6.Conclusions

Kin-chuen Hui 许健泉

Professor Department of Automation and Com

puter-Aided Engineering, CUHK

http://www2.acae.cuhk.edu.hk/~kchui/

Overall View of the Paper What problem does the paper

solve?

Freeform deformation of 3D shapes.

The essence of the paper:

Construct a local coordinate frame by a curve-pair.

Previous Work

Free-form deformation(FFD), Sederberg and Parry

Initially propose Skeleton-based technique,Burtnyk

Paper Link Using wires for deformation, Singh and Fiume

Paper Link Axial deformation techinque, Lazarus

Paper Link

Axial deformation technique

1. Basic idea of the technique

2. Axial Space—— A(C,l)

Defined by a curve C(t), and a local coordinate system l(t)=[lx(t), ly(t), lz(t)] on the curve.

P = (t, u, v, w)

3. Instance of an axial space

t = t0,the local coordinate frame.

4. Conversion of a point P in A(C,l) to 3D

f: R4 → R3 P = f(t,u,v,w)= C(t)+ulx(t)+vly(t)+wlz(t)

5. Reverse conversion: f-1

f-1 : R3 → R4

The value of t is generally decided by PN

where PN is closest to P,

lz(t) is the direction of the tangent at C(t), hence:

The point P in A(C,l) is expressed as

The major problem of the axial curve deforamtion:

Lack of control on the local coordinate frame of the axial curve

Cannot be twisted by manipulating the axial curve.

Framing a curve1. Frenet Frame

No user control of the orientation of the

C’’(t) vanishes.

2.Direction curve approach, Lossing and Eshleman

Axial curve-pair technique

Cannot be control intuitively

3. Local coordinate frame of a curve-pair

the coordinate frame at PN

C(t): Primary curve

CD(s): Orientation curve

PD: the intersection of CD(s) with a plane passing through PN and having a normal direction C’(t).

Problem of the Coordinate frame:

Considerable amount of computation for getting PN.

Improvement:

PD is obtained by projecting the point CD(t) to the plane

Local coordinate frame of a curve-pair

Axial curve-pair

An ordered pair (C, CD), | C(t) - CD(t)|≤ r

The construction of orientation curve

The orientation curve lies within a circular tube

Similar to construct an offset of the primary curve

Primary curve C(t) is a B-Spline curve

The process of construction is below:

(a).

(b).

(c).

The detailed process is the same to the process of

adjusting the local coordinate frame.

Manipulating axial curve-pairs

Primary curve C(t)

Orientation curve CD(t)

where

Simple approach to adjusting CD(t) when moving C(t)

→

→

Problem of the simple approach

(a).

(b). Overlapping BACK

New approach

the local coordinate of the vector relative to Pi keep constant while relocating Pi .

The local coordinate frame at Pi is specified with

a polygon tangent at Pi

a vector normal to the polygon tangent.

Polygon tangent

Give a polygon with vertices Pi , 0<i<n, the polygon tangent ti at Pi is

Local coordinate frame at a control point

The frame at Pi is given by the unit vectors

Where ti is the polygon tangent at Pi ,

Configuration of a curve-pair

The set of all the tuples

where

Specify the new position of qi after moving Pi

where

Comparing effect

Twisting the curve-pair

Rotation of qi about ti

Keep the configuration

The axial skeletal representation

The hierarchy of axially represented shapes.

Axial Skeletal Representation(ASR) of the object.

Implementation and results

Single axial ASR

The deformed dolphin model

A vase with the dolphin as decorative component

Construction of a ribbon knot

Construction of a leave pattern

Deformation of a squirrel shaped brooch

Conclusions

(a). Propose a new method to construct the local coordinate frame.

(b). Using a hierarchy of axial curve-pairs to constitute a complex object.

Thank you!Thank you!

Supplementary1.Burtnyk N, Wein M. Interactive skeleton

techniques for enhancing motion dynamics in key frame animation. CACM 1976; Oct:546-69.

2. Singh K, Fiume E. Wires: a geometric deformation technique. Proc.SIGGRAPH 98 1998:405-14.

3.Lazarus F, Coquillart S, Jancene P. Axial deformations: an intuitive deformation technique. CAD 1994:26(8):607-13.

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