Quaternion 靜宜大學資工系 蔡奇偉副教授 2010. 大綱 History of Quaternions Definition...
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Transcript of Quaternion 靜宜大學資工系 蔡奇偉副教授 2010. 大綱 History of Quaternions Definition...
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Quaternion靜宜大學資工系蔡奇偉副教授
2010
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大綱 History of Quaternions Definition of Quaternion Operations Unit Quaternion Operation Rules Quaternion Transforms Matrix Conversion
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History of QuaternionsIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i2 = j2 = k2 = i j k = −1 & cut it on a stone of this bridge
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Quaternions
Extension of imaginary numbers Avoids gimbal lock that the Euler could produce Focus on unit quaternions:
wzyx
wzyxwv
qkqjqiq
qqqqq
),,,(),(ˆ qq
1)ˆ( 2222 wzyx qqqqn q
A unit quaternion is:
ˆ (sin ,cos ) where 1q q q u u
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Compact (4 components) Can show that represents a rotation of 2f radians around uq of p
Unit quaternions are perfect for rotations!
1ˆ ˆ ˆ qpq
ˆ (sin ,cos )qf fq u
That is: a unit quaternion represent a rotation as a rotation axis and an angle
OpenGL: glRotatef(ux,uy,uz,angle); Interpolation from one quaternion to another is much
simpler, and gives optimal results
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Definition of Quaternion
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Operations - 1
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Operations - 2
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Operations - 3
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Unit Quaternion
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Operations - 4
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Operation Rules
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Quaternion Transforms
0wp Note:
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Proof:
See http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
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