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Quaternion

Quaternion. 靜宜大學資工系 蔡奇偉副教授 2010. 大綱. History of Quaternions Definition of Quaternion Operations Unit Quaternion Operation Rules Quaternion Transforms Matrix Conversion. History of Quaternions.

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Quaternion

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  1. Quaternion 靜宜大學資工系 蔡奇偉副教授 2010

  2. 大綱 History of Quaternions Definition of Quaternion Operations Unit Quaternion Operation Rules Quaternion Transforms Matrix Conversion

  3. History of Quaternions In 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

  4. Quaternions • Extension of imaginary numbers • Avoids gimbal lock that the Euler could produce • Focus on unit quaternions: • A unit quaternion is:

  5. Unit quaternions are perfect for rotations! • Compact (4 components) • Can show that represents a rotation of 2fradians around uqof p • 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

  6. Definition of Quaternion

  7. Operations - 1

  8. Operations - 2

  9. Operations - 3

  10. Unit Quaternion

  11. Operations - 4

  12. Operation Rules

  13. Quaternion Transforms Note:

  14. Proof: See http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

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