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 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 i 2 = j 2 = k 2 = i j k = −1 & cut it on a stone of this bridge

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

Compact (4 components) Can show that represents a rotation of 2  radians around u q of p Unit quaternions are perfect for rotations! 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

Definition of Quaternion

Operations - 1

Operations - 2

Operations - 3

Unit Quaternion

Operations - 4

Operation Rules

Quaternion Transforms Note:

Proof: See