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.

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 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

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

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

Definition of Quaternion

Operations - 1

Operations - 2

Operations - 3

Unit Quaternion

Operations - 4

Operation Rules

Quaternion Transforms

0wp Note:

Proof:

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