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