Quaternion - PowerPoint Presentation

Slide1

Quaternion

靜宜大學資工系

蔡奇偉副教授

2010Slide2

大綱

History of

Quaternions

Definition of Quaternion

Operations

Unit Quaternion

Operation Rules

Quaternion Transforms

Matrix ConversionSlide3

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 bridgeSlide4

Quaternions

Extension of imaginary numbers

Avoids

gimbal lock

that the Euler could produce

Focus on unit quaternions:

A unit quaternion is:Slide5

Compact (4 components)

Can show that r

epresents

a

rotation

of 2

f 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 resultsSlide6

Definition of QuaternionSlide7
Slide8

Operations - 1Slide9

Operations - 2Slide10

Operations - 3Slide11

Unit QuaternionSlide12

Operations - 4Slide13

Operation RulesSlide14

Quaternion Transforms

Note:Slide15

Proof:

See http

://en.wikipedia.org/wiki/Quaternions_and_spatial_rotationSlide16