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 ID: 377865
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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 QuaternionSlide7Slide8
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