Quaternions William Rowan Hamilton (1805-1865) Algebraic couples - - PowerPoint PPT Presentation

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Dec 15, 2022 501 likes •606 views

Quaternions William Rowan Hamilton (1805-1865) Algebraic couples (complex number) 1833 x iy 2 where i 1 Quaternions William Rowan Hamilton (1805-1865) Algebraic couples (complex number) 1833 x iy 2

SLIDE 1

Quaternions

William Rowan Hamilton (1805-1865)

– Algebraic couples (complex number) 1833

iy x  1

  i

where

SLIDE 2

Quaternions

William Rowan Hamilton (1805-1865)

– Algebraic couples (complex number) 1833 – Quaternions 1843

iy x  1

  i

where

kz jy ix w    j ik i kj k ji j ki i jk k ij ijk k j i               , , , , 1

2 2 2

where

SLIDE 3

Quaternions

William Thomson

“… though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.”

Arthur Cayley

“… which contained everything but had to be unfolded into another form before it could be understood.”

SLIDE 4

Unit Quaternions

Unit quaternions represent 3D rotations

) , ( ) , , , ( v q w z y x w kz jy ix w      

1

2 2 2 2

    z y x w

3

S

SLIDE 5

v ˆ



Rotation about an Arbitrary Axis

Rotation about axis by angle

1 

  qpq p ) , , , ( z y x  p

where

) , , ( z y x

) , , ( z y x   

Pur urely y Im Imaginary Qua uaternion

v ˆ 

       2 sin ˆ , 2 cos   v q

SLIDE 6

Unit Quaternion Algebra

Identity
Multiplication
Inverse
Unit quaternion space is

– closed under multiplication and inverse, – but not closed under addition and subtraction

) , ( ) , )( , (

2 1 1 2 2 1 2 1 2 1 2 2 1 1 2 1

v v v v v v v v q q        w w w w w w

) /( ) , , , ( ) /( ) , , , (

2 2 2 2 2 2 2 2 1

z y x w z y x w z y x w z y x w            



q

) , , , 1 (  q

SLIDE 7

Unit Quaternion Algebra

Antipodal equivalence

– q and –q represent the same rotation – 2-to-1 mapping between S and SO(3) – Twice as fast as in SO(3)

) ( ) ( p p

q q 

 R R

3

SLIDE 8

Rotation Composition

Rotation by a matrix
Rotation by a unit quaternion
Composition of Matrices (or Unit quaternions) is

simple multiplication

1 

  qvq v Mv Mv v   v M M v

1 2

 

1 2 1 1 1 2  

  q vq vq q q v

SLIDE 9

Acknowledgement

Acknowledgement: Some materials come from the lecture slides of

–

Prof. Jehee Lee, SNU, http://mrl.snu.ac.kr/courses/CourseGraphics/index_2017spring.html