3D Game Engine Design 1

3D Game Engine Design3D Game Engine DesignCh. 2.3.Ch. 2.3.

3D MAP LAB

3D Game Engine Design 2

QuaternionQuaternion

Quaternion Algebra Def >

A Quaternion is defined by q = w + xi + yj + zk where w, x, y, z are in R and

Def> Let

then

3D Game Engine Design 3

Quaternion(cont’d)Quaternion(cont’d)

2222* zyxwqq

3D Game Engine Design 4

Quaternion(cont’d)Quaternion(cont’d)

Thm>Let then and let then quaternion multiplication can be defined using vector dot product and cross product

Thm>A quaternion q may be also viewed as a 4D vector( w, x, y, z).then the dot product of two quaternion is

vwq ˆzkyjxiv ˆ ),,(ˆ zyxv

kxyyxzwzwjzxxzywywiyzzyxwxwzzyyxxww

xyyxzxxzyzzyzwywxwzwywxwzzyyxxww

vvvwvwvvwwvwvwqq

)()()()(

),,(),,(),,()(

ˆˆˆˆ)ˆˆ()ˆ)(ˆ(

1010011010100110

1010011010101010

101010101010

01010110101010101010

1001101010110010

)( *101010101010 qqWzzyyxxwwqq

3D Game Engine Design 5

Quaternion(cont’d)Quaternion(cont’d)

Def> A unit quaternion is a quaternion q for which

Thm> The inverse of unit quaternion and the product of unit quat

ernion are themselves unit quaternion.

Unit quaternion can be represented by

1)( qN

kw

zjw

yiw

xuandwwwhere

uq

kw

zjw

yiw

xwwq

zyxwzkyjxiwq

222

2

222

2

2222

111ˆ,1sin,cos

,sinˆcos

)111

(1

1,

3D Game Engine Design 6

Quaternion(cont’d)Quaternion(cont’d)

has length 1. However observe that the quaternion product similar to unit-length complex numbers

Euler’s identity for complex numbers generalizes to quaternion.

Thm> The power of unit quaternion

The logarithm of unit quaternion

),,(ˆ 210210 uuuvectorandkujuiuu

1ˆˆ uu sincos iz

sinˆcos:

sincos:'ˆ ueformGeneral

ieIdnetitysEuleru

i

)sin(ˆ)cos()sinˆ(cos tutuq tt

ueuq ueee ˆ)(log)sinˆ(coslog)(log ˆ

3D Game Engine Design 7

Quaternion(cont’d)Quaternion(cont’d)

Caution : standard identities are not allowed because of non-commutative of quaternion.

That is, )log(loglog qpqpandeee qpqp

3D Game Engine Design 8

Quaternion(cont’d)Quaternion(cont’d) Relationship to quaternion to rotation Thm>

Unit quaternion represents the rotation of the 3D vector by an angle about the 3D axis . And rotated vector is represented by . Proof>

it is enough to show that R(v) is 3D vector R(v) is length preserving function. R(v) is linear transformation. R(v) doesn’t have reflection component. Rotation axis is really u. ( 1) Rotation angle (2) We will prove (1),(2)

sinˆcos uq

v 2 u*ˆ)ˆ( qvqvR

2

3D Game Engine Design 9

Quaternion(cont’d)Quaternion(cont’d)

(1)

(2)

uu

uu

uuuuu

ququR

ˆ)sin(cosˆsinˆsincossincoscosˆ

)sinˆ)(cossincosˆ()sinˆ(cosˆ)sinˆ(cos

ˆ)ˆ(

22

22

*

)ˆ(ˆ)cos(ˆˆ

ˆˆˆ,ˆˆˆ,ˆˆˆ,0ˆˆˆˆˆˆ,.ˆˆ,ˆ

*

*

qvqvthenqvqtoangleanbyrotatedvvectorthethatassum

vuwanduwvwvuandwvwuvuisthatvectorslorthonormaofsethandedrightabewandvulet

u

v w

2

)ˆ( *qvq

3D Game Engine Design 10

Quaternion(cont’d)Quaternion(cont’d)

2)2cos(

])(sin)[(cos)]cossin2(ˆ)(sin)[(cos

)]cossin)ˆˆˆˆ()(sin)[(cos)cos(,

,ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ]cossin)ˆˆˆˆ()(sin)[(cos

])(sin)ˆˆ(cossinˆˆˆcossinˆˆ)(cosˆ[)]sinˆˆcosˆ)(sinˆˆcosˆ[(

)]sinˆ(cosˆ)sinˆ(cosˆ[)ˆˆ()ˆ(ˆ

22

22

22

22

22222

***

proofthecompletethis

WuW

vuvuWlyconsequent

uvwvwvwvuvandwuvuvuvuvnowvuvuW

uvvuvuvvWuvvuvvW

uvuvWqvqvWqvqv

3D Game Engine Design 11

Quaternion(cont’d)Quaternion(cont’d)

Conversion between various rotation representations

Angle-axis rotation

Quaternion rotation Rotation matrix

3D Game Engine Design 12

Quaternion(cont’d)Quaternion(cont’d)

Conversion between angle-axis and rotation matrix Thm> angle-axis to rotation matrix

00

0,)cos1()(sin

ˆ

01

02

12

2

2,1,0

uuuuuu

S

andmatrixidentitytheisIwhereSSIR

ismatrixrotationingcorrespondthenrotation

ofaxislengthunitthebeuuuuandrotationofangleanbeLet

3D Game Engine Design 13

Quaternion(cont’d)Quaternion(cont’d) Thm> Rotation matrix to angle-axis

00

0,)sin2(

2/)1)((cos:)(

??

01

02

12

221100

222120

121110

020100

uuuuuu

SwhereSRR

RtracethenrrrRtracedefine

axisanglethenrrrrrrrrr

R

T

=0 or 180(??)

3D Game Engine Design 14

Quaternion(cont’d)Quaternion(cont’d)

Conversion between quaternion and angle-axis Thm> angle-axis to quaternion

Thm> quaternion to angle-axis

).2/sin(),2/sin(),2/sin(),2/cos(,

),,(1),)(2/sin()2/cos(

210

210

2222210

uzanduyuxwarequaterniontheofcomponenttheaxisandangletheGiven

uuuuaxistheaboutbyrotationrepresentzyxwkujuiuzkyjxiwq

21

2222210

1/),,()(cos2

1),)(2/sin()2/cos(

wzyxuandw

zyxwkujuiuzkyjxiwq

3D Game Engine Design 15

Quaternion(cont’d)Quaternion(cont’d)

Conversion between quaternion and rotation matrix Thm> quaternion to rotation matrix

Thm> rotation matrix to quaternion

22

22

22

221222222221222222221

??

yxwxyzwyxzwxyzzxwzxywyxzwzxyzy

R

matrixRotationthenzkyjxiwq

)4/()(),4/()(),4/()(2/1)(

2/)1)((cos:)(

???

100102202112

221100

222120

121110

020100

wrrzwrrywrrxRtracewthen

RtracethenrrrRtracedefine

zkyjxiwqthenrrrrrrrrr

R