Game Physics Engine Development, 2nd Edition

502 Appendix C Mathematics Summary

A3×3matrix,

M =

⎡

⎢

⎣

abc

def

ghi

⎤

⎥

⎦

has its inverse given by

M

−1

=

1

det M

⎡

⎢

⎣

ei − fh ch −bi bf − ce

fg −di ai −cg cd − af

dh − eg bg − ah ae −bd

⎤

⎥

⎦

where det A is the determinant of the matrix,

det M =aei + dhc +gbf −ahf −gec − dbi

Aquaternion,

θ

ˆ

=

⎡

⎢

⎢

⎢

⎣

w

x

y

z

⎤

⎥

⎥

⎥

⎦

represents the same rotation as the following matrix:

 =

⎡

⎢

⎣

1 −(2y

2

+2z

2

) 2xy + 2zw 2xz −2yw

2xy −2zw 1 −(2x

2

+2z

2

) 2yz +2xw

2xz +2yw 2yz −2xw 1 − (2x

2

+2y

2

)

⎤

⎥

⎦

A transformation matrix M

t

can be changed into a new c oordinate system, using

a transform M

b

to the new coordinate system according to

M



t

=M

b

M

t

M

−1

b

C.4 Integration

To update an object’s position,

p



=p +

˙

pt +

1

2

¨

pt

2