Frames of Reference in Space - Rotation Transforms for Computer Graphics - page 153

which agrees perfectly. Naturally, this can be combined with a translation transform
using

⎡ ⎤

a 2 K +

cos α bK + c sin αacK − b sin α 0

⎣ ⎦

2 K

− + +

abK c sin α cos α cK a sin α 0

R − 1

α, n T − 1

t x ,t y ,t z = 2 K

acK + b sin αbcK − a sin α + cos α 0

0 0 0 1

⎡
⎣

⎤
⎦

−

100 t x

−

010 t y

×

.

001 t z

000 1

−

10.7 Summary

Hopefully, this chapter has covered most of the scenarios involving rotated and
translated frames of reference in 3D space. Although composite rotation transforms
offer a powerful mechanism for creating complex rotations, they are difficult to
visualise and suffer from gimbal lock. Perhaps, the most useful transform is for ro-
tating a frame about an arbitrary axis. For completeness, the important transforms
are summarised below.

10.7.1 Summary of Transforms

Translating a frame

⎡
⎣

⎤
⎦

100 − t x

010 − t y

T − 1

t x ,t y ,t z =

.

001 t z

000 1

−

Rotating a frame about a Cartesian axis

⎡ ⎤

1 0 0

R − 1

⎣ ⎦

α,x =

0 cos α sin α

0 − sin α cos α

⎡ ⎤

cos α sin α

01 0
sin α

0 −

R − 1

⎣ ⎦

α,y =

0 cos α

⎡ ⎤

cos α sin α 0

⎣ ⎦ .

R − 1

α,z =

− sin α cos α 0

0 0 1