Frames of Reference in the Plane - Rotation Transforms for Computer Graphics - page 113

sin β e 12 ) (x t y ) e 2

p =

+ −

t x ) e 1 +

−

( cos β (y

= −

(x cos β t x cos β) e 1 + (y cos β − t y cos β) e 2

− (x sin β − t x sin β) e 2 + (y sin β − t y sin β) e 1

= (x cos β + y sin β − t x cos β − t y sin β) e 1

+ ( − x sin β + y cos β + t x sin β − t y cos β) e 2

or in matrix form
⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

x
y 1

− −

cos β sin β t x cos β t y sin β x

y
1

− sin β cos βt x sin β − t y cos β

0 0 1

which is identical to ( 8.1 ).

8.6 Summary

In this chapter we have discovered that if a transform such as T t x ,t y or R β is used

for moving points, whilst keeping the frame fixed, their inverses T − 1

t x ,t y and R − β can
be used for moving the frame, whilst keeping the point fixed. It goes without saying
that the converse also holds, in that we could have declared a transform for moving
a frame, and its inverse could be used for moving a point.
We have also seen how geometric algebra provides an alternative approach to
transforms based upon vectors, bivectors and rotors, and can undertake the same
tasks.
In order to show the patterns that exist between these two mathematical ap-
proaches, all the commands are summarised.

8.6.1 Summary of Matrix Transforms

Given

⎡ ⎤

10 t x
01 t y
00 1

⎣ ⎦

T t x ,t y =

⎡ ⎤

cos β − sin β 0

⎣ ⎦ .

R β = sin β cos β 0

0 0 1

Translate frame

⎡ ⎤

10 − t x

⎣ ⎦ .

T − 1

t x ,t y =

01 − t y

00 1