Frames of Reference in the Plane - Rotation Transforms for Computer Graphics - page 104

Let's assume that a tree is located in a 3D frame of reference labelled XYZ and
the train's frame of reference is labelled X Y Z . When the two frames are coinci-
dent, the tree has identical coordinates in both frames. However, if the train's frame
X Y Z is translated by the transform T relative to XYZ , the tree's coordinates rel-
ative to the train have effectively moved in the opposite direction given by T − 1 .
Similarly, let's assume that a desk is located in a frame of reference labelled
XYZ and a chair's frame of reference is labelled X Y Z . When the two frames are
coincident, the desk has identical coordinates in both frames. However, if the chair's
frame X Y Z is rotated by the transform R relative to XYZ , the desk's coordinates
relative to the chair have effectively rotated in the opposite direction given by R − 1 .

8.3 Matrix Transforms

Having established the equivalence between transforms for moving points in a fixed
frame, and the inverse transforms for fixed points in a moving frame, let's explore
how we construct the transforms for translated and rotated frames of reference in
the plane. As in the previous chapter, we will show how matrices and multivectors
offer two approaches to this problem.
In computer graphics most frame of reference transforms are expressed by a
translation or a rotation, or a combination of both. We will explore these different
scenarios and develop transforms for converting coordinates in the original frame of
reference to coordinates in the second frame.

8.3.1 Translated Frame of Reference

Figure 8.1 shows two coincident 2D frames of reference X Y and XY , where a
point in one frame has identical coordinates in the other. Therefore, the two frames
of reference are related as follows
⎡

⎤ ⎡ ⎤

x
y 1

x
y
1

⎣ ⎦ = ⎣ ⎦

I

where I is the identity transform

⎡ ⎤

100
010
001

⎣ ⎦

I =

which ensures that P = P .