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9

Transformations

The theory of transformations concerns itself with changes in the coordinates
and shapes of objects upon the action of geometrical operations, dynamical
boosts, or other operators. In this chapter, we deal only with linear transfor-
mations, using examples from both plane geometry and relativistic dynamics
(space-time geometry). We also show how transformation techniques play an
important role in image processing. We formulate both the problems and
their solutions in the language of matrices. Matrices are still denoted by bold-
face type and matrix multiplication by an asterisk.

9.1 Two-Dimensional (2-D) Geometric Transformations

We first concern ourselves with the operations of inversion about the origin
of axes, reflection about the coordinate axes, rotation around the origin, scal-
ing, and translation. But prior to going into the details of these transforma-
tions, we need to learn how to draw closed polygonal figures in MATLAB so
that we can implement and graph the different cases.

9.1.1 Polygonal Figures Construction
Consider a polygonal figure whose vertices are located at the points:

,
( ,xy ), (x y ), … , (x y )
,
1 1 2 2 n n
The polygonal figure can then be thought off as line segments (edges) con-
necting the vertices in a given order, including the edge connecting the last
point to the initial point to ensure that we obtain a closed figure. The imple-
mentation of the steps leading to the drawing of the figure follows:
1. Label all vertex points.
2. Label the path you follow.

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© 2000 by CRC Press LLC
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