Affine Geometry - Computer Graphics and Geometric Modeling - page 94

It is easy to check that this motion sends the points A , B , C to A ¢, B ¢, C ¢.

The astute reader may have noticed just by looking at Figure 2.16 that there are
easier ways to solve Problem 2.2.8.5. For example, the motion M can also be obtained
by translating A to A ¢ and then reflecting about the x-axis. However, to emphasize a
point made earlier, using frames is a systematic approach that can be programmed
on a computer. Computers cannot “look.”

2.3 Similarities

A map S : R n Æ R n is called a similarity transformation , or simply a sim-

Definition.

ilarity , if

()() = SS pq r pq

for all p , q ΠR n

and some fixed positive constant r.

Clearly, motions are similarities, because they correspond to the case where r is
1 in the definition. On the other hand, the map S( p ) = 2 p is a similarity but not a
motion. In fact, S an example of a simple but important class of similarities.

A map R : R n Æ R n Definition. of the form R( p ) = r p , r > 0, is called a radial

transformation .

2.3.1. Theorem. Radial transformations are similarities.

Proof. Exercise.

The next theorem shows that similarities are not much more complicated than
motions.

2.3.2. Theorem. If S is a similarity, then S = MR, where M is a motion and R is a
radial transformation. Conversely, any map of the form MR, where M is a motion and
R is a radial transformation, is a similarity.

Proof. This is easy because if we use the notation in the definitions for a similarity
and a radial transformation, then R -1 S is a motion M.

2.3.3. Corollary. Every similarity in the plane can be expressed by equations of the form

x x ym ¢= + +

( ) +

y ¢=± - bx + ay n, (2.24)

2 2

where (a,b) π (0,0). (The r in the definition of a similarity is ab + in this case.)

Conversely, every map defined by such equations is a similarity.

2.3.4. Theorem. Similarity transformations