Graphics Reference
In-Depth Information
(1) preserve the betweenness relation,
(2) preserve collinearity and noncollinearity,
(3) map lines onto lines, and
(4) are one-to-one transformations of
R
n
onto itself.
Proof.
This theorem follows from Theorem 2.3.2 and some obvious facts about
radial transformations. In the planar case, it can also be proved directly like it was
done in the case of motions.
2.3.5. Theorem.
The similarity transformations form a group that contains the
group of motions as a subgroup.
Proof.
Obvious.
2.3.6. Theorem.
A similarity transformation in the plane is completely specified by
its action on three noncollinear points.
Proof.
Use Theorem 2.3.2.
2.3.7. Theorem.
Similarity transformations in the plane preserve angles.
Proof.
By Theorem 2.3.2, since motions preserve angles, it suffices to show that
radial transformations preserve angles, which is easy.
2.4
Affine Transformations
A one-to-one and onto mapping T :
R
n
Æ
R
n
Definition.
that maps lines onto lines
is called an
affine transformation
.
Actually, one can characterize affine transformations in a slightly stronger fashion.
Any one-to-one and onto map of
R
n
2.4.1. Theorem.
onto itself that preserves
collinearity is an affine transformation.
Proof.
The only thing that needs to be shown is that lines get mapped
onto
lines.
This is shown in a way similar to what was done in the proof of Lemma 2.2.4 and left
as an exercise.
The set of affine transformations in
R
n
2.4.2. Theorem.
forms a group that con-
tains the similarities as a subgroup.
Proof.
Exercise.
Affine transformations, like motions and similarities, have a simple analytic
description. Before we get to the main result for these maps in the plane, we analyze
transformations with equations of the form
