Maps and Distances between Spaces - Mathematical Tools for Shape Analysis and Description - page 45

Affinities do not preserve, in general, angles and lengths, though they have the property that
parallel lines remain parallel after transformation.
Examples of affine transformations include translation, geometric contraction, expansion,
homothety, reflection, rotation, shear mapping, similarity transformation, spiral similarities and
compositions of them. Some examples are shown in Figure 6.2 .

Figure 6.2: Some examples of affine transformations of an object (a), from left to right: congruence
(identity) (b), reflection (c), rotation (d), homothety (e), and shear (f ) [ 217 ].

An alternative definition of affinities involving linear algebra is based on affine combina-
tions of points. In affine spaces, affine combinations of points are defined as linear combinations
in which the sum of the coefficients is 1. Let fa i g i2I be a family of points in an affine space X ,
and f i g i2I be coefficients such that

P

i2I i D1 . en, WX!Y is an affine map if and only

if it holds:

X !

X

f i a i D i .a i / :

i2I i2I

6.1.3 MÖBIUS TRANSFORMATION
A Möbius transformation is any map of the form:

.z/D azCb
czCd

where z is a complex variable and a , b , c , d are complex numbers satisfying adbc¤0 .
e main properties of Möbius transformations (also called fractional linear transforma-
tions, projective linear transformations, homographies, homographic transformations, linear frac-
tional transformations, bilinear transformations) are that they are conformal , that is, they preserve
angles; they map every straight line to a line or circle; and they map every circle to a line or circle
(see Figure 6.3 ).
e set of all Möbius transformations forms a group under composition called the Möbius
group.