Maps and Distances between Spaces - Mathematical Tools for Shape Analysis and Description - page 44

if, for any x;y2X , it holds:

d Y ..x/;.y//Dd X .x;y/: (6.1)

e metric spaces X and Y are called isometric if there is a bijective isometry from X to Y . A
property of metric spaces which is invariant with respect to isometries is called a metric property
(or metric invariant ).
In the Euclidean plane R 2 and the Euclidean space R 3 , an example of isometries are geo-
metric congruences, which include rigid motions (translations, rotations) and reflections. If d X is
the intrinsic metric of X , isometries preserve geodesic distances. In Figure 6.1 , the hand bending
is the result of an isometric transformation , and the geodesic distance between points p and q
on the hand on the left is the same as the geodesic distance of their images p0 and q0 on the hand
on the right, after bending.

Figure 6.1: Isometric transformation of the space X into Y .

6.1.2 AFFINE TRANSFORMATIONS
An affine transformation (also called affine map or affinity ) is a transformation between affine
spaces ¹ which preserves:

• the collinearity between points: points which lie on a straight line (called collinear points)
before the transformation, still lie on a straight line after transformation;

• the ratio of distances between collinear points: for distinct collinear points p 1 ; p 2 ; p 3 , the
ratio of

!
f.p 1 /f.p 2 / and

!
f.p 2 /f.p 3 / ;

! !

p 1 p 2 and p 2 p 3 is the same as that of

• the barycenters of weighted collections of points; for example, the midpoint of a line seg-
ment remains the midpoint after transformation.

¹Informally, affine spaces are a generalization of vector spaces, where there is no a single point that serves as an origin.