# Similitude illustration

In geometry , a branch of mathematics , a similarity mapping (or similarity ) is an affinity that leaves route relationships and angle sizes unchanged, but generally changes the lengths of routes . The term is therefore only meaningful in those affine spaces in which there is an angle term and a length term. In most cases, these are affine point spaces to which a real Euclidean space is assigned as the space of the connection vectors (see Euclidean space # The Euclidean point space ). Figures that can be mapped to one another through a similarity map are called similar to one another.

In geodesy and astrometry , mapping is referred to as a similarity transformation . Its four transformation parameters are a rotation angle, a scale factor and two displacement values. They are used for simple coordinate transformations , for example for a small-scale survey to connect to the national coordinates, or for astrographs to reduce the plate to two or more connecting stars .

## Similarities as special affinities

The amount of the similarities in a affine space forms a subset of the affinities on . Is the dimension of greater than or equal to 2, then there are also affinities that no similarities are. In terms of concatenation, the similarities even form a subgroup of this group of affinities. ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A}$

All congruence mappings are also part of the similarities (they form a - generally real - subgroup), since they are, among other things, angle and ratio-accurate, i.e. they leave angles and distance relationships invariant. If only similarities are meant that are not congruence maps, one speaks of real similarities .

## classification

There are two types of similarity maps:

## Coordinate representation

In analytic geometry , a similarity mapping is based on the choice of an Euclidean affine coordinate system by means of a mapping equation of the shape

${\ displaystyle {\ vec {X '}} = m \ cdot A {\ vec {X}} + {\ vec {b}}}$

where is a real number and an orthogonal matrix . If it is the same direction Homothecy, so has the determinant of the value of 1, otherwise -1. ${\ displaystyle m> 0}$${\ displaystyle A}$${\ displaystyle A}$