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:

• Rotational stretchings are true to orientation (ie they leave the direction of rotation of polygons unchanged). They consist of a centric stretch and a rotation , and they are characterized by the stretch factor and the rotation angle. If the stretching factor is equal to 1, a pure rotation occurs, which is the special case of a congruence mapping.
• Folded extensions reverse the orientation and consist of a reflection on a hyperplane (a straight line reflection , if the affine space is two-dimensional, a plane reflection , if it is three-dimensional) and a centric extension . If the stretching factor is equal to 1, it is a matter of a pure mirroring, which means that the special case of a congruence mapping is also present here.

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. ${\displaystyle m>0}$ ${\displaystyle A}$ ${\displaystyle A}$

See also

• Similarity (matrix)

literature

• Hermann Schaal: Linear Algebra and Analytical Geometry - Volume 1 , Vieweg-Verlag Braunschweig, ISBN 3528030569
• Heribert Kahmen: Surveying , 18th edition, de Gruyter textbook, Berlin 1993