Scaling (geometry)

Under a central extension is understood in the geometry of an image that enlarges all routes in a certain given ratio or reduced, the image lines are parallel respectively to the original routes. Centric dilations are special similarity illustrations.

Definition

Given a point of the plane of the drawing or of the room, and a real number. The central dilation with center and stretching factor (Figure factor ) is the image of the plane or the space in itself, in which the image point of a point has the following properties:

  • , And lie on a straight line.
  • For lie on the same side of, for on different pages.
  • The route length is equal to times the length of the route.

The two sketches show the application of two centric dilations ( and with ) each have a triangle ABC.

Properties

  • Centric dilations are straight, circular and conformal.
  • The length ratios are preserved.
  • The photo spread any track, the times the length.
  • Any geometric figure is mapped to a character with the fold area.
  • Any body is mapped to a body with the fold volume.
  • The central dilations with a specific center form algebraically seen a group.
  • The image of a straight line is parallel to the straight line.
  • In vectorial notation, the central dilation is described by the center and stretching factor by
  • Thus, a central extension, the affinity, which is described by the matrix and the translation vector.
  • Also, the identity mapping is counted as stretching the stretch factor of the dilations. A non-identical aspect has exactly one fixed point, that is their expanded center, and their Fixgeraden are exactly the lines that pass through this center.

Special cases

For there is the identity map (identity), for a point reflection. The case is not allowed, otherwise all points have the same image point, namely the center.

Generalizations

  • The central dilation is an example of a dilation. In the axiomatic built affine geometry that term is defined by using the parallelism.
  • The central dilation is the special case of a rotary extension with rotation angle 0
  • In place of the affine 2 - or 3- dimensional space over the real numbers, one can define central dilations also more generally in any finite-dimensional affine space over an arbitrary field, and even over an arbitrary skew field. The " vector " representation is the same as in the real case, however, form the parallel shifts, which are stretched from one center, in general, only a left vector space over the coordinates chief body.
  • In the flat, two-dimensional case is somewhat more general even then spoken by a central extension if the parallel shifts are stretched (as coordinate " vectors" ) of an affine translation plane over a quasi body with a " scalar " from the core of the quasi body.

In the two latter cases, one can speak neither angle nor length of relationship loyalty in general, since neither a square nor must exist a measure of length. Here are the centric dilations but always to dilatation and the affinities and for fixed points and Fixgeraden the same applies as in the real case.

Function (mathematics) Group theory Translation (geometry)
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