Similarity (geometry)

In geometry, two figures are similar to each other if and only if it by a similarity mapping ( even this figure is often called similarity) can be transformed into one another. That is, there is a geometric figure, resulting from centric dilations and Kongruenzabbildungen (ie translations, rotations, reflections ) can be put together and which maps a figure on the other. Thus, similarity extends the congruence of figures to the possibility of extension.

Properties

Angle and distance relationships are consistent in similar figures; so all circles and in each case all regular polygons of the same number of corners as equilateral triangles and squares, similar to each other.

It is considered that congruent figures are similar always. The reverse, however, is wrong: Similar figures are not necessarily congruent, since they may differ in size.

As a mathematical sign for geometric similarity (the tilde ), for example: means that the triangles are similar. If you want to express contrast, congruence, then (a " blend " with the equal sign ) can be used instead of or.

Similarity in triangles

Triangles play a crucial role, as can be traced back to such a large number of figures. The following applies:

Two triangles are similar to each other when

  • They coincide in two (and thus in all three ) angles; or
  • They coincide in all ratios of corresponding sides; or
  • To match at an angle and in ratio of the adjacent sides; or
  • They coincide in the ratio of two sides and the angle opposite the larger side.

These sets are called similarity sets.

Ray sets

The radiation rates to worry about the ratios of triangle sides of similar triangles certain important statements.

Similarity in the fractal geometry

Scale-invariant similarity in broken, " fractal " dimensions is the subject of fractal geometry.

The similarity is the result of the recursion nonlinear algorithms. A well-known example is the Mandelbrot set, similar to the adjacent sections having their boundary line at each point of all sizes.

  • Geometric Figure
Translation (geometry) Rotation (mathematics)
36058
de