# Similarity (geometry)

In terms of geometry , two figures are exactly similar to one another if they can be converted into one another by means of a similarity mapping (this mapping is also often referred to as similarity ). This means that there is a geometric mapping that can be composed of centric stretching and congruence mapping (i.e. shifts , rotations , reflections ) and that maps one figure onto the other. Similarity extends the congruence ( congruence ) of figures by the possibility of stretching.

In the table are the first three congruence maps. Note that a reflection reverses orientations. Only centric stretches change lengths.

## properties

Angle and route relationships match in similar figures; thus all circles and all regular polygons with the same number of corners, such as equilateral triangles and squares , are similar to one another.

It is true that congruent figures are always similar. The reverse is wrong, however: Similar figures are not necessarily congruent, as they can be of different sizes.

(The tilde ) is used as a mathematical symbol for geometric similarity , for example: means that the triangles and are similar. If, on the other hand, you want to express congruence, or (a “mixture” with the equal sign ) can be used instead. ${\ displaystyle \ sim}$ ${\ displaystyle \! \ \ Delta ABC \ sim \ Delta A'B'C '}$ ${\ displaystyle \ Delta ABC}$ ${\ displaystyle \ Delta A'B'C '}$ ${\ displaystyle \ simeq}$ ${\ displaystyle \ cong}$ ### Similarity in triangles

Triangles play a central role here, as many figures can be traced back to them. The following applies:

Two triangles are similar to each other if

• they coincide in two (and thus in all three) angles; or
• they match in all proportions of the corresponding sides; or
• they coincide at one angle and in proportion to the adjacent sides; or
• they match in the ratio of two sides and in the opposite angle of the larger side.

These sentences are called similarity sentences .

### Ray theorems

The ray theorems make important statements about the proportions of the triangle sides of certain similar triangles.

### Similar conic sections

• Two non-degenerate conic sections (ellipse, hyperbola, parabola) are similar if they have the same eccentricity.

The similarity of all parabolas (their eccentricity is 1) is shown in the article Parabolas .

An ellipse / hyperbola with semi-axes has the eccentricity. Stretching by the factor at the center point does not change the eccentricity. ${\ displaystyle a, b}$ ${\ displaystyle \ varepsilon = {\ frac {\ sqrt {a ^ {2} \ mp b ^ {2}}} {a}} \.}$ ${\ displaystyle c}$ ### Self-similarity of logarithmic spirals Examples for and${\ displaystyle a = 1,2,3,4,5}$ ${\ displaystyle k = \ tan 20 ^ {\ circ}}$ The logarithmic spiral can be understood on the one hand as a picture of the spiral under the stretching at the zero point with the factor , but also as a picture of the rotation around the angle . ${\ displaystyle \; r = ae ^ {k \ varphi}}$ ${\ displaystyle \; r = e ^ {k \ varphi} \;}$ ${\ displaystyle a \ neq 0}$ ${\ displaystyle \; r = e ^ {k \ varphi}}$ ${\ displaystyle \ varphi _ {0} = - {\ tfrac {\ ln a} {k}}}$ A curve whose images are congruent to itself under centric stretching is called self-similar . So:

• The spiral is self-similar.${\ displaystyle \; r = e ^ {k \ varphi} \;}$ Pictured: the spirals for can also be obtained by rotating the red spiral around . ${\ displaystyle a = 2,3,4,5}$ ${\ displaystyle -109 ^ {\ circ}, - 173 ^ {\ circ}, - 218 ^ {\ circ}, - 253 ^ {\ circ}}$ ## Similarity in fractal geometry

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

The similarity is the result of the recursion of nonlinear algorithms. A well-known example is the Mandelbrot set , the borderline of which is similar at every point to the adjacent sections in all orders of magnitude.