Similarity (geometry)

Similar figures

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.

Shift.

rotation

reflection

Elongation

properties

All figures of the same color here are similar to one another.

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 .

Similarity in the theorems of rays

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

Extract from the crowd of Mandelbrot

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.