Similarity (geometry)


In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.
For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles.
If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.
Two congruent shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

Similar triangles

Two triangles, and are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional. It can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several criteria each of which is necessary and sufficient for two triangles to be similar:
  • Any two pairs of angles are congruent, which in Euclidean geometry implies that all three angles are congruent:
  • All the corresponding sides are proportional:
  • Any two pairs of sides are proportional, and the angles included between these sides are congruent:
Symbolically, we write the similarity and dissimilarity of two triangles and as follows:
There are several elementary results concerning similar triangles in Euclidean geometry:
  • Any two equilateral triangles are similar.
  • Two triangles, both similar to a third triangle, are similar to each other.
  • Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
  • Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs being in the same proportion.
Given a triangle and a line segment one can, with a ruler and compass, find a point such that. The statement that point satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent.
In the axiomatic treatment of Euclidean geometry given by George David Birkhoff the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.
Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry.

Other similar polygons

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Likewise, equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.
For given, all regular -gons are similar.

Similar curves

Several types of curves have the property that all examples of that type are similar to each other. These include:

In Euclidean space

A similarity of a Euclidean space is a bijection from the space onto itself that multiplies all distances by the same positive real number, so that for any two points and we have
where is the Euclidean distance from to. The scalar has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When a similarity is called an isometry. Two sets are called similar if one is the image of the other under a similarity.
As a map a similarity of ratio takes the form
where is an orthogonal matrix and is a translation vector.
Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it.
The similarities of Euclidean space form a group under the operation of composition called the similarities group. The direct similitudes form a normal subgroup of and the Euclidean group of isometries also forms a normal subgroup. The similarities group is itself a subgroup of the affine group, so every similarity is an affine transformation.
One can view the Euclidean plane as the complex plane, that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by
  • , and
  • ,
where and are complex numbers,. When, these similarities are isometries.

Similarity with a center

If a similarity has exactly one invariant point: a point that the similarity keeps unchanged, then this only point is called "center" of the similarity.
On the first image below the title, on the left, one or another similarity shrinks a regular polygon into a concentric one, the vertices of which are each on a side of the previous polygon. This rotational reduction is repeated, so the initial polygon is extended into an abyss of regular polygons. The center of the similarity is the common center of the successive polygons. A red segment joins a vertex of the initial polygon to its image under the similarity, followed by a red segment going to the following image of vertex, and so on to form a spiral. Actually we can see more than three direct similarities on this first image, because every regular polygon is invariant under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular pentagon under a homothety of negative ratio, which is a similarity of ±180° angle and a positive ratio equal to.
Below the title on the right, the second image shows a similarity decomposed into a rotation and a homothety. Similarity and rotation have the same angle of +135 degrees modulo 360 degrees. Similarity and homothety have the same ratio of multiplicative inverse of the ratio of the inverse similarity. Point is the common center of the three transformations: rotation, homothety and similarity. For example point is the image of under the rotation, and point is the image of under the homothety, more briefly
by naming, and the previous rotation, homothety and similarity, with “" like "Direct".
This direct similarity that transforms triangle into triangle can be decomposed into a rotation and a homothety of same center in several manners. For example,, the last decomposition being only represented on the image. To get we can also compose in any order a rotation of −45° angle and a homothety of ratio
With "" like "Mirror" and "" like "Indirect", if is the reflection with respect to line, then is the indirect similarity that transforms segment like into segment, but transforms point into and point into itself. Square is the image of under similarity of ratio Point is the center of this similarity because any point being invariant under it fulfills only possible if, otherwise written.
How to construct the center of direct similarity from square, how to find point center of a rotation of +135° angle that transforms ray into ray ? This is an inscribed angle problem plus a question of orientation. The set of points such that is an arc of circle that joins and, of which the two radius leading to and form a central angle of. This set of points is the blue quarter of circle of center inside square. In the same manner, point is a member of the blue quarter of circle of center inside square. So point is the intersection point of these two quarters of circles.

In general metric spaces

In a general metric space, an exact similitude is a function from the metric space into itself that multiplies all distances by the same positive scalar, called 's contraction factor, so that for any two points and we have
Weaker versions of similarity would for instance have be a bi-Lipschitz function and the scalar a limit
This weaker version applies when the metric is an effective resistance on a topologically self-similar set.
A self-similar subset of a metric space is a set for which there exists a finite set of similitudes with contraction factors such that is the unique compact subset of for which
These self-similar sets have a self-similar measure with dimension given by the formula
which is often equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the are "small", we have the following simple formula for the measure:

Topology

In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer.
The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
  1. Positive defined:
  2. Majored by the similarity of one element on itself :
More properties can be invoked, such as:Reflectivity: orFiniteness:
The upper value is often set at 1.
Note that, in the topological sense used here, a similarity is a kind of measure. This usage is not the same as the similarity transformation of the and sections of this article.

Psychology

The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings. Certain perceptual categorization models in psychology are based on geometric similarity, assuming that learning involves the storage of specific instances in memory. The categorization of another object is subsequently based on the similarity of the object to the instances in memory.