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 (enlarging or reducing), 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 other uses, see Similarity (disambiguation) and Similarity transformation (disambiguation).
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.
Two triangles, △ABC and △A'B'C' 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.[1] It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem.[2] 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.[3]
There are several criteria each of which is necessary and sufficient for two triangles to be similar:
Symbolically, we write the similarity and dissimilarity of two triangles △ABC and △A'B'C' as follows:[8]
There are several elementary results concerning similar triangles in Euclidean geometry:[9]
Given a triangle △ABC and a line segment DE one can, with a ruler and compass, find a point F such that △ABC ~ △DEF. The statement that point F satisfying this condition exists is Wallis's postulate[11] and is logically equivalent to Euclid's parallel postulate.[12] In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent.
In the axiomatic treatment of Euclidean geometry given by George David Birkhoff (see Birkhoff's axioms) 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.[7]
Similar triangles provide the basis for many synthetic (without the use of coordinates) 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.[13]
Several types of curves have the property that all examples of that type are similar to each other. These include:
(direct similitudes), and
(opposite similitudes),
A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have
where d(x,y) is the Euclidean distance from x to y.[16] The scalar r has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When r = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity.
As a map a similarity of ratio r takes the form
where is an n × n orthogonal matrix and is a translation vector.
Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.[17] Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it.[18]
The similarities of Euclidean space form a group under the operation of composition called the similarities group S.[19] The direct similitudes form a normal subgroup of S and the Euclidean group E(n) of isometries also forms a normal subgroup.[20] The similarities group S 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,[b] 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
where a and b are complex numbers, a ≠ 0. When |a|= 1, these similarities are isometries.
Reflectivity: or
Finiteness:
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 (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance).
The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
More properties can be invoked, such as:
The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
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 § In Euclidean space and § In general metric spaces sections of this article.
Self-similarity[edit]
Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set {..., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ...} of numbers of the form {2i, 3·2i} where i ranges over all integers. When this set is plotted on a logarithmic scale it has one-dimensional translational symmetry: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.