Similarity (geometry)


In Euclidean geometry, two objects are similar if they have the same shape, or 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.
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.
This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but 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 statements each of which is necessary and sufficient for two triangles to be similar:
This is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.
When two triangles and are similar, one writes
There are several elementary results concerning similar triangles in Euclidean geometry:
Given a triangle and a line segment one can, with ruler and compass, find a point such that. The statement that the 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 G.D. 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 n, all regular n-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:
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 = 1 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.

Ratios of sides, of areas, and of volumes

The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures. The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length and an altitude drawn to that side of length then a similar triangle with corresponding side of length will have an altitude drawn to that side of length. The area of the first triangle is,, while the area of the similar triangle will be. Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.
The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures.
Galileo's square–cube law concerns similar solids. If the ratio of similitude between the solids is, then the ratio of surface areas of the solids will be, while the ratio of volumes will be.

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. :
  3. Majored by the similarity of one element on itself :
  4. :
More properties can be invoked, such as reflectivity or finiteness. 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.

Self-similarity

means that a pattern is non-trivially similar to itself, e.g., the set of numbers of the form where 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.

Psychology

The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.