Similarity (geometry)

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Similar figures

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.

File:TranslationL.svg

Translation

File:RotationL.svg

Rotation

File:SymétrieL.svg

Reflection

File:SimilitudeHomothétieL.svg

Scaling

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.

File:Similar-geometric-shapes.svg

Figures shown in the same color are similar

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.Template:Citation needed

Similar trianglesEdit

Two triangles, Template:Math and Template:Math 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.Template:Sfn 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.Template:Sfn 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.<ref>For instance, Template:Harvnb and Template:Harvnb.</ref>

There are several criteria each of which is necessary and sufficient for two triangles to be similar:

Any two pairs of angles are congruent,<ref>Euclid's Elements, Book VI, Proposition 4.</ref> which in Euclidean geometry implies that all three angles are congruent:Template:Efn

If Template:Math is equal in measure to Template:Math and Template:Math is equal in measure to Template:Math then this implies that Template:Math is equal in measure to Template:Math and the triangles are similar.

All the corresponding sides are proportional:<ref>Euclid's Elements, Book VI, Proposition 5.</ref>

<math display=block>\frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}} = \frac{\overline{AC}}{\overline{A'C'}}.</math>

This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

Any two pairs of sides are proportional, and the angles included between these sides are congruent:<ref>Euclid's Elements, Book VI, Proposition 6.</ref>

<math display=block> \frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}}, \quad \angle ABC \cong \angle A'B'C'.</math>

This is known as the SAS similarity criterion.Template:Sfn 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.

Symbolically, we write the similarity and dissimilarity of two triangles Template:Math and Template:Math as follows:<ref name=PL>Template:Cite book</ref>

<math display=block>\begin{align} \triangle ABC &\sim \triangle A'B'C' \\ \triangle ABC &\nsim \triangle A'B'C' \end{align}</math>

There are several elementary results concerning similar triangles in Euclidean geometry:Template:Sfn

Any two equilateral triangles are similar.
Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles).
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.<ref>Template:Cite book</ref> 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 (sides) being in the same proportion.

Given a triangle Template:Math and a line segment Template:Math one can, with a ruler and compass, find a point Template:Mvar such that Template:Math. The statement that point Template:Mvar satisfying this condition exists is Wallis's postulate<ref>Named for John Wallis (1616–1703)</ref> and is logically equivalent to Euclid's parallel postulate.Template:Sfn 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.Template:Sfn

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.Template:Sfn

Other similar polygonsEdit

File:SimilarRectangles.svg

Similar rectangles

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) 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 (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

For given Template:Mvar, all [[regular polygon|regular Template:Mvar-gons]] are similar.

Similar curvesEdit

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

Lines (any two lines are even congruent)
Line segments
Circles
Parabolas<ref>a proof from academia.edu</ref>
Hyperbolas of a specific eccentricity<ref name="uluc">The shape of an ellipse or hyperbola depends only on the ratio b/a</ref>
Ellipses of a specific eccentricity<ref name="uluc" />
Catenaries
Graphs of the logarithm function for different bases
Graphs of the exponential function for different bases
Logarithmic spirals are self-similar

In Euclidean spaceEdit

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection Template:Mvar from the space onto itself that multiplies all distances by the same positive real number Template:Mvar, so that for any two points Template:Mvar and Template:Mvar we have

where Template:Math is the Euclidean distance from Template:Mvar to Template:Mvar.Template:Sfn The scalar Template:Mvar has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When Template:Math 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 Template:Tmath a similarity of ratio Template:Mvar takes the form

where Template:Tmath is an Template:Math orthogonal matrix and Template:Tmath is a translation vector.

Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.Template:Sfn Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it.Template:Sfn

The similarities of Euclidean space form a group under the operation of composition called the similarities group Template:Mvar.Template:Sfn The direct similitudes form a normal subgroup of Template:Mvar and the Euclidean group Template:Math of isometries also forms a normal subgroup.Template:Sfn The similarities group Template:Mvar 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,Template:Efn 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

<math>f(z) = az + b</math> (direct similitudes), and
<math>f(z) = a\overline z + b</math> (opposite similitudes),

where Template:Mvar and Template:Mvar are complex numbers, Template:Math. When Template:Math, these similarities are isometries.

Area ratio and volume ratioEdit

File:A proportion to conceive square root of 5.svg

Template:AnchorThe tessellation of the large triangle shows that it is similar to the small triangle with an area ratio of 5. The similarity ratio is <math>\tfrac{5}{h} = \tfrac{h}{1} = \sqrt 5.</math> This can be used to construct a non-periodic infinite tiling.

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length Template:Mvar and an altitude drawn to that side of length Template:Mvar then a similar triangle with corresponding side of length Template:Mvar will have an altitude drawn to that side of length Template:Mvar. The area of the first triangle is <math>A = \tfrac{1}{2}bh,</math> while the area of the similar triangle will be <math display=block>A' =\frac{1}{2} \cdot kb \cdot kh = k^2A.</math> 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 (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed).

Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is Template:Mvar, then the ratio of surface areas of the solids will be Template:Math, while the ratio of volumes will be Template:Math.

Similarity with a centerEdit

Template:Multiple image

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 Template:Mvar, which is a similarity of ±180° angle and a positive ratio equal to Template:Mvar.

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 Template:Tmath multiplicative inverse of the ratio Template:Tmath (square root of 2) of the inverse similarity. Point Template:Mvar is the common center of the three transformations: rotation, homothety and similarity. For example point Template:Mvar is the image of Template:Mvar under the rotation, and point Template:Mvar is the image of Template:Mvar under the homothety, more briefly <math display=block>T = H(W) = (R(F)) = (H \circ R)(F) = D(F),</math> by naming Template:Mvar, Template:Mvar and Template:Mvar the previous rotation, homothety and similarity, with “Template:Mvar" like "Direct".

This direct similarity that transforms triangle Template:Math into triangle Template:Math can be decomposed into a rotation and a homothety of same center Template:Mvar in several manners. For example, Template:Math, the last decomposition being only represented on the image. To get Template:Mvar we can also compose in any order a rotation of −45° angle and a homothety of ratio Template:Tmath

With "Template:Mvar" like "Mirror" and "Template:Mvar" like "Indirect", if Template:Mvar is the reflection with respect to line Template:Mvar, then Template:Math is the indirect similarity that transforms segment Template:Mvar like Template:Mvar into segment Template:Mvar, but transforms point Template:Mvar into Template:Mvar and point Template:Mvar into Template:Mvar itself. Square Template:Mvar is the image of Template:Mvar under similarity Template:Mvar of ratio Template:Tmath Point Template:Mvar is the center of this similarity because any point Template:Mvar being invariant under it fulfills <math>AK = \tfrac{AK}{\sqrt 2},</math> only possible if Template:Math, otherwise written Template:Math.

How to [[#caption2OfSection|construct the center Template:Mvar]] of direct similarity Template:Mvar from square Template:Mvar, how to find point Template:Mvar center of a rotation of +135° angle that transforms ray Template:Tmath into ray Template:Tmath? This is an inscribed angle problem plus a question of orientation. The set of points Template:Mvar such that <math>\overset{}{ \overrightarrow{PE}, \overrightarrow{PA} = +135^\circ }</math> is an arc of circle Template:Mvar that joins Template:Mvar and Template:Mvar, of which the two radius leading to Template:Mvar and Template:Mvar form a central angle of Template:Math. This set of points is the blue quarter of circle of center Template:Mvar inside square Template:Mvar. In the same manner, point Template:Mvar is a member of the blue quarter of circle of center Template:Mvar inside square Template:Mvar. So point Template:Mvar is the intersection point of these two quarters of circles.

In general metric spacesEdit

File:Sierpinski deep.svg

Sierpiński triangle. A space having self-similarity dimension <math>\tfrac{\log 3}{\log 2} = \log_2 3,</math> which is approximately 1.58. (From Hausdorff dimension.)

In a general metric space Template:Math, an exact similitude is a function Template:Mvar from the metric space Template:Mvar into itself that multiplies all distances by the same positive scalar Template:Mvar, called Template:Mvar 's contraction factor, so that for any two points Template:Mvar and Template:Mvar we have

Weaker versions of similarity would for instance have Template:Mvar be a bi-Lipschitz function and the scalar Template:Mvar 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 Template:Math is a set Template:Mvar for which there exists a finite set of similitudes Template:Math with contraction factors Template:Math such that Template:Mvar is the unique compact subset of Template:Mvar for which

File:Epi17.png

A self-similar set constructed with two similitudes: <math>\begin{align} z' &= 0.1[(4+i)z+4] \\ z' &= 0.1[(4+7i)z^* + 5 - 2i] \end{align}</math>

<math display=block>\bigcup_{s\in S} f_s(K)=K.</math>

These self-similar sets have a self-similar measure Template:Math with dimension Template:Mvar given by the formula

which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the Template:Math are "small", we have the following simple formula for the measure:

<math display=block>\mu^D(f_{s_1}\circ f_{s_2} \circ \cdots \circ f_{s_n}(K)) = (r_{s_1}\cdot r_{s_2}\cdots r_{s_n})^D.\,</math>

TopologyEdit

Template:More citations needed section 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

Positive defined:

<math display=block>\forall (a,b), S(a,b)\geq 0</math>

Majored by the similarity of one element on itself (auto-similarity):

<math display=block>S (a,b) \leq S (a,a) \quad \text{and} \quad \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b</math>

More properties can be invoked, such as:

Reflectivity: <math>\forall (a,b)\ S (a,b) = S (b,a),</math> or
Finiteness: <math>\forall (a,b)\ S(a,b) < \infty.</math>

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 Template:Section link and Template:Section link sections of this article.

Self-similarityEdit

Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set Template:Math of numbers of the form Template:Math where Template:Mvar 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.

PsychologyEdit

Template:Expand section The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.<ref>Template:Cite thesis</ref> Certain perceptual categorization models in psychology are based on geometric similarity, assuming that learning involves the storage of specific instances (i.e. of general object specifications) in memory. The categorization of another object ist subsequently based on the similarity of the object to the instances in memory.<ref>Template:Cite book</ref>