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In Euclidean geometry, two objects are similar if they have the same
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
, 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 (enlarging or reducing), possibly with additional
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s 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,
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s are not all similar to each other,
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
s are not all similar to each other, and
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s 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. 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 (''equiangular triangles'') 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 congruent angles, which in Euclidean geometry implies that their all three angles are congruent: ::If is equal in measure to , and is equal in measure to , then this implies that is equal in measure to and the triangles are similar. *All the corresponding sides are proportional: :: . 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: :: and is equal in measure to . 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. Symbolically, we write the similarity and dissimilarity of two triangles and as follows: :ABC\sim A'B'C' :ABC\nsim A'B'C' 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 ( 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. 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 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
(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 In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protrac ...
) 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 Synthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to: Science * Synthetic chemical or compound, produced by the process of chemical synthesis * 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 In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of ...
, the
geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states ...
,
Ceva's theorem In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are ...
,
Menelaus's theorem Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respe ...
and the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
. Similar triangles also provide the foundations for right triangle trigonometry.


Other similar polygons

The concept of similarity extends to
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s 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
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
s would be similar). 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: * Lines (any two lines are even congruent) *
Line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s *
Circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s *
Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
s *
Hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s of a specific eccentricityThe shape of an ellipse or hyperbola depends only on the ratio b/a
/ref> *
Ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s of a specific eccentricity *
Catenaries In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superfi ...
*Graphs of the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
function for different bases *Graphs of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
for different bases * Logarithmic spirals are self-similar


In Euclidean space

A similarity (also called a similarity transformation or similitude) of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from the space onto itself that multiplies all distances by the same positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, so that for any two points and we have :d(f(x),f(y)) = r\, d(x,y), \, where "" is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
(
rigid transformation In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The 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 takes the form :f(x) = rAx + t, 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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of and the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
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 In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
. One can view the Euclidean plane as the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, 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 (direct similitudes) and (opposite similitudes), where and are complex numbers, . When , these similarities are isometries.


Area ratio and volume ratio

The ratio between the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
s 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 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
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
s 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 , then the ratio of surface areas of the solids will be , while the ratio of volumes will be .


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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
under the similarity, followed by a red segment going to the following image of vertex, and so on to form a 
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...
under a homothety of negative \text-k,\, which is a similarity of ±180° angle and a positive ratio \textk. Below the title on the right, the second image shows a similarity decomposed into a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
and a homothety. Similarity and rotation have the same angle of +135 degrees modulo 360 degrees. Similarity and homothety have the same ratio \text\,\frac, multiplicative inverse of the \,\text\sqrt\; ( square root of 2) of the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
 similarity. Point ''S'' is the common center of the three transformations: rotation, homothety and similarity. For example point ''W'' is the image of ''F'' under the rotation, and point ''T'' is the image of ''W'' under the homothety, more briefly ''T '' = (''W '') = ( ''F'' )) = ( ''F'' ) = ( ''F ''), by naming R,\;H\textD\, the previous rotation, homothety and similarity, with \textD\,\text\, This direct similarity that transforms triangle ''EFA'' into triangle ''ATB'' can be decomposed into a rotation and a homothety of same center ''S'' in several manners. For example, D\,=\,R\,\circ\,H\,=\,H\,\circ\,R, the last decomposition being only represented on the image. To get D we can also compose in any order a rotation \text-\,45^ angle and a homothety \,\text\,\frac. With \textM\,\text\, and \textI\,\text\, if M\, is the reflection with respect to line (''CW'' ), then M\,\circ\,D\,=\,I\; is the indirect similarity that transforms segment  'BF'' \textD\, into segment  'CT''  but transforms point ''E'' into ''B'' and point ''A'' into ''A'' itself. Square ''ACBT'' is the image of ''ABEF'' under similarity I\,\text\;1\,/\sqrt.\; Point ''A'' is the center of this similarity because any point ''K'' being invariant under it fulfills AK\,=\,AK/\sqrt,\; only possible \text\,AK\,=\,0,\; otherwise written A\,=\,K. How to construct the center ''S'' of direct similarity D \text\,ABEF,\, how to find point ''S'' center of a rotation of +135° angle that transforms ray  'SE'' )_into ray [''SA'' )? This is an_inscribed angle problem_plus a_question_of orientation_(vector_space).html" ;"title="inscribed angle.html" ;"title="'SE'' ) into ray [''SA'' )? This is an inscribed angle">'SE'' ) into ray [''SA'' )? This is an inscribed angle problem plus a question of orientation (vector space)">orientation. The set of points P\,\text\;\angle\;\,=\,+\,135^\, is an arc of circle \overset \; that joins ''E'' and ''A'', of which the two radius leading to ''E'' and ''A'' form a central angle \text\;\;2\,(180^\,-\,135^)\,=\,2\,\times\,45^\,=\,90^.\, This set of points is the blue quarter of circle of center ''F'' inside square ''ABEF''. In the same manner, point ''S''  is a member of the blue quarter of circle of center ''T'' inside square ''BCAT''. So point ''S'' is the 
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
point of these two quarters of circles.


In general metric spaces

In a general
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, 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 :d(f(x),f(y)) = r d(x,y).\, \, Weaker versions of similarity would for instance have be a bi-
Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
function and the scalar a limit :\lim \frac = r. 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 :]\bigcup_ f_s(K)=K. \, These self-similar sets have a self-similar measure (mathematics), measure with dimension given by the formula :\sum_ (r_s)^D=1 \, which is often (but not always) equal to the set's
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
and packing dimension. If the overlaps between the are "small", we have the following simple formula for the measure: :\mu^D(f_\circ f_ \circ \cdots \circ f_(K))=(r_\cdot r_\cdots r_)^D.\,


Topology

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
can be constructed by defining a similarity instead of a
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
. 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: #:\forall (a,b), S(a,b)\geq 0 # Majored by the similarity of one element on itself (auto-similarity): #:S (a,b) \leq S (a,a) \quad \text \quad \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b More properties can be invoked, such as reflectivity (\forall (a,b)\ S (a,b) = S (b,a)) or finiteness (\forall (a,b)\ S(a,b) < \infty). 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 (mathematics), measure. This usage is not the same as the ''similarity transformation'' of the and sections of this article.


Self-similarity

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 A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
it has one-dimensional
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
: 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.


See also

* Congruence (geometry) * Hamming distance (string or sequence similarity) *
Helmert transformation The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a geometric transformation method within a three-dimensional space. It is frequently used in geodesy to produce datum transformations between datums. Th ...
*
Inversive geometry Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotion Emotions are mental states brought on by neurophysiological changes, variou ...
* Jaccard index * Proportionality * Basic proportionality theorem * Semantic similarity * Similarity search * Similarity (philosophy) * Similarity space on
numerical taxonomy Numerical taxonomy is a classification system in biological systematics which deals with the grouping by numerical methods of taxonomic units based on their character states. It aims to create a taxonomy using numeric algorithms like cluster ...
* Homoeoid (shell of concentric, similar ellipsoids) *
Solution of triangles Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. A ...


Notes


References

* * * * * * * *


Further reading

* * Coxeter, H. S. M. (1969) 961 "§5 Similarity in the Euclidean Plane". pp. 67–76. "§7 Isometry and Similarity in Euclidean Space". pp. 96–104. ''Introduction to Geometry''.
John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...
. * *


External links

{{commons category, Similarity (geometry)
Animated demonstration of similar triangles
Geometry Equivalence (mathematics) Euclidean geometry Triangle geometry