Similarity (geometry)
Overview
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
. More precisely, either one is congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
to the result of a uniform scaling
Scaling (geometry)
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
(enlarging or shrinking) of the other. This means that either object can be rescaled and repositioned so as to coincide precisely with the other object.
Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.
Unanswered Questions
Encyclopedia
Two geometrical objects are called similar if they both have the same shape
. More precisely, either one is congruent
to the result of a uniform scaling
(enlarging or shrinking) of the other. This means that either object can be rescaled and repositioned so as to coincide precisely with the other object.
Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation
and reflection
— i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circle
s are similar to each other, all square
s are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipse
s are not all similar to each other, nor are hyperbola
s 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.
This article assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.
In particular, similar triangles are triangles that have the same shape and are identical to one another except for scale. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserves their measures.
Formally speaking, two triangles and are said to be similar if either of the following equivalent conditions holds:
1. Corresponding sides have lengths in the same ratio:
2. is equal in measure to , and is equal in measure to . This also implies that is equal in measure to .
When two triangles and are similar, one writes
The 'is similar to' symbol can also be expressed as three vertical lines: lll
s 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 (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).
) of a Euclidean space
is a function
f from the space into itself that multiplies all distances by the same positive scalar
r, so that for any two points x and y we have
where "d(x,y)" is the Euclidean distance
from x to y. Two sets are called similar if one is the image of the other under such a similarity.
A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry
. Therefore, in general Euclidean spaces every similarity is an affine transformation
, because the Euclidean group
E(n) is a subgroup of the affine group
.
Viewing the complex plane
as a 2-dimensional space over the reals
, the 2D similarity transformations expressed in terms of the complex plane are and , and all affine transformation
s are of the form (a, b, and c complex).
(X, d), an exact similitude is a function
f from the metric space X into itself that multiplies all distances by the same positive scalar
r, called f's contraction factor, so that for any two points x and y we have
Weaker versions of similarity would for instance have f be a bi-Lipschitz
function and the scalar r 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 (X, d) is a set K for which there exists a finite set of similitudes with contraction factors such that K is the unique compact subset of X for which
These self-similar sets have a self-similar measure
with dimension D 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 are "small", we have the following simple formula for the measure:
, 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 reflectivity () or finiteness (). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
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, ..}. When this set is plotted on a logarithmic scale
it has translational symmetry
.
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
. More precisely, either one is congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
to the result of a uniform scaling
Scaling (geometry)
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
(enlarging or shrinking) of the other. This means that either object can be rescaled and repositioned so as to coincide precisely with the other object.
Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
— i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s are similar to each other, all square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s are not all similar to each other, nor are hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
s 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.
This article assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.
Similar triangles
To understand the concept of similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale.In particular, similar triangles are triangles that have the same shape and are identical to one another except for scale. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserves their measures.
Formally speaking, two triangles and are said to be similar if either of the following equivalent conditions holds:
1. Corresponding sides have lengths in the same ratio:
- i.e. . This is equivalent to saying that one triangle is an enlargement of the other.
2. is equal in measure to , and is equal in measure to . This also implies that is equal in measure to .
When two triangles and are similar, one writes
The 'is similar to' symbol can also be expressed as three vertical lines: lll
Angle/side similarities
The following three criteria are sufficient to prove that a pair of triangles are similar. The first two state that if triangles have the same shape (AA criterion) then they are similar, and that if they are to scale (SSS criterion) then they are similar. The third criterion, SAS, combines some of the information used by each of the first two.- AA: if two triangles have two corresponding pairs of angles with the same measure then they are similar. Sometimes this criterion is also referred to as AAA because equality across triangles of two angles implies equality of the third. This criterion means that if a triangle is copied to preserve the shape, then the copy is to scale.
- SSS (Three sides proportional): If the ratio of corresponding sides of two triangles does not depend on the pair of corresponding sides chosen, then the triangles are similar. This means that any triangle copied to scale is also copied in shape.
- SAS (Ratio of two sides, included angle): if two sides in a triangle are proportional to two corresponding sides in another triangle, and the angles included between these sides have the same measure in each triangle, then the triangles are similar. This means that to enlarge a triangle, it is sufficient to copy one angle, and scale just the two sides that form the angle.
Other similar polygons
The concept of similarity extends to polygonPolygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
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).
Similar curves
Several types of curves have the property that all examples of that type are similar to each other. These include:- CircleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s - ParabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
s - HyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
s all having the same eccentricityEccentricity (mathematics)In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,... - EllipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s all having the same eccentricity - CatenariesCatenaryIn physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...
- Graphs of the logarithmLogarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
function for different bases - Logarithmic spiralLogarithmic spiralA logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous...
s
Similarity in Euclidean space
One of the meanings of the terms similarity and similarity transformation (also called dilationDilation (mathematics)
In mathematics, a dilation is a function f from a metric space into itself that satisfies the identityd=rd \,for all points where d is the distance from x to y and r is some positive real number....
) of a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f from the space into itself that multiplies all distances by the same positive scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
r, so that for any two points x and y we have
where "d(x,y)" is the Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
from x to y. Two sets are called similar if one is the image of the other under such a similarity.
A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
. Therefore, in general Euclidean spaces every similarity is an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
, because the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
E(n) is a subgroup of the affine group
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
.
Viewing the complex plane
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
as a 2-dimensional space over the reals
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, the 2D similarity transformations expressed in terms of the complex plane are and , and all affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s are of the form (a, b, and c complex).
Similarity in general metric spaces
In a general metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(X, d), an exact similitude is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f from the metric space X into itself that multiplies all distances by the same positive scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
r, called f's contraction factor, so that for any two points x and y we have
Weaker versions of similarity would for instance have f be a bi-Lipschitz
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
function and the scalar r 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 (X, d) is a set K for which there exists a finite set of similitudes with contraction factors such that K is the unique compact subset of X for which
These self-similar sets have a self-similar measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
with dimension D given by the formula
which is often (but not always) equal to the set's Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
and packing dimension
Packing dimension
In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset,...
. If the overlaps between the are "small", we have the following simple formula for the measure:
Topology
In topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
can be constructed by defining a similarity instead of a distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical 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:
- Majored by the similarity of one element on itself (auto-similarity): and
More properties can be invoked, such as reflectivity () or finiteness (). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
Self-similarity
Self-similaritySelf-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
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, ..}. When this set is plotted on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
it has translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
.
See also
- Congruence (geometry)Congruence (geometry)In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
- Hamming distanceHamming distanceIn information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...
(string or sequence similarity) - inversive geometry
- Jaccard indexJaccard indexThe Jaccard index, also known as the Jaccard similarity coefficient , is a statistic used for comparing the similarity and diversity of sample sets....
- ProportionalityProportionality (mathematics)In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
- Semantic similaritySemantic similaritySemantic similarity or semantic relatedness is a concept whereby a set of documents or terms within term lists are assigned a metric based on the likeness of their meaning / semantic content....
- Similarity searchNearest neighbor searchNearest neighbor search , also known as proximity search, similarity search or closest point search, is an optimization problem for finding closest points in metric spaces. The problem is: given a set S of points in a metric space M and a query point q ∈ M, find the closest point in S to q...
- Similarity space on Numerical taxonomyNumerical taxonomyNumerical 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 analysis rather than using subjective evaluation of...
- HomoeoidHomoeoidA homoeoid is a shell bounded by two concentric, similar ellipses or ellipsoids.When the thickness of the shell becomes negligible, it is called a thin homoeoid.-Mathematical definition:If the outer shell is given by...
(shell of concentric, similar ellipsoids) - Solution of trianglesSolution of trianglesIn trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information...