Two geometrical objects are called
similar if they both have the same
shapeThe 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
congruentIn 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
scalingIn 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
rotationA 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
reflectionIn 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
circleA 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
squareIn 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,
ellipseIn 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
hyperbolaIn 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
polygonIn 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
proportionalIn 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:
- 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
- Parabola
In 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
- 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 having the same eccentricityIn 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,...
- 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 all having the same eccentricity
- Catenaries
In 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 logarithm
The 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 spiral
A 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
dilationIn 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 spaceIn 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
functionIn 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
scalarIn 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 distanceIn 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
isometryIn 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 transformationIn 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 groupIn 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 groupIn 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 planeA 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
realsIn 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 transformationIn 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 spaceIn 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
functionIn 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
scalarIn 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-
LipschitzIn 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
measureIn 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 dimensionthumb|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 dimensionIn 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 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 spaceIn 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
distanceDistance 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-similarityIn 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 scaleA 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 symmetryIn 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)
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 distance
In 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 index
The Jaccard index, also known as the Jaccard similarity coefficient , is a statistic used for comparing the similarity and diversity of sample sets....
- Proportionality
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 similarity
Semantic 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 search
Nearest 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 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 analysis rather than using subjective evaluation of...
- Homoeoid
A 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 triangles
In 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...
External links