- Similarity (geometry)
= Geometry =

Two geometrical objects are called

**similar**if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the sameshape , or additionally themirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, allcircle s are similar to each other, all squares are similar to each other, and allparabola s are similar to each other. On the other hand,ellipse s are "not" all similar to each other, "nor" arehyperbola s all similar to each other. Two triangles are similarif and only if they have the same threeangle s, the so-called "AAA" condition. However, since the sum of the interior angles in a triangle is fixed in an euclidean plane, as long as two angles are the same, all three are, called "AA".**imilar triangles**If triangle "ABC" is similar to triangle "DEF", then this relation can be denoted as:$riangle\; ABC\; sim\; riangle\; DEF.,$In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.

Suppose that triangle "ABC" is similar to triangle "DEF" in such a way that the angle at vertex "A" is congruent with the angle at vertex "D", the angle at "B" is congruent with the angle at "E", and the angle at "C" is congruent with the angle at "F". Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following::$\{AB\; over\; BC\}\; =\; \{DE\; over\; EF\},$

:$\{AB\; over\; AC\}\; =\; \{DE\; over\; DF\},$

:$\{AC\; over\; BC\}\; =\; \{DF\; over\; EF\},$

:$\{AB\; over\; DE\}\; =\; \{BC\; over\; EF\}\; =\; \{AC\; over\; DF\}.$

This idea can be extended to similar

polygon s with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional.**Angle/side similarities**A concept commonly taught in high school mathematics is that of proving the "angle" and "side" theorems, which can be used to define two triangles as similar (or indeed, congruent).

In each of these three-letter acronyms, "A" stands for equal angles, and "S" for equal sides. For example,

**ASA**refers to an angle, side and angle that are all equal andadjacent , in that order.*

**AAA**- Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as**AA**) implies similarity as well."See also:"

Congruence (geometry) **imilarity in Euclidean space**One of the meanings of the terms

**similarity**and**similarity transformation**(also called dilation) of aEuclidean 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:$d(f(x),f(y))\; =\; r\; d(x,y),\; ,$

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 anisometry .Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of the complex plane are $f(z)=az+b$ and $f(z)=aoverline\; z+b$, and all

affine transformation s are of the form $f(z)=az+boverline\; z+c$ ("a", "b", and "c" complex).**imilarity in general metric spaces**In a general

metric space ("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:$d(f(x),f(y))\; =\; r\; d(x,y).,\; ,$

Weaker versions of similarity would for instance have "f" be a bi-Lipschitz function and the scalar "r" a limit

:$lim\; frac\{d(f(x),f(y))\}\{d(x,y)\}\; =\; r.$

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 ("X", "d") is a set "K" for which there exists a finite set of similitudes $\{\; f\_s\; \}\_\{sin\; S\}$ with contraction factors $0leq\; r\_s\; <\; 1$ such that "K" is the unique compact subset of "X" for which

:$igcup\_\{sin\; S\}\; f\_s(K)=K.\; ,$

These self-similar sets have a self-similar measure $mu^D$with dimension "D" given by the formula

:$sum\_\{sin\; S\}\; (r\_s)^D=1\; ,$

which is often (but not always) equal to the set's

Hausdorff dimension andpacking dimension . If the overlaps between the $f\_s(K)$ are "small", we have the following simple formula for the measure::$mu^D(f\_\{s\_1\}circ\; f\_\{s\_2\}\; circ\; cdots\; circ\; f\_\{s\_n\}(K))=(r\_\{s\_1\}cdot\; r\_\{s\_2\}cdots\; r\_\{s\_n\})^D.,$

**Topology**In

topology , ametric space can be constructed by defining a**similarity**instead of adistance . 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**): $S\; (a,b)\; leq\; S\; (a,a)$ and $forall\; (a,b),\; S\; (a,b)\; =\; S\; (a,a)\; Leftrightarrow\; a=b$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).**elf-similarity**Self-similarity 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 alogarithmic scale it hastranslational symmetry .**ee also***

Congruence (geometry)

*Hamming distance (string or sequence similarity)

* inversive geometry

*Jaccard index

* Proportionality

*Semantic similarity

* Similarity search

*Similarity space onNumerical taxonomy **External links*** [

*http://www.plainmath.net/index.php?page=conandsim Similarity on PlainMath.Net*]

* [*http://www.mathopenref.com/similartriangles.html Animated demonstration of similar triangles*]

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