similarity in geometry
Two figures $K$ and ${K}^{\prime}$ in a Euclidean plane^{} or space (http://planetmath.org/EuclideanVectorSpace) are similar^{} iff there exists a bijection^{} $f$ from the set of points of $K$ onto the set of points of ${K}^{\prime}$ such that, for any $P,Q\in K$, the ratio
$$\frac{{P}^{\prime}{Q}^{\prime}}{PQ}$$ 
of the lengths of the line segments^{} ${P}^{\prime}{Q}^{\prime}$ and $PQ$ is always the same number $k$, where ${P}^{\prime}=f(P)$ and ${Q}^{\prime}=f(Q)$.
The number $k$ is called the ratio of similarity or the line ratio of the figure ${K}^{\prime}$ with respect to the figure $K$ (N.B. the in which the figures are mentioned!). The similarity of $K$ and ${K}^{\prime}$ is often denoted by
$${K}^{\prime}\sim K(\text{or}K\sim {K}^{\prime}).$$ 
Examples

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All squares are similar.

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All cubes are similar.

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All circles are similar.

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All parabolas^{} are similar.

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All sectors of circle with equal central angle^{} are similar.

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All spheres are similar.

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All equilateral triangles^{} are similar.
Nonexamples

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Not all rectangles^{} are similar.

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Not all rhombi are similar.

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Not all rectangular prisms are similar.

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Not all ellipses are similar.

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Not all ellipsoids^{} are similar.

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Not all triangles^{} are similar.

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The corresponding angles (consisting of corresponding points) of two similar figures are equal.

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The lengths of any corresponding arcs of two similar figures are proportional in the ratio $k$.

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The areas of two similar regions are proportional in the ratio ${k}^{2}$ when $k$ is the line ratio of the regions.

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The volumes of two similar solids are proportional in the ratio ${k}^{3}$ when $k$ is the line ratio of the solids.
Remarks

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In any Euclidean space $E$, the relation^{} (http://planetmath.org/Relation) of similarity (denoted $\sim $) on the set of figures in $E$ is an equivalence relation^{}.

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If one pair of corresponding line segments in the similar figures $K$ and ${K}^{\prime}$ are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal : They are congruent (http://planetmath.org/Congruence^{}) (${K}^{\prime}\cong K$).
Title  similarity in geometry 
Canonical name  SimilarityInGeometry 
Date of creation  20130322 17:08:48 
Last modified on  20130322 17:08:48 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  19 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 51F99 
Classification  msc 51M05 
Classification  msc 5100 
Synonym  similarity 
Synonym  similitude 
Related topic  Homothety 
Related topic  ProportionEquation 
Related topic  HarmonicMeanInTrapezoid 
Defines  similar 
Defines  ratio of similarity 
Defines  similitude ratio 
Defines  line ratio 