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

• All squares are similar.

• All cubes are similar.

• All circles are similar.

• All parabolas are similar.

• All sectors of circle with equal central angle are similar.

• All spheres are similar.

• All equilateral triangles are similar.

Nonexamples

• Not all rectangles are similar.

• Not all rhombi are similar.

• Not all rectangular prisms are similar.

• Not all ellipses are similar.

• Not all ellipsoids are similar.

• Not all triangles are similar.

Properties

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

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

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

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

Remarks

• 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.

• 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	2013-03-22 17:08:48
Last modified on	2013-03-22 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 51-00
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