similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not 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. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

Two congruent shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

Similar triangles

Two triangles, and , are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional. It can be shown that two triangles having congruent angles (''equiangular triangles'') are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.

There are several criteria each of which is necessary and sufficient for two triangles to be similar:

*Any two pairs of congruent angles, which in Euclidean geometry implies that their all three angles are congruent:

::If is equal in measure to , and is equal in measure to , then this implies that is equal in measure to and the triangles are similar.

*All the corresponding sides are proportional:

:: . This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

*Any two pairs of sides are proportional, and the angles included between these sides are congruent:
:: and is equal in measure to .
This is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.

Symbolically, we write the similarity and dissimilarity of two triangles and as follows:

:$ABC\sim A'B'C'$
:$ABC\nsim A'B'C'$

There are several elementary results concerning similar triangles in Euclidean geometry:
* Any two equilateral triangles are similar.
* Two triangles, both similar to a third triangle, are similar to each other ( transitivity of similarity of triangles).
* Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
* Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion.

Given a triangle and a line segment one can, with ruler and compass, find a point such that . The statement that the point satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent.

In the axiomatic treatment of Euclidean geometry given by George David Birkhoff (see Birkhoff's axioms) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.

Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry.

Other similar polygons

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) 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). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

For given ''n'', all regular ''n''-gons are similar.

Similar curves

Several types of curves have the property that all examples of that type are similar to each other. These include:
* Lines (any two lines are even congruent)
*Line segments
*Circles
* Parabolas
* Hyperbolas of a specific eccentricityThe shape of an ellipse or hyperbola depends only on the ratio b/a
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* Ellipses of a specific eccentricity
*Catenaries
*Graphs of the logarithm function for different bases
*Graphs of the exponential function for different bases
* Logarithmic spirals are self-similar

In Euclidean space

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection from the space onto itself that multiplies all distances by the same positive real number , so that for any two points and we have

:$d(f(x),f(y)) = r\, d(x,y), \,$

where "" is the Euclidean distance from to . The scalar has many names in the literature including; the ''ratio of similarity'', the ''stretching factor'' and the ''similarity coefficient''. When = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity.

As a map , a similarity of ratio takes the form

:$f(x) = rAx + t,$

where is an