Similarity (geometry)

More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.

If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

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 elementary results concerning similar triangles in Euclidean geometry:[9] Given a triangle △ABC and a line segment DE one can, with a ruler and compass, find a point F such that △ABC ~ △DEF.

[12] In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent.

[7] Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry.

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.

A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

These include: A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have where d(x,y) is the Euclidean distance from x to y.

The altitudes of similar triangles are in the same ratio as corresponding sides.

A red segment joins a vertex of the initial polygon to its image under the similarity, followed by a red segment going to the following image of vertex, and so on to form a spiral.

For example we see the image of the initial regular pentagon under a homothety of negative ratio –k, which is a similarity of ±180° angle and a positive ratio equal to k. Below the title on the right, the second image shows a similarity decomposed into a rotation and a homothety.

Point S is the common center of the three transformations: rotation, homothety and similarity.

by naming R, H and D the previous rotation, homothety and similarity, with “D" like "Direct".

⁠ With "M" like "Mirror" and "I" like "Indirect", if M is the reflection with respect to line CW, then M ○ D = I is the indirect similarity that transforms segment BF like D into segment CT, but transforms point E into B and point A into A itself.

Square ACBT is the image of ABEF under similarity I of ratio ⁠

only possible if AK = 0, otherwise written A = K. How to construct the center S of direct similarity D from square ABEF, how to find point S center of a rotation of +135° angle that transforms ray ⁠

is an arc of circle EA that joins E and A, of which the two radius leading to E and A form a central angle of 2(180° – 135°) = 2 × 45° = 90°.

This set of points is the blue quarter of circle of center F inside square ABEF.

In the same manner, point S is a member of the blue quarter of circle of center T inside square BCAT.

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

Weaker versions of similarity would for instance have f be a bi-Lipschitz 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 { fs}s∈S with contraction factors 0 ≤ rs < 1 such that K is the unique compact subset of X for which

In topology, a metric space can be constructed by defining a similarity instead of a distance.

The definition of the similarity can vary among authors, depending on which properties are desired.

More properties can be invoked, such as: The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

Note that, in the topological sense used here, a similarity is a kind of measure.

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, ...} of numbers of the form {2i, 3·2i} where i ranges over all integers.

The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.