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 same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other.

For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition.

03:35, 14 December 2006 (UTC)68.187.36.27

Similarity in Euclidean space

One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean 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))=rd(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 an isometry.

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 $f(z)=a{\overline {z}}+b$ , and all affine transformations are of the form $f(z)=az+b{\overline {z}}+c$ (a, b, and c complex).

Similarity 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))=rd(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 $\{f_{s}\}_{s\in S}$ with contraction factors $0\leq r_{s}<1$ such that K is the unique compact subset of X for which

\cup _{s\in S}f_{S}(K)=K\,

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

\sum _{s\in S}(r_{s})^{D}=1\,

which is often (but not always) equal to the set's Hausdorff dimension and Packing 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}}\cdots f_{s_{n}}(K))=(r_{s_{1}}\cdot r_{s_{2}}\cdots r_{s_{n}})^{D}

Topology

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

Self-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 a logarithmic scale it has translational symmetry.

External links