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.

Similar triangles

If triangle ABC is similar to triangle DEF, then this relation can be denoted as

\triangle ABC\sim \triangle DEF

.

In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.

Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following:

{AB \over BC}={DE \over EF},

{AB \over AC}={DE \over DF},

{AC \over BC}={DF \over EF},

{AB \over DE}={BC \over EF}={AC \over DF}.

This idea can be extended to similar polygons with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional.

Angle/side similarities

A concept commonly taught in high school mathematics is that of proving the "angle" and "side" theorems, which can be used to define two triangles as similar (or indeed, congruent).

In each of these three-letter acronyms, A stands for equal angles, and S for equal sides. For example, ASA refers to an angle, side and angle that are all equal and adjacent, in that order.

AAA - Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as AA) implies similarity as well.

See also: Congruence (geometry)

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.

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