Similarity (geometry)

Shapes shown in the same color are similar

Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, or 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 equilateral triangles 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. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar.

This article assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.

Similar triangles

In order to understand the concept of similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale.

If you were to draw a map, you would probably try to preserve the shape of what you are mapping, while you would make your picture at a unit rate that is in proportion to the original size or value.

In particular, similar triangles are triangles that have the same shape and are up to scale of one another. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserve their measures.

Formally speaking, we say that two triangles $\triangle ABC$ and $\triangle DEF$ are similar if either of the following conditions holds:

1. Corresponding sides have lengths in the same ratio:

i.e.

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

. This is equivalent to saying that one triangle is an enlargement of the other.

2. $\angle BAC$ is equal in measure to $\angle EDF$ , and $\angle ABC$ is equal in measure to $\angle DEF$ . This also implies that $\angle ACB$ is equal in measure to $\angle DFE$ .

When two triangles $\triangle ABC$ and $\triangle DEF$ are similar, we write

\triangle ABC\sim \triangle DEF\,

The 'is similar to' symbol can also be expressed as three vertical lines: lll

This idea extends to similar polygons with more sides. Given any two similar polygons, corresponding sides are proportional. However, proportionality of corresponding sides is not sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Corresponding angles must also be equal in measure.

Angle/side similarities

The following three criteria are sufficient to prove that a pair of triangles are similar. In summary, they state that if triangles have the same shape then they are to scale (AA criterion), and that if they are to scale then they have the same shape (SSS). Another extra criterion, SAS, will also be explained below.

AA: if two triangles have two corresponding pairs of angles with the same measure then they are similar. Sometimes this criterion is also referred to as AAA because two angles of equal measure implies equality of the third. This criterion means that if a triangle is copied to preserve the shape, then the copy is "to scale.

SSS: If the ratio of corresponding sides of two triangles does not depend on the sides chosen, then the triangles are similar. This means that if any triangle copied to scale is also copied in shape.

SAS: if two sides are taken in a triangle, that are proportional to two corresponding sides in another triangle, and the angles included between these sides have the same measure, then the triangles are similar. This means that in order to enlarge a triangle, it is sufficient to copy one angle, and to scale just the two sides that form the angle.

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. Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E(n) is a subgroup of the affine group.

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

\bigcup _{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}}\circ \cdots \circ 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

Animated demonstration of similar triangles