**Similarity Classes**

All similar triangles have the same shape. These shapes can be classified using complex numbers in a method advanced by J.A. Lester and Rafael Artzy. For example, an equilateral triangle can be expressed by complex numbers 0, 1, (1 + i √3)/2. Lester and Artzy call the ratio

- S(
*u,v,w*) = (*u*−*w*)/(*u*−*v*) the**shape**of triangle (*u, v, w*). Then the shape of the equilateral triangle is - (0–(1+ √3)/2)/(0–1) = ( 1 + i √3)/2 = cos(60°) + i sin(60°) = exp(i π/3).

For any affine transformation of the Gaussian plane, *z* mapping to *a z + b, a* ≠ 0, a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry. The shape *p* = S(*u,v,w*) depends on the order of the arguments of function S, but permutations lead to related values. For instance,

- Also

Combining these permutations gives Furthermore,

- These relations are "conversion rules" for shape of a triangle.

The shape of a quadrilateral is associated with two complex numbers *p,q*. If the quadrilateral has vertices *u,v,w,x*, then *p* = S(*u,v,w*) and *q* = S(*v,w,x*). Artzy proves these propositions about quadrilateral shapes:

- If then the quadrilateral is a parallelogram.
- If a parallelogram has |arg
*p*| = |arg*q*|, then it is a rhombus. - When
*p*= 1 + i and*q*= (1 + i)/2, then the quadrilateral is square. - If and sgn
*r*= sgn(Im*p*), then the quadrilateral is a trapezoid.

A polygon has a shape defined by *n* – 2 complex numbers The polygon bounds a convex set when all these shape components have imaginary components of the same sign.

Read more about this topic: Shape

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