Shape - Similarity Classes

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) = (uw)/(uv) 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:

  1. If then the quadrilateral is a parallelogram.
  2. If a parallelogram has |arg p| = |arg q|, then it is a rhombus.
  3. When p = 1 + i and q = (1 + i)/2, then the quadrilateral is square.
  4. 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.

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