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,
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.
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