Editing Shape (section)

===Similarity classes===
All [[similar triangles]] have the same shape. These shapes can be classified using [[complex number]]s {{mvar|u}}, {{mvar|v}}, {{mvar|w}} for the vertices, in a method advanced by J.A. Lester<ref>J.A. Lester (1996) "Triangles I: Shapes", ''[[Aequationes Mathematicae]]'' 52:30–54</ref> and [[Rafael Artzy]]. For example, an [[equilateral triangle]] can be expressed by the complex numbers 0, 1, {{nowrap|(1 + i√3)/2}} representing its vertices. Lester and Artzy call the ratio
<math display="block">S(u, v, w) = \frac{u - w}{u - v} </math>
the '''shape''' of triangle {{math|(''u'', ''v'', ''w'')}}. Then the shape of the equilateral triangle is
<math display="block">\frac{0 - \frac{1 + i \sqrt{3}}{2}}{0 - 1} = \frac{1 + i\sqrt{3}}{2} = \cos(60^\circ) + i\sin(60^\circ) = e^{i\pi/3}.</math>
For any [[affine transformation]] of the [[complex plane]], <math>z \mapsto a z + b,\quad a \ne 0,</math> &nbsp; a triangle is transformed but does not change its shape. Hence shape is an [[invariant (mathematics)|invariant]] of [[affine geometry]].
The shape {{math|1=''p'' = S(''u'',''v'',''w'')}} depends on the order of the arguments of function S, but [[permutation]]s lead to related values. For instance,
<math display="block">1 - p = 1 - \frac{u-w}{u-v} = \frac{w-v}{u-v} = \frac{v-w}{v-u} = S(v,u,w).</math> Also <math display="block">p^{-1} = S(u,w,v).</math>
Combining these permutations gives <math>S(v,w,u) = (1 - p)^{-1}.</math> Furthermore,
<math display="block">p(1-p)^{-1} = S(u,v,w)S(v,w,u) = \frac{u-w}{v-w} = S(w,v,u). </math> These relations are "conversion rules" for shape of a triangle.

The shape of a [[quadrilateral]] is associated with two complex numbers {{mvar|p}}, {{mvar|q}}. If the quadrilateral has vertices {{math|''u''}}, {{math|''v''}}, {{math|''w''}}, {{math|''x''}}, then {{math|1=''p'' = S(''u'',''v'',''w'')}} and {{math|1=''q'' = S(''v'',''w'',''x'')}}. Artzy proves these propositions about quadrilateral shapes:
# If <math> p = (1-q)^{-1},</math> then the quadrilateral is a [[parallelogram]].
# If a parallelogram has {{math|1={{!}} arg ''p'' {{!}} = {{!}} arg ''q'' {{!}}}}, then it is a [[rhombus]].
# When {{math|1=''p'' = 1 + i}} and {{math|1=''q'' = (1 + i)/2}}, then the quadrilateral is [[square]].
# If <math>p = r(1-q^{-1})</math> and {{math|1=sgn ''r'' = sgn(Im ''p'')}}, then the quadrilateral is a [[trapezoid]].
A [[polygon]] <math> (z_1, z_2,...z_n)</math> has a shape defined by ''n'' − 2 complex numbers <math>S(z_j, z_{j+1}, z_{j+2}), \ j=1,...,n-2.</math> The polygon bounds a [[convex set]] when all these shape components have imaginary components of the same sign.<ref>[[Rafael Artzy]] (1994) "Shapes of Polygons", ''Journal of Geometry'' 50(1–2):11–15</ref>