Editing Shape (section)

== Equivalence of shapes ==
In geometry, two subsets of a [[Euclidean space]] have the same shape if one can be transformed to the other by a combination of [[translation (geometry)|translations]], [[rotation]]s (together also called [[rigid transformation]]s), and [[Scaling (geometry)|uniform scalings]]. In other words, the ''shape'' of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an [[equivalence relation]], and accordingly a precise mathematical definition of the notion of shape can be given as being an [[equivalence class]] of subsets of a Euclidean space having the same shape.

Mathematician and statistician [[David George Kendall]] writes:<ref>{{cite journal|
doi = 10.1112/blms/16.2.81|
author = Kendall, D.G.|
title = Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces|
journal = Bulletin of the London Mathematical Society|
year = 1984|
volume = 16|
issue = 2|
pages = 81–121|
url = http://image.diku.dk/imagecanon/material/kendall-shapes.pdf}}</ref>
<blockquote>In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale<ref>Here, scale means only [[uniform scaling]], as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).</ref> and rotational effects are filtered out from an object.’</blockquote>

Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size and placement in space of the object. For instance, a "'''<small>d</small>'''" and a "'''<big>p</big>'''" have the same shape, as they can be perfectly superimposed if the "'''<small>d</small>'''" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see [[Procrustes superimposition]] for details). However, a [[mirror image]] could be called a different shape. For instance, a "'''<big>b</big>'''" and a "'''<big>p</big>'''" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object is scaled non-uniformly. For example, a [[sphere]] becomes an [[ellipsoid]] when scaled differently in the vertical and horizontal directions. In other words, preserving axes of [[symmetry]] (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object.

===Congruence and similarity===
{{Main|Congruence (geometry)|Similarity (geometry)}}
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are [[Congruence (geometry)|congruent]]. An object is therefore congruent to its [[mirror image]] (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size.

Objects that have the same shape or mirror image shapes are called [[geometrically similar]], whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size.

=== Homeomorphism ===
{{Main|Homeomorphism}}
A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions.

One way of modeling non-rigid movements is by [[homeomorphism]]s. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus|donut]] are not. An often-repeated [[mathematical joke]] is that topologists cannot tell their coffee cup from their donut,<ref>{{cite book|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|first1=John H.|last1=Hubbard|first2=Beverly H.|last2=West|publisher=Springer|series=Texts in Applied Mathematics|volume=18|year=1995|isbn=978-0-387-94377-0|page=204|url=https://books.google.com/books?id=SHBj2oaSALoC&q=%22coffee+cup%22+topologist+joke&pg=PA204}}</ref> since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. 

A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper.

=== Shape analysis ===
{{main|Statistical shape analysis}}
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of [[statistical shape analysis]]. In particular, [[Procrustes analysis]] is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example [[Spectral shape analysis]]).

===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>