Editing Homogeneous coordinates (section)

{{Short description|Coordinate system used in projective geometry}}
[[File:RationalBezier2D.svg|thumb|upright=1.15|Rational Bézier curve &ndash; polynomial curve defined in homogeneous coordinates (blue) and its projection on plane &ndash; rational curve (red)]]

In [[mathematics]], '''homogeneous coordinates''' or '''projective coordinates''', introduced by [[August Ferdinand Möbius]] in his 1827 work {{lang|de|Der barycentrische Calcul}},<ref>August Ferdinand Möbius: ''Der barycentrische Calcul'', Verlag von Johann Ambrosius Barth, Leipzig, 1827.</ref><ref>{{MacTutor|class=Biographies|id=Mobius|title=August Ferdinand Möbius}}</ref><ref>
{{cite book |title=History of Modern Mathematics|first=David Eugene|last=Smith
|publisher=J. Wiley & Sons|year=1906|page=[https://archive.org/details/historymodernma00smitgoog/page/n58 53]
|url=https://archive.org/details/historymodernma00smitgoog}}</ref> are a [[system of coordinates]] used in [[projective geometry]], just as [[Cartesian coordinate system|Cartesian coordinates]] are used in [[Euclidean geometry]]. They have the advantage that the coordinates of points, including [[points at infinity]], can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including [[computer graphics]] and 3D [[computer vision]], where they allow [[affine transformation]]s and, in general, [[projective transformation]]s to be easily represented by a [[Transformation matrix|matrix]]. They are also used in fundamental [[elliptic curve cryptography]] algorithms.<ref>{{Cite web|url=https://datatracker.ietf.org/doc/html/rfc6090|title=Fundamental Elliptic Curve Cryptography Algorithms|date=February 2011 |last1=Igoe |first1=Kevin |last2=McGrew |first2=David |last3=Salter |first3=Margaret }}</ref>

If homogeneous coordinates of a point are multiplied by a non-zero [[Scalar (mathematics)|scalar]] then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the [[projective space]] being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
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This to too mathy for the lead section, try to merge with later section:
Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space ''V'' of dimension ''n''&nbsp;+&nbsp;1, introduce coordinates in ''V'' by choosing a basis, and use these in ''P''(''V''), the equivalence classes of proportional non-zero vectors in ''V''. -->