Editing Homogeneous coordinates (section)

==Use in computer graphics and computer vision==
{{See also|Transformation matrix}}

Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as [[Translation (geometry)|translation]], [[Rotation (mathematics)|rotation]], [[Scaling (geometry)|scaling]] and [[perspective projection]] to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple and efficient processing. By contrast, using Cartesian coordinates, translations and perspective projection cannot be expressed as matrix multiplications, though other operations can. Modern [[OpenGL]] and [[Microsoft Direct3D|Direct3D]] [[graphics card]]s take advantage of homogeneous coordinates to implement a [[vertex shader]] efficiently using [[vector processor]]s with 4-element registers.<ref>{{cite web|url=http://msdn.microsoft.com/en-us/library/bb206341(VS.85).aspx|title=Viewports and Clipping (Direct3D 9) (Windows)|website=msdn.microsoft.com|access-date=10 April 2018}}</ref><ref>Shreiner, Dave; Woo, Mason; Neider, Jackie; Davis, Tom; "OpenGL Programming Guide", 4th Edition, {{isbn|978-0-321-17348-5}}, published December 2004.  Page 38 and Appendix F (pp. 697-702) Discuss how [[OpenGL]] uses homogeneous coordinates in its rendering pipeline.  Page 2 indicates that OpenGL is a software interface to [[graphics card|graphics hardware]].</ref>

For example, in perspective projection, a position in space is associated with the line from it to a fixed point called
the ''center of projection''. The point is then mapped to a plane by finding the point of intersection of that plane and
the line. This produces an accurate representation of how a three-dimensional object appears to the eye. In the simplest
situation, the center of projection is the origin and points are mapped to the plane {{nowrap|<math>z = 1</math>}}, working for
the moment in Cartesian coordinates. For a given point in space, {{nowrap|<math>(x, y, z)</math>}}, the point where the
line and the plane intersect is {{nowrap|<math>(x/z, y/z, 1)</math>}}. Dropping the now superfluous <math>z</math> coordinate,
this becomes {{nowrap|<math>(x/z, y/z)</math>}}. In homogeneous coordinates, the point {{nowrap|<math>(x, y,
z)</math>}} is represented by {{nowrap|<math>(xw, yw, zw, w)</math>}} and the point it maps to on the plane is
represented by {{nowrap|<math>(xw, yw, zw)</math>}}, so projection can be represented in matrix form as
<math display="block">\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}</math>
Matrices representing other geometric transformations can be combined with this and each other by matrix multiplication.
As a result, any perspective projection of space can be represented as a single matrix.<ref>{{cite book |title=Mathematics for Computer Graphics Applications| url=https://archive.org/details/mathematicsforco00mort|url-access=limited|first=Michael E.|last=Mortenson |publisher=Industrial Press Inc.|year=1999| isbn=0-8311-3111-X|page=[https://archive.org/details/mathematicsforco00mort/page/n330 318]}}</ref><ref>{{cite book| title=Computer Graphics: Theory into Practice|first=Jeffrey J.|last=McConnell|publisher=Jones & Bartlett Learning|year=2006|isbn=0-7637-2250-2|page=[https://archive.org/details/computergraphics0000mcco/page/120 120]|url=https://archive.org/details/computergraphics0000mcco/page/120}}</ref>