Homogeneous 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 transformations and, in general, projective transformations to be easily represented by a matrix.The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity.Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction.[8] The discussion in the preceding section applies analogously to projective spaces other than the plane.[9] The use of real numbers gives homogeneous coordinates of points in the classical case of the real projective spaces, however any field may be used, in particular, the complex numbers may be used for complex projective space.Homogeneous coordinates for projective spaces can also be created with elements from a division ring (a skew field).Another definition of the real projective plane can be given in terms of equivalence classes.Lines in this space are defined to be sets of solutions of equations of the formThe origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane.This produces a variation on the definition, namely the projective plane is defined as the set of lines inSo the projective space of dimension n can be defined as the set of lines through the origin indefined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous.In general, there is no difference either algebraically or logically between homogeneous coordinates of points and lines.This leads to the concept of duality in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will also be a theorem.A useful method, due to Julius Plücker, creates a set of six coordinates as the determinantsThe Plücker embedding is the generalization of this to create homogeneous coordinates of elements of any dimension[15][16] The homogeneous form for the equation of a circle in the real or complex projective plane iswhich has two solutions over the complex numbers, giving rise to the points with homogeneous coordinatesMöbius's original formulation of homogeneous coordinates specified the position of a point as the center of mass (or barycenter) of a system of three point masses placed at the vertices of a fixed triangle.There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of, resulting in a different system of homogeneous coordinates with the same triangle of reference.[18] Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, 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 Direct3D graphics cards take advantage of homogeneous coordinates to implement a vertex shader efficiently using vector processors with 4-element registers.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 planeMatrices 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.