Homogeneous Coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry like 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 transformations and, in general, projective transformations to be easily represented by a matrix.

If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general, 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 on the projective plane.

Read more about Homogeneous CoordinatesIntroduction, Homogeneity, Other Dimensions, Alternative Definition, Elements Other Than Points, Duality, Plücker Coordinates, Application To Bézout's Theorem, Circular Points, Change of Coordinate Systems, Barycentric Coordinates, Trilinear Coordinates, Use in Computer Graphics

Other articles related to "homogeneous coordinates, coordinates, coordinate, homogeneous":

Conic Section - Homogeneous Coordinates
... In homogeneous coordinates a conic section can be represented as Or in matrix notation The matrix is called the matrix of the conic section ...
Homogeneous Coordinates - Use in Computer Graphics
... See also Transformation matrix Homogeneous coordinates are ubiquitous in computer graphics because they allow common operations such as translation, rotation ... and points are mapped to the plane z = 1, working for the moment in Cartesian coordinates ... Dropping the now superfluous z coordinate, this becomes (x/z, y/z) ...
Translation (geometry) - Matrix Representation
... Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication Write the 3-dimensional vector w = (wx, wy, wz ... To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) can be multiplied by this translation matrix As shown below, the multiplication will give the expected result The ...
Transformation Matrix - Other Kinds of Transformations - Affine Transformations
... represent affine transformations with matrices, we can use homogeneous coordinates ... Using transformation matrices containing homogeneous coordinates, translations can be seamlessly intermixed with all other types of transformations. 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear) ...

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