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 Coordinates: Introduction, 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":

**Homogeneous Coordinates**

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**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

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### Famous quotes containing the word homogeneous:

“If we Americans are to survive it will have to be because we choose and elect and defend to be first of all Americans; to present to the world one *homogeneous* and unbroken front, whether of white Americans or black ones or purple or blue or green.... If we in America have reached that point in our desperate culture when we must murder children, no matter for what reason or what color, we don’t deserve to survive, and probably won’t.”

—William Faulkner (1897–1962)