In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as 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. They are also used in fundamental elliptic curve cryptography algorithms. If homogeneous coordinates of a point are multiplied by a non-zero 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.
Introduction 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. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point ( x , y ) {\displaystyle (x,y)} on the Euclidean plane, for any non-zero real number Z {\displaystyle Z} , the triple ( x Z , y Z , Z ) {\displaystyle (xZ,yZ,Z)} is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, ( x , y , 1 ) {\displaystyle (x,y,1)} is such a system of homogeneous coordinates for the point ( x , y ) {\displaystyle (x,y)} . For example, the Cartesian point ( 1 , 2 ) {\displaystyle (1,2)} can be represented in homogeneous coordinates as ( 1 , 2 , 1 ) {\displaystyle (1,2,1)} or ( 2 , 4 , 2 ) {\displaystyle (2,4,2)} . The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. The equation of a line through the origin ( 0 , 0 ) {\displaystyle (0,0)} may be written n x + m y = 0 {\displaystyle nx+my=0} where n {\displaystyle n} and m {\displaystyle m} are not both 0 {\displaystyle 0} . In parametric form this can be written x = m t , y = − n t {\displaystyle x=mt,y=-nt} . Let Z = 1 / t {\displaystyle Z=1/t} , so the coordinates of a point on the line may be written ( m / Z , − n / Z ) {\displaystyle (m/Z,-n/Z)} . In homogeneous coordinates this becomes ( m , − n , Z ) {\displaystyle (m,-n,Z)} . In the limit, as t {\displaystyle t} approaches infinity, in other words, as the point moves away from the origin, Z {\displaystyle Z} approaches 0 {\displaystyle 0} and the homogeneous coordinates of the point become ( m , − n , 0 ) {\displaystyle (m,-n,0)} . Thus we define ( m , − n , 0 ) {\displaystyle (m,-n,0)} as the homogeneous coordinates of the point at infinity corresponding to the direction of the line n x + m y = 0 {\displaystyle nx+my=0} . As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates. To summarize:
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