# Homogeneous Coordinates

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## homogeneous coordinates

[‚hä·mə′jē·nē·əs kō′ȯrd·ən·əts]
(mathematics)
To a point in the plane with cartesian coordinates (x,y) there corresponds the homogeneous coordinates (x1, x2, x3), where x1/ x3= x, x2/ x3= y; any polynomial equation in cartesian coordinates becomes homogeneous if a change into these coordinates is made.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Homogeneous Coordinates

of a point, line, and so on, coordinates that have the property that the object determined by them does not change when all coordinates are multiplied by a nonzero number. For example, the homogeneous coordinates of a point M in the plane are three numbers X, Y, and Z, related by the equation X:Y:Z = x:y:1, where x and y are its Cartesian coordinates. The introduction of homogeneous coordinates makes it possible to extend the class of points of the Euclidean plane by the addition of points whose third homogeneous coordinate is zero (ideal points, or points at infinity). This is important in projective geometry.

References in periodicals archive ?
Let M = [[[x.sub.w] [y.sub.w] [z.sub.w] 1].sup.T] denote the homogeneous coordinates of a 3D point and m = [[u v 1].sup.T] denote the homogeneous coordinates of the corresponding image point.
The matrix maps a vector expressed in homogeneous coordinates with respect of the OUVW coordinate system to the reference coordinate OXYZ system.
It was proved that results similar to those of the unit ball hold for projective spaces, but in this case the symbols that yield commutative [C.sup.*]-algebras are those that depend only on the radial part of the homogeneous coordinates.
The homogeneous coordinates of [B.sub.i] in the local-coordinate system oxyz can be described, respectively, as [sup.o][b.sub.1] = [(0 b 0 1).sup.T], [sup.o][b.sub.2] = [(-([square root of (3)]/2)b -(1/2)b 0 1).sup.T], and [sup.o][b.sub.3] = [(([square root of (3)]/2)b -(1/2)b 0 1).sup.T]; the homogeneous coordinates of A; in the global-coordinate system OXYZ can be described, respectively, as [sup.O][a.sub.1] = [(0 a 0 1).sup.T], [sup.O][a.sub.2] = [(-([square root of (3)]/2)a -(1/2)a 0 1).sup.T], and [sup.O][a.sub.3] = [(([square root of (3)]/2)a -(1/2)a 0 1).sup.T].
Therefore, the transformational relationship of the space point P between homogeneous coordinates [([x.sub.z], [y.sub.z], [z.sub.z], 1).sup.T] in grasping point coordinates [O.sub.Z][X.sub.Z][Y.sub.Z][Z.sub.Z] and homogeneous coordinates [([x.sub.c], [y.sub.c], [z.sub.c], 1).sup.T] in the camera coordinate system [O.sub.C][X.sub.C][Y.sub.C][Z.sub.C] is
Homogeneous coordinates using the pinhole method of camera projection are
Hermes, "Homogeneous coordinates and continuous asymptotically stabilizing feedback controls," in Differential Equations, vol.
Firstly, assume ([x.sub.f], [y.sub.f], 1) to be any pixel with homogeneous coordinates in the panoramic coordinate system with a corresponding spherical homogeneous coordinate ([x.sub.Ps], [y.sub.Ps], [z.sub.Ps], 1).
To manage homogeneous coordinates, DSD entails replacing the final row of the summed quadric matrix Q with [0 0 0 1] and determining the [Q.sup.-1] to obtain the optimal vertex position.
where d and o are the homogeneous coordinates of the distorted and original input images.
Points in homogeneous coordinates can be written as [x'.sub.i] = [Hx.sub.i], where H represents the transformation of 3x3 homography matrix:

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