# 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.

## 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 ?
Therefore, the new homogeneous coordinate of each node on fluid meshes can be calculated based on Eq.
However, DSD can cause destruction of the appearance characteristics of the model because it lacks complete homogeneous coordinate transformation.
The corresponding affine transformation can be represented by a 4 x 4 matrix T(v) in a homogeneous coordinate system as follows:
proposed a linear homogeneous coordinate transformation method [19].
Here we note that Kempf had proved (using different methods than those used in this paper) that the homogeneous coordinate ring of a Grassmanian in its Plucker embedding is wonderful (see [5]).
Based on features of geometric error and homogeneous coordinate transformation matrixes, coordinate transformation of the component of CNC machine tool in the local coordinate systems can be expressed with the product of homogeneous coordinate transformation matrix.
0] is translation matrix from left camera to right camera; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the homogeneous coordinate of the projection point position in left camera; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the homogeneous coordinate of the projection point position in right camera; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the homogeneous coordinate of a 3-D point.
Indeed, the homogeneous coordinates of the point T in the W-frame can be written as a function of [beta]:
Points in homogeneous coordinates can be written as [x'.
It is a nice illustration of a utilization of matrices in geometry, application of homogeneous coordinates, homogeneous transformation matrices and matrix algebra (Many similar exercises can be found in [Cox, 1998], [Craig, 1986], [Tsai, 1999]).
where d and o are the homogeneous coordinates of the distorted and original input images.
T] are the homogeneous coordinates of point m and M , and P is a 3 x 4 is camera projection matrix.

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