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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.
Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a ...
The camera matrix derived in the previous section has a null space which is spanned by the vector = This is also the homogeneous representation of the 3D point which has coordinates (0,0,0), that is, the "camera center" (aka the entrance pupil; the position of the pinhole of a pinhole camera) is at O.
In computer vision, the fundamental matrix is a 3×3 matrix which relates corresponding points in stereo images.In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point x′ on the other image must lie.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis , vertices , tangents and the pole and polar relationship between points and lines of the plane determined by the conic.
Not every triangulation method assures invariance, at least not for general types of coordinate transformations. For a homogeneous representation of 3D coordinates, the most general transformation is a projective transformation, represented by a matrix . If the homogeneous coordinates are transformed according to
where and ′ are homogeneous representations of the 2D image coordinates and ~ and ~ ′ are proper 3D coordinates but in two different coordinate systems. Another consequence of the normalized cameras is that their respective coordinate systems are related by means of a translation and rotation.