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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 ...
Furthermore, not all six components can be zero. Thus the Plücker coordinates of L may be considered as homogeneous coordinates of a point in a 5-dimensional projective space, as suggested by the colon notation. To see these facts, let M be the 4×2 matrix with the point coordinates as columns.
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.
The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean , the higher dimensional space is a real projective space . The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices.
A point in the plane may be represented in homogeneous coordinates by a triple (x, y, z) where x/z and y/z are the Cartesian coordinates of the point. [10] This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the ...
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 ...
By using homogeneous coordinates, the intersection point of two implicitly defined lines can be determined quite easily. In 2D, every point can be defined as a projection of a 3D point, given as the ordered triple (x, y, w). The mapping from 3D to 2D coordinates is (x′, y′) = ( x / w , y / w ).
Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2, K) by y = M x T, where x and y are points in K 3 (vectors) and M is an invertible 3 × 3 matrix over K. [10] Two matrices represent the same projective transformation if one is a constant multiple of the other.