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The coordinate p 0 of p is commonly referred to as the homogenizing coordinate or the weight of p. Consequently, the set of points in 3-space, which is projectively closed by adding points at infinity, is identified with the set of all one-dimensional subspaces in ℝ 4. The homogeneous coordinate vectors pand λ pdescribe the same point for any constant factor λ ≠ 0. If the 0-th component satisfies p 0≠ 0, we may obtain the corresponding Cartesian coordinates p - = ( p - 1, p - 2, p - 3 ) T ∈ ℝ 3of the very point p from p - i = p i / p o ,where i = 1, 2, 3. (6) correspond to geometric transformations as follows: The t i represent translations, the u i represent non uniform scaling, the s i j represent shears, and the p i represent projections, where i, j ∈. This requires projective and sometimes perspective transformations which are easily handled in homogeneous coordinates and sufficiently simple as to be implemented in hardware, because the same matrix framework is used irrespective of the details of the transformation. Homogeneous coordinates are extensively used in computer graphics for computing transformations such as projection of a 3D scene onto a viewing plane (such as a computer display). Only at the end of a set of transformations is it necessary to recover the Euclidean coordinates by dividing the ( x, y, z ) values by the weight w. The weight of a point may change as a result of transformations. Multiple transformations are achieved by successive matrix multiplications or by pre multiplying the matrices to yield a single overall transformation.