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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 ...
with the notation : defined to mean the signed ratio of the displacement from W to X to the displacement from Y to Z. For colinear displacements this is a dimensionless quantity. If the displacements themselves are taken to be signed real numbers, then the cross ratio between points can be written
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 ).
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.
Switching to homogeneous coordinates using the embedding (a, b) ↦ (a, b, 1), the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). [9]
InPage is a word processor and page layout software by Concept Software Pvt. Ltd., an Indian information technology company. It is used for languages such as Urdu, Arabic, Balti, Balochi, Burushaski, Pashto, Persian, Punjabi, Sindhi and Shina under Windows and macOS.
CRULP (Center for research for Urdu language processing) has been working on phonetic keyboard designs for URDU and other local languages of Pakistan. Their Urdu Phonetic Keyboard Layout v1.1 for Windows is widely used and considered as a standard for typing Urdu on Microsoft platform.