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There are several point ↔line dualities (such as the very useful Hough transform) in the plane that do not generalize well for higher dimensions. This is because the natural duality is point ↔hyperplane in projective N-space ℙN with point ↔line when N = 2. However, in ∥-coords there is a useful and direct generalization, which is the subject of this chapter. At first the basic idea for lines in ℝN, rather than ℙN, is derived intuitively, paving the way for the general case, which is treated subsequently (as in  and ).
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