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Projective Space

All of the concepts that we have discussed for the projective plane, \ensuremath{{\cal P}^2}, have analogies in projective space, \ensuremath{{\cal P}^3}. For example, there is a duality between points and planes, lines are self-dual, a pencil of planes is a two-dimensional projective space,3 the cross ratio between planes is invariant, quadrics play the same role as conics, the absolute conic remains invariant under similarity transformations, and the Laguerre formula can be used to find the angle between two projection rays. For more details, see [2].

A point in \ensuremath{{\cal P}^3} is represented by a 4-tuple $\ensuremath{{\bf p}} = (X,Y,Z,W)$, and similarly for a plane $\ensuremath{{\bf n}} $. Not surprisingly, a point lies in a plane if and only if $\ensuremath{{\bf p}} ^T \ensuremath{{\bf n}} =0$. Slightly more difficult are the tasks of finding the plane which passes through three given points or of finding the intersections of planes. To answer these questions, we must first define a representation for lines.


Stanley Birchfield