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In a projective space, a collineation can be defined by its
matrix M with respect to some fixed basis. If X and X' are
the coordinate vectors of the original and transformed points, we have
X' = MX.
This maps hyperplanes of points to transformed hyperplanes, and we would
like to express this as transformation of dual (hyperplane) coordinates.
Let A and A' be the original and transformed hyperplane coordinates.
For all points X we have:
The correct transformation is therefore A'M = A or:
A' = A M-1
or if we choose to represent hyperplanes by column vectors:
A't = (M-1)t At
Of course, all of this is only defined up to a scaling factor. The
M* = (M-1)t is sometimes called the dual of M.