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** Up:** Hyperplanes and Duality
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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} *A*^{t}

Of course, all of this is only defined up to a scaling factor. The
matrix
*M*^{*} = (*M*^{-1})^{t} is sometimes called the *dual* of *M*.

*Bill Triggs*

*1998-11-13*