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##

Lines in the Plane and Incidence

We will develop the theory of lines in the projective plane
.
Most of the results can also be extended to higher dimensions.
Let
and
be homogeneous
representatives for two distinct points in the plane (*i.e.* *M* and
*N* are linearly independent 3-vectors:
). Let
*L*=(*a*,*b*,*c*) be a the dual coordinates of a line (hyperplane) in the
plane. By the definition of a hyperplane, *M* lies on *L* if and only
if the dot product
vanishes, *i.e.* if and only if the
3-vector *L* is orthogonal to the 3-vector *M*. The line *MN* through
*M* and *N* must be represented by a 3-vector *L* orthogonal to both *M*and *N*, and hence proportional to the cross product :

Since
is bilinear, the mapping
for fixed *N*is a linear mapping defined by the matrix

The vector *N* generates the kernel of this mapping.
Another way to characterize the line *MN* is as the set of points
for arbitrary
.
Evidently

Dually, if *L* and *L*' are two lines defined by their dual coordinates,
then
is some other line through the intersection
*X* of *L* and *L*' (since
implies
). As
varies, this line traces out the
entire pencil of lines through *X*. By duality,
.

One further way to approach lines is to recall that if *M*, *N* and *P*are collinear, each point is a linear combination of the two others. In
particular, the
determinant |*MNP*| vanishes. If *M* and *N*are fixed, this provides us with a linear constraint that *P* must
satisfy if it is to lie on the line:
.

** Next:** The Fixed Points of
** Up:** Linear Algebra and Homogeneous
** Previous:** Linear Algebra and Homogeneous
*Bill Triggs*

*1998-11-13*