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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 Mand N, and hence proportional to the cross product :

Since is bilinear, the mapping for fixed Nis 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 Pare collinear, each point is a linear combination of the two others. In particular, the determinant |MNP| vanishes. If M and Nare 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