# Projective Geometry

Eric Bainville - Oct 2007

## Lines II

A line is represented by a matrix A in the set L defined in the previous section. If λ ≠ 0, then λA represents the same line. Equivalently, a line is represented by a pair (a,b) of vectors in R3, verifying the conditions (a,b) ≠ (0,0) and <a,b> = 0. As with points and planes, we will say "the line A" instead of "the line represented by A" when there is no ambiguity.

The line A (with A ∈ L) is the set of all points represented by Ker(A).

The line A (with A ∈ L) is the intersection of all planes represented by Ker(A').

We have to link these definitions with the previous "constructive" ones: line through two points, and line intersection of two planes.

Given two distinct points p and q, the line pq through points p and q is represented by A=(pqT-qpT)' (note the ').

Given two distinct planes u and v, the line uv intersection of planes u and v is represented by A=uvT-vuT.

Consider a line A, and a point p not on the line. Then u = Ap is a non zero vector verifying <u,p> = 0, so u represents a plane containing p. Since u is in the image of A, it is in the orthogonal of Ker(A), and is orthogonal to all vectors of Ker(A). It means u contains all points of the line A. This can be summarized as:

Given a line A, and a point p; compute the vector u = Ap. If u = 0, then p is on the line. Otherwise, u represents the unique plane containing line A and point p.

Given a line A, and a plane u; compute the vector p = A'u. If p = 0, then u contains the line. Otherwise, p represents the unique intersection point of line A and plane u.