Projective Geometry

Eric Bainville - Oct 2007

Lines I

Given two distinct points represented by vectors p and q, the line pq is the set of all points whose representants are λ p + μ q, for all couples (λ,μ) in R² except (0,0).

Given two distinct planes represented by vectors u and v, the line uv is the intersection of all planes whose representants are λ u + μ v, for all couples (λ,μ) in R² except (0,0).

A line is a set of points whose representants lie in a 2-dimensional vector space.

A line is the intersection of all planes whose representants lie in a 2-dimensional vector space.

Note that we don't have yet defined any coordinates or equations for lines. Such coordinates can be defined, and are called Plücker-Grassman coordinates. They are usually defined using the six 2x2 determinants between the coordinates of two points. I will present here an equivalent 4x4 matrix definition, providing simple expressions to the main constructions and queries.

In the next chapter, I will introduce a special class of 4x4 matrices, and some operators on them. Then these matrices will be used to define line coordinates.