Projective Geometry

Eric Bainville - Oct 2007

Points and Planes

Here I give several definitions to point and plane. They all are equivalent.

A point (i.e. real 3D projective point) is represented by a non-zero vector of R⁴. Two non-zero vectors p and q represent the same point if and only if there exists a non zero scalar λ such that p = λ q. A vector representing a point is usually referred to as coordinates of this point. The 0 vector does not represent a point.

A plane (i.e. real 3D projective plane) is represented by a non-zero vector of R⁴. Two non-zero vectors u and v represent the same plane if and only if there exists a non zero scalar λ such that u = λ v. A vector representing a plane is usually referred to as an equation of this plane. The 0 vector does not represent a plane.

The space (i.e. the real 3D projective space) is the set of all points.

The dual space (i.e. the real 3D projective dual space) is the set of all planes.

A point is a set of points whose representants lie in a 1-dimensional vector space.

A plane is a set of points whose representants lie in a 3-dimensional vector space.

A point is the intersection of all planes whose representants lie in a 3-dimensional vector space.

A plane is the intersection of all planes whose representants lie in a 1-dimensional vector space.

Let <x,y> = ∑ x_i y_i denote the usual dot product in R⁴.

The plane u is the set of all points p such that <u,p> = 0.

The point p is the intersection of all planes u such that <p,u> = 0.

Note that we did not define exactly what a point is. We can define a point as the set of all the vectors representing it (i.e. an equivalence class for the relation associating all proportional vectors), and the projective space is the quotient of R⁴-{0} by this equivalence relation. Here, the only way we will ever handle a point is through one of its representants, so we don't really need a very formal definition of a point. To make things easier, we can say "the point (1,2,3,4)", which means "the point represented by (1,2,3,4). The point (-1,-2,-3,-4) is the same point, and so is (2,4,6,8). The same holds for planes.