# 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 R^{4}.
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 R^{4}.
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 R^{4}.

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 R^{4}-{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.

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