📽️ Projective Geometry

Homogeneous Coordinates §

A point on the image plane can be extended to homogeneous coordinates with

where the third coordinate allows us to represent rays in 3-dimensional space. The third dimensions is also used in satisfy certain equations in transformations across 🗺️ Coordinate Systems.

All points along the ray are projectively equivalent. That is, they’re equivalent if they satisfy

This equivalence is also expressed as

These equivalences define equivalence classes in , each of which constitutes a ray. The projective plane is the set of all rays.

To go back from to , we must find the spot on the ray with the third coordinate equal to . Thus,

Notice that this injection requires . Rays where (of the form ) are called points at infinity. These points cannot be projected onto the image plane; geometrically, the rays are parallel to the image plane and thus will never intersect it.

Projective Lines §

A line on the image plane is defined as

In projective space, this line becomes a plane defined by orthogonal vector , and a ray is on the plane if .

Lines From Points §

Given two points and on the image plane, the line connecting them satisfies , so we can find the orthogonal vector as

There are two special forms of :

1. represents , so passes through the origin.
2. is only orthogonal to points , which are points at infinity. Thus, this line contains all points at infinity.

Points From Lines §

Two lines intersect at a point. Since the point must be orthogonal to both vectors and , we have point

Notice that if and are parallel, and , their intersection is

which is a point at infinity.