Real life usage of the projective plane theory

Question

I'm learning about the theory of the projective plane. Very generally speaking, it is an extension of the plane, which includes additional points which are defined as the intersection points of two parallel lines. In the projective plane, every two lines have an interesection point. Whether they're parallel or not. Every point in the projective plane can be represented by three numbers (you actually need less than that, but nevemind now).

Is there any real life application which uses the projective plane? I can think that, for instance, a software which needs to find the intersections of a line, can benefit from always having an intersection point which might lead to simpler code, but is it really used?

Answer 1

A very real use of the projective plane is in the field of camera calibration, or more precisely in camera resectioning. The object there is to find the transformation from 3D space to the 2D camera image. Using projective geometry, one can define this transformation as a linear mapping from 4D to 3D. Thus, the transformation is given by the 3x4 camera matrix. Wikipedia, as usual, has a nice article on this to get you started.

Answer 2

Projective geometry is used extensively in computer vision, essentially because taking a picture (a 2D perspective image of a 3D world) exactly corresponds to a projective transformation. The spatial information that can be recovered from a planar image is thus subject to projective constraints. Stereo vision, camera calibration, shape from motion, and many other computer vision techniques utilize projective geometry.

It also has many applications in computational geometry, mostly arising due to duality.

Answer 3

Elliptic curves defined in the projective plane have real life applications in cryptography.