What is projective geometry and cross ratio in computer vision?

Question

Please explain a layman's way. I was reading this tutorial http://lear.inrialpes.fr/people/triggs/pubs/isprs96/node27.html but i am not able to understand.

Answer 1

A geometry can be thought of as describing a set of operations that are valid to perform in order to preserve certain relationships between shapes and figures. The more rigid the geometry, the fewer operations it allows you to do, but the more relationships it can preserve.

Euclidean geometry -- what we typically think of when we hear the word "geometry" -- is very rigid. The only things you're allowed to do to a shape are to rotate or translate it. If you change anything else, it's not the same shape anymore. Projective geometry, by contrast, allows much more powerful transformations, but some of those relationships can't be guaranteed anymore. The three relationships that are preserved are:

• type: Points remain points, and lines remain lines.
• incidence: If a point was on a line before, it will still be on a line.
• cross-ratio: If you have four points (a, b, c, d) that lie on the same line, a certain relationship of their distances to each other will be preserved.

The reason that's useful is that projective geometry is a much more powerful tool for modeling the way cameras and imaging works. Euclidean geometry is applicable only to idealized, flat, two-dimensional surfaces, and as such is confined to very specific cases.