Projective Geometry - Homogeneous Coordinates

Birchfield, S, 1998: An Introduction to Projective Geometry for Computer Vision, Unpublished

These unpublished notes introduce projective geometry from four different points of view and give a good introduction to Plücker coordinates of a line in projective 3-space and the geometry of image formation. An HTML version of these notes can be found here.

Blinn, J, 1977: A Homogeneous Formulation for Lines in 3 Space, Proceedings of ACM SIGGRAPH'77

This paper presents a homogeneous formulation for lines in 3 dimensions as an anti-symmetric 4x4 matrix which transforms as a tensor. This tensor actually exists in both covariant and contravariant forms, both of which are useful in different situations. The derivation of these forms and their use in solving various geometrical problems is described.

Blinn, J, 1993: The Homogeneous Perspective Transform, IEEE Computer Graphics and Applications, May 1993, pp 75-80

The perspective transform basically turns space inside out. Most people don't have an intuitive feel for what this does to a shape, so this paper provides one.

Blinn, J, & Newell, M, 1978: Clipping Using Homogeneous Coordinates, Computer Graphics vol 12, no 3, pp 245-251

The clipping problem is usually solved without consideration for the differences between Euclidean space and the space represented by homogeneous coordinates. For some constructions, this leads to errors in picture generation which show up as lines marked invisible when they should be visible. This paper will examine these cases and present techniques for correctly clipping the line segments.

Bloomenthal, J, & Rokne, J, 1994: Homogeneous Coordinates, The Visual Computer, vol 11, pp 15-26

Our aim here is to provide an intuitive yet theoretically based discussion that assembles the key features of homogeneous coordinates and their applications to Computer Graphics. These applications include affine transformations, perspective projection, line intersections, clipping, and rational curves and surfaces.

Davis, T R, 2000: Homogeneous Coordinates, Unpublished

When it comes to projective geometry, the calculations are simple, but may appear mysterious or even somewhat magical the first time through. What in the heck is the w coordinate? Where on earth did the formula for perspective come from? A complete de-mystification requires both a knowledge of the calculations (a left-brain task), and some good geometric mental images for your right brain. This article relates the calculations to a good mental model of projective geometry.

Davis, T R, 2001a: Homogeneous Coordinates and Computer Graphics, Unpublished

We begin the study of homogeneous coordinates by describing a set of problems from three-dimensional computer graphics that at first seem to have unrelated solutions. We will then show that with certain “tricks”, all of them can be solved in the same way. Finally, we will show that this “same way” is in fact just a recasting of the original problems in terms of projective geometry.

Davis, T R, 2001b: Projective Geometry, Unpublished

If you are good at the mechanics of painting, but have no sense at all of how to render a scene in perspective, there is a completely mechanical way to get a highly accurate rendition. Instead of a canvas, use a piece of glass. If you imagine the lines that light follows as it moves from the various objects to your eye through the glass, light rays from the top and bottom of an object will make an angle that basically determines the size of the object's image on the glass. If the same object is further away, the angle will be smaller, so the image on the glass will also be smaller. This is the basic idea behind perspective drawing. It is also the basic idea behind projective geometry, which tells us how the drawings of objects on the glass are related to the positions of the objects in the real world, to the position of the glass, and to the position of the eye.