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We are all familiar with Euclidean geometry and with the fact that it describes our three-dimensional world so well. In Euclidean geometry, the sides of objects have lengths, intersecting lines determine angles between them, and two lines are said to be parallel if they lie in the same plane and never meet. Moreover, these properties do not change when the Euclidean transformations (translation and rotation) are applied. Since Euclidean geometry describes our world so well, it is at first tempting to think that it is the only type of geometry. (Indeed, the word geometry means ``measurement of the earth.'') However, when we consider the imaging process of a camera, it becomes clear that Euclidean geometry is insufficient: Lengths and angles are no longer preserved, and parallel lines may intersect.

Euclidean geometry is actually a subset of what is known as projective geometry. In fact, there are two geometries between them: similarity and affine. To see the relationships between these different geometries, consult Figure 1. Projective geometry models well the imaging process of a camera because it allows a much larger class of transformations than just translations and rotations, a class which includes perspective projections. Of course, the drawback is that fewer measures are preserved -- certainly not lengths, angles, or parallelism. Projective transformations preserve type (that is, points remain points and lines remain lines), incidence (that is, whether a point lies on a line), and a measure known as the cross ratio, which will be described in section 2.4.

Figure 1: The four different geometries, the transformations allowed in each, and the measures that remain invariant under those transformations.
\begin{figure}\begin{tabular}{l\vert cccc}
& Euclidean & similarity & affine & p...
... & X \cr
\hspace{1em} cross ratio & X & X & X & X \cr

Projective geometry exists in any number of dimensions, just like Euclidean geometry. For example the projective line, which we denote by $\ensuremath{{\cal P}^1} $, is analogous to a one-dimensional Euclidean world; the projective plane, $\ensuremath{{\cal P}^2} $, corresponds to the Euclidean plane; and projective space, $\ensuremath{{\cal P}^3} $, is related to three-dimensional Euclidean space. The imaging process is a projection from $\ensuremath{{\cal P}^3} $ to $\ensuremath{{\cal P}^2} $, from three-dimensional space to the two-dimensional image plane. Because it is easier to grasp the major concepts in a lower-dimensional space, we will spend the bulk of our effort, indeed all of section 2, studying $\ensuremath{{\cal P}^2} $, the projective plane. That section presents many concepts which are useful in understanding the image plane and which have analogous concepts in $\ensuremath{{\cal P}^3} $. The final section then briefly discusses the relevance of projective geometry to computer vision, including discussions of the image formation equations and the Essential and Fundamental matrices.

The purpose of this monograph will be to provide a readable introduction to the field of projective geometry and a handy reference for some of the more important equations. The first-time reader may find some of the examples and derivations excessively detailed, but this thoroughness should prove helpful for reading the more advanced texts, where the details are often omitted. For further reading, I suggest the excellent book by Faugeras [2] and appendix by Mundy and Zisserman [5].

next up previous
Next: The Projective Plane Up: An Introduction to Projective Previous: An Introduction to Projective
Stanley Birchfield