Projective geometry

The aim of this dossier is to introduce the basics of two- and three-dimensional projective geometry. Our starting point for this presentation is affine geometry. Projective geometry has a number of advantages over affine geometry, without losing any of its basic concepts, which can easily be recovered, since affine space is contained within projective space. The major advantage of projective geometry is that its formulation is more uniform and homogeneous than that of affine geometry. For example, in the projective plane, there is no notion of line parallelism, as two lines always intersect at a point. So, to represent a projective transformation, all you need is a matrix. It does not require a translation vector like its affine counterpart. We'll also see that projective geometry encompasses many geometries in addition to affine geometry, such as Euclidean geometry and non-Euclidean geometries, known as Cayley-Klein geometries.

In particular, we deal with :

• notions of projective reference points, coordinates and transformations;

• the notion of the biratio of four aligned points, the projective analogue of the affine ratio of three aligned points;

• the principle of duality, which maps points and lines in the projective plane (respectively points and planes in three-dimensional projective space), making it possible to obtain new theorems from known theorems by dualizing them;

• classical theorems, such as the theorems of Desargues, Pappus and Pascal, as well as their dual correspondents;

• Felix Klein's Erlangen program identifying "geometry" with "the theory of invariants of a group of transformations" and the role of projective geometry for Euclidean and non-Euclidean geometries.

These concepts have many applications, particularly in the fields of CAD (computer-aided design), computer graphics and physics.

In CAD, the initial standard for curves and surfaces was polynomial representations, see for example [8]. Thanks to the methods of projective geometry, rational curves and surfaces have now been integrated into the majority of CAD programs under the term "NURBS", which stands for "Non-Uniform Rational B-Splines". For an introduction to NURBS and its links with projective geometry, see [7]. Many current research papers in the field of CAD use results from projective geometry, for example:

• to satisfy projective constraints such as incidence, collinearity and intersection, in the context of computer-aided perspective drawing [22] ;

• to recover geometric quantities of conics...