§ 2. Projective Geometry

The physical space is the Euclidean 3-D space E3, a real 3-dimensional affine space endowed with the inner product.

Our ambient space is the projective 3-D space P3, obtained by completing E3 with a projective plane, known as plane at infinity Π∞. In this ideal plane lie the intersections of the planes parallel in E3.

The projective (or homogeneous) coordinates of a point in P3 are 4-tuples defined up to a scale factor. We write

where ≃ indicates equality to within a multiplicative factor.

The affine points are those of P3 which do not belong to Π∞. Their projective coordinates are of the form x,y,z,1, where x,y,z are the usual Cartesian coordinates.

Π∞ is defined by its equation t=0.

The linear transformations of a projective space into itself are called collineations or homographies. Any collineation of P3 is represented by a generic 4×4 invertible matrix.

Affine transformations are the subgroup of collineations of P3 that preserves the plane at infinity (i.e., parallelism).

Similarity transformations are the subgroup of affine transformations that leave invariant a very special curve, the absolute conic, which is in the plane at infinity and whose equation is:

Similarity transformations preserves the angles.

The space is therefore stratified into more and more specialized structures:

• projective

• affine (knowing the plane at infinity)

• Euclidean (knowing the absolute conic)

The stratification reflects the amount of knowledge that we possess about the scene and the sensor.