§ 2. Projective Geometry
The physical space is the Euclidean 3-D space
Our ambient space is the projective 3-D space
The projective (or homogeneous) coordinates of a point in
(1) |
where
The affine points are those of
The linear transformations of a projective space into itself are called
collineations or homographies. Any collineation of
Affine transformations are the subgroup of
collineations of
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:
(2) |
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.