## 2.1 Projective geometry

Projective geometry serves as a mathematical framework for 3D multi-view imaging and 3D computer graphics. It is used to model the image-formation process, generate synthetic images, or reconstruct 3D objects from multiple images. To model lines, planes or points in a 3D space, usually the

*Euclidean geometry*is employed. However, a disadvantage of the Euclidean geometry is that points at infinity cannot be modeled and are considered as a special case. This special case can be illustrated by using a

*perspective drawing*of two parallel lines. In perspective, two parallel lines, e.g., a railroad track, meet at infinity at the vanishing point (see Figure 2.1). However, the intersection of the parallel lines at infinity is not easily modeled by the Euclidean geometry. A second disadvantage of Euclidean geometry is that projecting a 3D point onto an image plane requires a perspective scaling operation. As the scale factor is a parameter, the perspective scaling requires a division that becomes a non-linear operation. Therefore, the use of the Euclidean geometry should be avoided.

Projective geometry constitutes an attractive framework to circumvent the
above disadvantages of the Euclidean geometry. In Euclidean space, a point
defined in three dimensions is represented by a 3-element vector (X,Y,Z)^{T }. In
the projective space, the same point is described using a 4-element vector
(X_{1},X_{2},X_{3},X_{4})^{T } such that

_{4}≠0. Usually, the coordinates (X,Y,Z)

^{T }and (X

_{1},X

_{2},X

_{3},X

_{4})

^{T }are called

*inhomogeneous coordinates*and

*homogeneous coordinates*, respectively.

As a generalization, the mapping from a point in the n-dimensional Euclidean space to an (n + 1)-dimensional projective space can be written as

| (2.2) |

where λ≠0 corresponds to a free scaling parameter. This free scaling parameter λ is often called the homogeneous scaling factor. Using the presented projective geometry framework, we now introduce the pinhole camera model.