The camera is modelled by its optical centre and its retinal plane (or image plane) . In each camera, a 3-D point in world coordinates (where the world coordinate frame is fixed arbitrarily) is projected into an image point in camera coordinates, where is the intersection of with the line containing and . In projective (or homogeneous) coordinates, the transformation from to is modelled by the linear transformation
The points for which S=0 define the focal plane and are projected to infinity.
Each pinhole camera is therefore modelled by its perspective projection matrix (PPM) , which can be decomposed into the product
The matrix gathers the intrinsic parameters of the camera, and has the following form:
where are the focal lengths in vertical and horizontal pixels, respectively, and are the coordinates of the principal point. The matrix is composed by a rotation matrix and a vector , encoding the camera position and orientation (extrinsic parameters) in the world reference frame, respectively:
Let us write the PPM as
The plane (S=0) is the focal plane, and the two planes and intersect the retinal plane in the vertical (U=0) and horizontal (V=0) axis of the retinal coordinates, respectively.
The optical centre is the intersection of the three planes introduced in the previous paragraph; therefore
The optical ray associated to an image point is the line , i.e. the set of points . The equation of this ray can be written in parametric form as
with an arbitrary real number.