Raytracing and the camera matrix – a connection

In this section we derive the algebraic equations commonly used in the computer graphics technique of raytracing, where the simulation of geometric propagation of rays in an illuminated scene is used to produce highly realistic renderings of three-dimensional virtual scenes. Somewhat counter-intuitively, but for reasons of efficiency, in practical raytracer implementations, ray trajectories originate not from the illumination source but rather from the camera center, proceeding into the scene where the ray may encounter objects or illumination sources. For the purpose of raytracing, it is convenient to describe the properties of the camera in a manner that simplifies the problem of projecting rays from the camera’s center of projection (COP) into the three-dimensional scene. In particular, the following parameters are sufficient to describe the geometric projection characteristics of the pinhole camera: (1) the COP or camera center C, (2) the principal, or optical axis p̂, (3) the horizontal axis of the image plane û, (4) the vertical axis of the image plane v̂, and (5) the focal length f . Vectors are typeset in bold face, and unit vectors are denoted with a hat. We make the simplifying assumption, which is true for most cameras, that p̂, û and v̂ form an orthonormal set. The camera geometry and parameters are illustrated in Fig. (1), where the camera’s COP is positioned in the three-dimensional world-coordinate systems denoted by the X, Y and Z axes. The principal axis p̂ intersects the focal-plane at the principal point at the origin of the (u, v)