Collinearity equation

The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the central projection of a point of the object through the optical centre of the camera to the image on the sensor plane.
Definition.
Let x,y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by formula_1, the coordinates of the image point of P on the sensor plane by "x" and "y" and the coordinates of the projection (optical) centre by formula_2. As a consequence of the projection method there is the same fixed ratio formula_3 between formula_4 and formula_5, formula_6 and formula_7, and the distance of the projection centre to the sensor plane formula_8 and formula_9. Hence:
Solving for formula_3 in the last equation and entering it in the others yields:
The point P is normally given in some coordinate system "outside" the camera by the coordinates "X", "Y" and "Z", and the projection centre by formula_16. These coordinates may be transformed through a rotation and a translation to the system on the camera. The translation doesn't influence the differences of the coordinates, and the rotation, often called camera transform, is given by a 3×3-matrix "R", transforming formula_17 into:
and 
Substitution of these expressions, leads to a set of two equations, known as the "collinearity equations":
The most obvious use of these equations is for images recorded by a camera. In this case the equation describes transformations from object space (X, Y, Z) to image coordinates (x, y). It forms the basis for the equations used in bundle adjustment. 
They indicate that the image point (on the sensor plate of the camera), the observed point (on the object) and the projection center of the camera were aligned when the picture was taken.