Combining equations (3.2), (3.3) and (3.6) the following expression is obtained for a camera with some specific intrinsic calibration and with a specific position and orientation:
Using (3.8) the plane corresponding to a back-projected image line can also be obtained: Since
The projection of the outline of a quadric can also be obtained. For a line in an image to be tangent to the projection of the outline of a quadric, the corresponding plane should be on the dual quadric. Substituting equation (3.9) in (2.17) the following constraint
is obtained for to be tangent to the outline. Comparing this result with the definition of a conic (2.10), the following projection equation is obtained for quadrics (this results can also be found in ).