The projection matrix

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:

$$\left[\begin{array}{c} x \\ y \\ 1 \end{array}\right] \sim \... ...left[\begin{array}{c} X \\ Y \\ Z \\ 1 \end{array}\right] \, ,$$

which can be simplified to

$${\tt m} \sim {\bf K} [ {\bf R}^\top \, \mbox{-}{\bf R}^\top {\tt t} ] {\tt M}$$ (C7)

or even

$${\tt m} \sim {\bf P} {\tt M} \enspace .$$ (C8)

The $3 \times 4$ matrix ${\bf P}$ is called the camera projection matrix.

Using (3.8) the plane corresponding to a back-projected image line ${\tt l}$ can also be obtained: Since ${\tt l}^\top {\tt m} \sim {\tt l}^\top {\bf P} {\tt M} \sim {\tt\Pi}^\top {\tt M}$,

$${\tt\Pi} \sim {\bf P}^\top {\tt l}$$ (C9)

The transformation equation for projection matrices can be obtained as described in paragraph 2.2.3. If the points of a calibration grid are transformed by the same transformation as the camera, their image points should stay the same:

$${\tt m} \sim {\bf P'} {\tt M'} \sim {\bf P} {\bf T}^{-1} {\bf T} {\tt M} \sim {\bf P} {\tt M}$$ (C10)

and thus

$${\bf P} \mapsto {\bf P}' \sim {\bf P} {\bf T}^{-1}$$ (C11)

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 ${\tt l}^\top {\bf P} {\bf Q}^* {\bf P}^\top {\tt l}= 0$ is obtained for ${\tt l}$ 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 [83]). :

$${\bf C}^* \sim {\bf PQ}^*{\bf P}^\top\enspace .$$ (C12)