Triangulation
Equations (24) and (25) express the coordinates of both points \(m = \left( \begin{array}{cc} u & v \end{array} \right)\) and \(m' = \left( \begin{array}{cc} u' & v' \end{array} \right)\), respectively the left and right projections of a point \(M\) of the scene.
Asking:
\(T = \left[ \begin{array}{cccc} r_{11} & r_{12} & r_{13} & t_{x} \\ r_{21} & r_{22} & r_{23} & t_{y} \\ r_{31} & r_{32} & r_{33} & t_{z} \\ 0 & 0 & 0 & 1 \end{array} \right] \mbox{ and } T' = \left[ \begin{array}{cccc} r'_{11} & r'_{12} & r'_{13} & t'_{x} \\ r_{21} & r'_{22} & r'_{23} & t'_{y} \\ r'_{31} & r'_{32} & r'_{33} & t'_{z} \\ 0 & 0 & 0 & 1 \end{array} \right]\)
We can write the following four-equation system:
\(\begin{array}{cc} u = f_{x} \frac{r_{11}X+r_{12}Y+r_{13}Z+t_{x}}{r_{31}X+r_{32}Y+r_{33}Z+t_{z}}+C_{x} & v = f_{y} \frac{r_{21}X+r_{22}Y+r_{23}Z+t_{y}}{r_{31}X+r_{32}Y+r_{33}Z+t_{z}}+C_{y} \\ u' = f'_{x} \frac{r'_{11}X+r'_{12}Y+r'_{13}Z+t'_{x}}{r'_{31}X+r'_{32}Y+r'_{33}Z+t'_{z}}+C'_{x} & v' = f'_{y} \frac{r'_{21}X+r'_{22}Y+r'_{23}Z+t'_{y}}{r'_{31}X+r'_{32}Y+r'_{33}Z+t'_{z}}+C'_{y} \end{array}\)