Assume that we have parametric lines \(L_0\) and \(L_1\), the challenge is to find the points \(s_0\) and \(s_1\) of closest approach. First, we must test the assumption that our lines are skew, meaining they are not parallel and do not intersect. To frame this, we take the forms of \(L_0\) and \(L_1\) and simplify them by thinking of them as parametic vectors where \(C_0\) and \(C_1\), represent the camera position vectors and \(v_0\) and \(v_1\) represent the vector was previously calculated from the subtraction of match coordinates with the camera vector. We make the simple equations:
\[L_0 = v_0 t_0 + C_0 \hspace{1cm} L_1 = v_1 t_1 + C_1\]To make sure that the lines are not parallel, which is unlikely, we must verify that their cross product is not zero. if \(v_0 \times v_1 = 0\) then we have a degenerate case with infinitely many solutions. As long as we know this is not the case we can proceed. We know that the cross product of the two vectors \(c = v_0 \times v_1\) is perpendicular to the lines \(L_0\) and \(L_1\). We know that the plane \(P\), formed by the translation of \(L1\) along \(c\), contains \(C_1\). We also know that the point \(C_1\) is perpendicular to the vector \(n_0 = v_1 \times (v_0 \times v_1)\). Thus, the intersection of \(L_0\) with \(P\) is also the point, \(s_0\), that is nearest to \(L_1\), given by the equation:
\[s_0 = C_0 + \frac{(C_1 - C_0) \cdot n_0}{v_0 \cdot n_0} \cdot v_0\]This also holds for the second line \(L_1\), the point \(s_1\), and vector \(n_1 = v_0 \times (v_1 \times v_0)\). with the equation:
\[s_1 = C_1 + \frac{(C_0 - C_1) \cdot n_1}{v_1 \cdot n_1} \cdot v_1\]Now, given to points that represent the closest points of approach, we simply find the midpoint \(m\):
\[m = \begin{bmatrix} (s_0[x] + s_1[x])/2\\ (s_0[y] + s_1[y])/2\\ (s_0[z] + s_1[z])/2 \end{bmatrix}\]
This is copied from a section of my thesis. If you found this useful to your research please consider using the following bibtex:
@mastersthesis{CalebAdamsMSThesis,
author={Caleb Ashmore Adams},
title={High Performance Computation with Small Satellites and Small Satellite Swarms for 3D Reconstruction},
school={The University of Georgia},
url={http://piepieninja.github.io/research-papers/thesis.pdf},
year=2020,
month=may
}