. 2020 Aug 4;9(9):6. doi: 10.1167/tvst.9.9.6

Table A1.

Derivation of the X, Y, and Z Components of $P_{3}^{W}$

Solving for the x-component:

\begin{matrix} \begin{matrix} P_{3}^{W} |_{X} = L_{P} |_{X} + \frac{z_{C} + r sin (θ_{E L E V}) - S_{H}}{z_{C} + r sin (θ_{E L E V})} ({P_{2} |}_{X} - {L_{P} |}_{X}) \\ P_{3}^{W} |_{X} = r cos (θ_{E L E V}) + \frac{z_{C} + r sin (θ_{E L E V}) - S_{H}}{z_{C} + r sin (θ_{E L E V})} (0 - r cos (θ_{E L E V})) \\ P_{3}^{W} |_{X} = r cos (θ_{E L E V}) - \frac{(z_{C} + r sin (θ_{E L E V}) - S_{H}) r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} \\ P_{3}^{W} |_{X} = \frac{(z_{C} + r sin (θ_{E L E V})) r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} - \frac{(z_{C} + r sin (θ_{E L E V}) - S_{H}) r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} \\ P_{3}^{W} |_{X} = \frac{(z_{C} + r sin (θ_{E L E V})) r cos (θ_{E L E V}) - (z_{C} + r sin (θ_{E L E V}) - S_{H}) r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} \\ P_{3}^{W} |_{X} = \frac{(z_{C} + r sin (θ_{E L E V}) - z_{C} - r sin (θ_{E L E V}) + S_{H}) r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} \\ P_{3}^{W} |_{X} = S_{H} \frac{r cos (θ_{E L E V})}{z_{C} + r sin (θ_{E L E V})} \end{matrix} (A 5) \end{matrix}

Solving for the y-component:

\begin{matrix} \begin{matrix} P_{3}^{W} |_{Y} = L_{P} |_{Y} + \frac{z_{C} + r sin (θ_{E L E V}) - S_{H}}{z_{C} + r sin (θ_{E L E V})} ({P_{2} |}_{Y} - {L_{P} |}_{Y}) \\ P_{3}^{W} |_{Y} = 0 + \frac{z_{C} + r sin (θ_{E L E V}) - S_{H}}{z_{C} + r sin (θ_{E L E V})} (y_{T} - 0) \\ P_{3}^{W} |_{Y} = y_{T} \frac{z_{C} + r sin (θ_{E L E V}) - S_{H}}{z_{C} + r sin (θ_{E L E V})} \end{matrix} (A 6) \end{matrix}

And by definition, the

P_{3}^{W}

lies on the z = S_H plane, thus

P_{3}^{W} |_{Z} = S_{H}