Central Angle - Calculation of TvL

Calculation of TvL

The calculation of and can be found using a common subroutine:

{}_{\color{white}.}\!\begin{pmatrix}\operatorname{sgn}(V)=|V|\!\cdot V^{-1};\quad\overrightarrow{\operatorname{sgn}}(V)=\operatorname{sgn}\big(\operatorname{sgn}(V)+\frac{1}{2}\big)\\{}_{(\,\operatorname{sgn}(0)=0;\qquad\overrightarrow{\operatorname{sgn}}(0)=+1\,)}\end{pmatrix}{}_{\color{white}.}\!\!\,\!
\begin{align}\phi_w=\phi_s;\; &\phi_c=\phi_f\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\\ &\widehat{\sigma}_s=\widehat{\sigma}_w\!\cdot\overrightarrow{\mbox{sgn}}(S\!B_w)+\pi\!\cdot\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w)\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(S\!B_w));\end{align}\,\!
\begin{align}\phi_w=\phi_f;\; &\phi_c=\phi_s\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\\ &\widehat{\sigma}_f=\widehat{\sigma}_w\!\cdot\overrightarrow{\mbox{sgn}}(-S\!B_w)+\pi\!\cdot\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w)\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(-S\!B_w))\\ &\qquad\qquad\qquad\qquad\quad+2\pi\!\cdot\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w-\widehat{\sigma}_s));\end{align}\,\!

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Each point has at least two values, both a forward and reverse value.

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    ““To my thinking” boomed the Professor, begging the question as usual, “the greatest triumph of the human mind was the calculation of Neptune from the observed vagaries of the orbit of Uranus.”
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