Central Angle - Occupying Great Circle

Occupying Great Circle

The arc path, tracing the great circle that a central angle occupies, is measured as that great circle's azimuth at the equator, introducing an important property of spherical geometry, Clairaut's constant:

From this and relationships to ,

\begin{align}\widehat{\Alpha} &=\Big|\arcsin\big(\cos(\phi_w)\sin(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arccos\left(\frac{\sin(\phi_w)}{\sin(\widehat{\sigma}_w)}\right)\Big|,\\ &=\Big|\arctan\big(\cos(\widehat{\sigma}_w)\tan(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arctan\big(\sin(\widehat{\alpha}_w)\sin(\widehat{\sigma}_w)\cot(\phi_w)\big)\Big|.\end{align}\,\!

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