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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Famous quotes containing the words occupying and/or circle:

    “... governing is occupying but not interesting, governments are occupying but not interesting ...”
    —Gertrude Stein (1874–1946)

    “There is all the difference in the world between departure from recognised rules by one who has learned to obey them, and neglect of them through want of training or want of skill or want of understanding. Before you can be eccentric you must know where the circle is.”
    —Ellen Terry (1847–1928)