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)

    “It is a good lesson—though it may often be a hard one—for a man who has dreamed of literary fame, and of making for himself a rank among the world’s dignitaries by such means, to step aside out of the narrow circle in which his claims are recognized, and to find how utterly devoid of all significance, beyond that circle, is all that he achieves, and all he aims at.”
    —Nathaniel Hawthorne (1804–1864)