# Central angle

Central angle

.

Coordinates

On a sphere or ellipsoid, the central angle is delineated along a great circle.The usually provided coordinates of a point on a sphere/ellipsoid is its common latitude ("Lat"), $phi,!$, and longitude ("Long"), $lambda,!$. The "point", $widehat\left\{sigma\right\},!$, is actually——relative to the great circle it is being measured on——the "transverse colatitude" ("TvL"), and the central angle/angular distance is the difference between two TvLs, $Deltawidehat\left\{sigma\right\},!$.

Calculation of TvL

The calculation of $widehat\left\{sigma\right\}_s,!$ and $widehat\left\{sigma\right\}_f,!$ can be found using a common subroutine:

:$V_s,V_f,V_w,V_c:mathrm\left\{;Standpoint, forepoint, working, coworking values\right\};,!$:::: $widehat\left\{alpha\right\}_w:mathrm\left\{;Orthodromic azimuth at widehat\left\{sigma\right\}_w\right\};,!$

:::

:$Deltalambda=lambda_f-lambda_s;,!$

:::$\left\{\right\}_\left\{color\left\{white\right\}.\right\}!left\left(mbox\left\{If \right\} phi_s=phi_f=0mbox\left\{, then \right\};widehat\left\{sigma\right\}_s=frac\left\{pi-|Deltalambda\left\{2\right\},;widehat\left\{sigma\right\}_f=frac\left\{pi+|Deltalambda\left\{2\right\} ight\right)\left\{\right\}_\left\{color\left\{white\right\}.\right\}!!,!$

:

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

Occupying great circle

The arc path, $scriptstyle\left\{widehat\left\{Alpha,!$, 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:

:::::$sin\left(widehat\left\{Alpha\right\}\right)=Big|cos\left(phi_w\right)sin\left(widehat\left\{alpha\right\}_w\right)Big|;,!$

From this and relationships to $widehat\left\{sigma\right\},!$,

:

Angular distance formulary

The angular distance can be calculated either directly as the TvL difference, or via the common coordinates (here, either SAw, SBw value set can be used):

and, using half-angles,

:

There is also a logarithmical form:

:$\left\{\right\}_\left\{color\left\{white\right\}.\right\};mathbb\left\{N\right\}=expleft\left(ln!left\left(frac\left\{cosleft\left(frac\left\{phi_f-phi_s\right\}\left\{2\right\} ight\right)\right\}\left\{sinleft\left(frac\left\{phi_s+phi_f\right\}\left\{2\right\} ight\right)\right\} ight\right)-lnleft\left( anBig\left(frac\left\{2\right\}Big\right) ight\right) ight\right);,!$ $\left\{\right\}_\left\{color\left\{white\right\}.\right\}quad!Deltawidehat\left\{sigma\right\}=2arctan!left\left(,left|expleft\left(ln!left\left(frac\left\{sin\left(arctan\left(mathbb\left\{N\right\}\right)\right)\right\}\left\{sin\left(arctan\left(mathbb\left\{D\right\}\right)\right)\right\} ight\right)+lnleft\left( anBig\left(frac\left\{2\right\}Big\right) ight\right) ight\right) ight|, ight\right).,!$

ee also

*inscribed angle

* [http://www.mathopenref.com/arccentralangle.html Central Angle of an Arc definition] With interactive animation
* [http://www.mathopenref.com/arccentralangletheorem.html Central Angle Theorem described] With interactive animation
* [http://www.cut-the-knot.org/Curriculum/Geometry/InscribedAngle.shtml Inscribed and Central Angles in a Circle]

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### Look at other dictionaries:

• central angle — noun : an angle formed by two radii of a circle * * * Geom. an angle formed at the center of a circle by two radii. [1900 05] * * * central angle noun (mathematics) An angle bounded by two radii of a circle • • • Main Entry: ↑centre …   Useful english dictionary

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• central angle — noun Date: 1904 an angle formed by two radii of a circle …   New Collegiate Dictionary

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