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", $widehat\{sigma\},!$, 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, $Deltawidehat\{sigma\},!$.

Calculation of TvL

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

:$V\_s,V\_f,V\_w,V\_c:mathrm\{;Standpoint,\; forepoint,\; working,\; coworking\; values\};,!$:::: $widehat\{alpha\}\_w:mathrm\{;Orthodromic\; azimuth\; at\; widehat\{sigma\}\_w\};,!$

:::

:$Deltalambda=lambda\_f-lambda\_s;,!$

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

:

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

Occupying great circle

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

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

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

:

Angular distance formulary

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

:S!A}^2+{S!B}^2}, ight),\&quad{}^{mathit{(can,only,find,the,first,quadrant,,i.e.,;up,to,90^circ)\&=arccos!Big(sin(phi_s)sin(phi_f)+cos(phi_s)cos(phi_f)cos(Deltalambda),Big),\&quad{}^{mathit{(not,recommended,for,small,angles,;due,to,rounding,error)\&=arctan!left(frac{sqrtS!A}^2+{S!B}^2{sin(phi_s)sin(phi_f)+cos(phi_s)cos(phi_f)cos(Deltalambda)} ight),\{}^{color{white}.}end{align},!

and, using half-angles,

:

There is also a logarithmical form:

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

ee also

*inscribed angle

External links

* [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]