Central angle

Central angle

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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};,!

:::{}_{color{white}.}!egin{pmatrix}operatorname{sgn}(V)=|V|!cdotV^{-1};quadoverrightarrow{operatorname{sgn(V)=operatorname{sgn}ig(operatorname{sgn}(V)+frac{1}{2}ig)\{}_{(,operatorname{sgn}(0)=0;qquadoverrightarrow{operatorname{sgn(0)=+1,)}end{pmatrix}{}_{color{white}.}!!,!

: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}.}!!,!

:egin{align}phi_w=phi_s;;&phi_c=phi_f!!:mbox{Get};widehat{sigma}_w!!:\&widehat{sigma}_s=widehat{sigma}_w!cdotoverrightarrow{mbox{sgn(S!B_w)+pi!cdotoverrightarrow{mbox{sgn(widehat{sigma}_w)mbox{sgn}(1-overrightarrow{mbox{sgn(S!B_w));end{align},!

:egin{align}phi_w=phi_f;;&phi_c=phi_s!!:mbox{Get};widehat{sigma}_w!!:\&widehat{sigma}_f=widehat{sigma}_w!cdotoverrightarrow{mbox{sgn(-S!B_w)+pi!cdotoverrightarrow{mbox{sgn(widehat{sigma}_w)mbox{sgn}(1-overrightarrow{mbox{sgn(-S!B_w))\&qquadqquadqquadqquadquad+2pi!cdotmbox{sgn}(1-overrightarrow{mbox{sgn(widehat{sigma}_w-widehat{sigma}_s));end{align},! _____________________________________________________________________: ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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},!,

:egin{align}widehat{Alpha}&=Big|arcsinig(cos(phi_w)sin(widehat{alpha}_w)ig)Big|!!!&&=Big|arccosleft(frac{sin(phi_w)}{sin(widehat{sigma}_w)} ight)Big|,\&=Big|arctanig(cos(widehat{sigma}_w) an(widehat{alpha}_w)ig)Big|!!!&&=Big|arctanig(sin(widehat{alpha}_w)sin(widehat{sigma}_w)cot(phi_w)ig)Big|.end{align},!

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):

:egin{align}{}_{color{white}.}\Deltawidehat{sigma}&=widehat{sigma}_f;-;widehat{sigma}_s,\&=arcsin!left(sqrtS!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,

: egin{align}{}_{color{white}.}\&=2arcsin!left(sqrt{sin^2!left(frac{phi_f-phi_s}{2} ight)+cos(phi_s)cos(phi_f)sin^2!left(frac{Deltalambda}{2} ight)}, ight),\&=2arccos!left(sqrt{cos^2!left(frac{phi_f-phi_s}{2} ight)-cos(phi_s)cos(phi_f)sin^2!left(frac{Deltalambda}{2} ight)}, ight),\&=2arctan!left(sqrt{frac{sin^2left(frac{phi_f-phi_s}{2} ight)+cos(phi_s)cos(phi_f)sin^2Big(frac{Deltalambda}{2}Big)}{cos^2left(frac{phi_f-phi_s}{2} ight)-cos(phi_s)cos(phi_f)sin^2!Big(frac{Deltalambda}{2}Big), ight).\{}^{color{white}.}end{align},!

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]


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