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A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is (by definition) equal to the central angle itself. It is also known as the arc segment's angular distance.

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{O}\,\!, 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, \Delta\widehat{O}\,\!.

Calculation of TvL

The calculation of \widehat{O_b}\,\! and \widehat{O_d}\,\! can be found using a common subroutine:

V_b,V_d,V_w,V_c:\mathrm{\;Basal,\ destination,\ working,\ coworking\ values};\,\!
 \widehat{\alpha}_w:\mathrm{\;Orthodromic\ azimuth\ at\ \widehat{O}_w};\,\!

\begin{pmatrix}\operatorname{sgn}(Q)=|Q|\times Q^{-1};\quad\operatorname{\widehat{sgn}}(Q)=\operatorname{sgn}(\operatorname{sgn}(Q)+\frac{1}{2})\\{}_{(\,\operatorname{sgn}(0)=0;\qquad\operatorname{\widehat{sgn}}(0)=+1\,)}\end{pmatrix}\,\!

\Delta\lambda=\lambda_d-\lambda_b;\,\!

\left(\mbox{If } \phi_b=\phi_d=0\mbox{, then }\;\widehat{O}_b=\frac{\pi-|\Delta\lambda|}{2},\;\widehat{O}_d=\frac{\pi+|\Delta\lambda|}{2}\right)\,\!

\begin{matrix}\phi_w=\phi_b;\;\phi_c=\phi_d\!\!:\mbox{Get}\;\widehat{O}_w\!\!:\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\qquad\;\;\,\widehat{O}_b=\widehat{O}_w\times\operatorname{\widehat{sgn}}(S\!B_w)+\pi\times\operatorname{\widehat{sgn}}(\widehat{O}_w)\operatorname{sgn}(1-\operatorname{\widehat{sgn}}(S\!B_w));\end{matrix}\,\!

\begin{matrix}\phi_w=\phi_d;\;\phi_c=\phi_b\!\!:\mbox{Get}\;\widehat{O}_w\!\!:\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\qquad\qquad\;\;\widehat{O}_d=\widehat{O}_w\times\operatorname{\widehat{sgn}}(-S\!B_w)+\pi\times\operatorname{\widehat{sgn}}(\widehat{O}_w)\operatorname{sgn}(1-\operatorname{\widehat{sgn}}(-S\!B_w))\\\qquad\qquad\qquad\qquad\qquad\qquad\quad+2\pi\times\operatorname{sgn}(1-\operatorname{\widehat{sgn}}(\widehat{O}_d-\widehat{O}_b));\end{matrix}\,\!
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{| \begin{matrix}S\!A_w=\cos(\phi_c)\sin(\Delta\lambda);\qquad\qquad\qquad\qquad\qquad\qquad\;\;\\S\!B_w=\sin(\phi_w+\phi_c)\sin(\frac{\Delta\lambda}{2})^2+\sin(\phi_c-\phi_w)\cos(\frac{\Delta\lambda}{2})^2;\end{matrix}\,\! \left(\,\sin(\Delta\widehat{O})^2={S\!A_w}^2+{S\!B_w}^2;\quad >\tan(\widehat{a}_w) \begin{matrix}\widehat{O}_w\!\!\!&=&\!\!\!\arctan( >\sec(\widehat{a}_w)
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Each point has at least two values, both a forward and reverse value.

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):

\begin{matrix}{}_{.}\\\Delta\widehat{O}\!\!&=&\!\!\!\!\widehat{O}_d\;-\;\widehat{O}_b\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\\&=&\!\!\!\!\!\!\!\!\!\arcsin\!\left(\sqrt\qquad\qquad\qquad\\&=&\!\!\!\arccos(\sin(\phi_b)\sin(\phi_d)+\cos(\phi_b)\cos(\phi_d)\cos(\Delta\lambda)),\\&&\!\!\!\!\!\!\!\!\!\!\!{}^{\mathit{(not\,recommended\,for\,small\,angles,\;due\,to\,rounding\,error)}}\qquad\\&=&\!\!\!\!\!\!\!\arctan\!\left(\frac{\sin(\Delta\widehat{O})}{\cos(\Delta\widehat{O})}\right),\qquad\qquad\qquad\qquad\qquad\qquad\quad\\{}^{.}\end{matrix}\,\!

and, using half-angles,

   \begin{matrix}{}_{.}\\&=&\!\!\!2\arcsin\!\left(\sqrt{\sin\!\left(\frac{\phi_d-\phi_b}{2}\right)^2+\cos(\phi_b)\cos(\phi_d)\sin\left(\frac{\Delta\lambda}{2}\right)^2}\,\right),\\\\&=&\!\!\!2\arccos\!\left(\sqrt{\cos\!\left(\frac{\phi_d-\phi_b}{2}\right)^2-\cos(\phi_b)\cos(\phi_d)\sin\!\left(\frac{\Delta\lambda}{2}\right)^2}\,\right),\\\\&=&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!2\arctan\!\left(\sqrt{\frac{\sin\left(\frac{\phi_d-\phi_b}{2}\right)^2+\cos(\phi_b)\cos(\phi_d)\sin\left(\frac{\Delta\lambda}{2}\right)^2}{\cos\left(\frac{\phi_d-\phi_b}{2}\right)^2-\cos(\phi_b)\cos(\phi_d)\sin\!\left(\frac{\Delta\lambda}{2}\right)^2}}\,\right).\\{}^{.}\end{matrix}\,\!

There is also a logarithmical form:

\begin{matrix}{}_{.}\\{}^{\cdot}\;\;\mathbb{N}=\exp\left(\ln\!\left(\frac{\cos\left(\frac{\phi_d-\phi_b}{2}\right)}{\sin\left(\frac{\phi_b+\phi_d}{2}\right)}\right)-\ln(\tan(\frac{|\Delta\lambda|}{2}))\right);\\{}^{.}\end{matrix}\,\!
\begin{matrix}{}_{.}\\{}^{\cdot}\;\;\mathbb{D}=\exp\left(\ln\!\left(\frac{\sin\left(\frac{|\phi_d-\phi_b|}{2}\right)}{\cos\left(\frac{\phi_b+\phi_d}{2}\right)}\right)-\ln(\tan(\frac{|\Delta\lambda|}{2}))\right);\\{}^{.}\end{matrix}\,\!

\begin{matrix}{}_{.}\\\Delta\widehat{O}=2\arctan\!\left(\,\left|\exp\left(\ln\!\left(\frac{\sin(\arctan(\mathbb{N}))}{\sin(\arctan(\mathbb{D}))}\right)+\ln(\tan(\frac{|\phi_d-\phi_b|}{2}))\right)\right|\,\right).\\{}^{.}\end{matrix}\,\!

See also

Elementary geometry | Angle

Kąt środkowy

 

This article is licensed under the GNU Free Documentation License. It uses material from the "Central angle".

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