Central angle

From Wikipedia, the free encyclopedia

Angle AOB forms a central angle of circle O

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.

1 Coordinates
2 Calculation of TvL
3 Angular distance formulary
4 See also
5 External links

[edit] 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, $\Delta\widehat{\sigma}\,\!$ .

[edit] Calculation of TvL

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

$V_b,V_d,V_w,V_c:\mathrm{\;Basal,\ destination,\ working,\ coworking\ values};\,\!$

$\widehat{\alpha}_w:\mathrm{\;Orthodromic\ azimuth\ at\ \widehat{\sigma}_w};\,\!$

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

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

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

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

$\begin{matrix}\phi_w=\phi_d;\;\phi_c=\phi_b\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\qquad\qquad\;\;\widehat{\sigma}_d=\widehat{\sigma}_w\times\operatorname{\widehat{sgn}}(-S\!B_w)+\pi\times\operatorname{\widehat{sgn}}(\widehat{\sigma}_w)\operatorname{sgn}(1-\operatorname{\widehat{sgn}}(-S\!B_w))\\\qquad\qquad\qquad\qquad\qquad\qquad\quad+2\pi\times\operatorname{sgn}(1-\operatorname{\widehat{sgn}}(\widehat{\sigma}_d-\widehat{\sigma}_b));\end{matrix}\,\!$

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$\begin{matrix}S\!A_w=\cos(\phi_c)\sin(\Delta\lambda);\qquad\qquad\qquad\qquad\qquad\qquad\;\;\\S\!B_w=\sin(\phi_w+\phi_c)\sin(\frac{\Delta\lambda}{2})^2+\sin(\phi_c-\phi_w)\cos(\frac{\Delta\lambda}{2})^2;\end{matrix}\,\!$
$\left(\,\sin(\Delta\widehat{\sigma})^2={S\!A_w}^2+{S\!B_w}^2;\quad\|\tan(\widehat{a}_w)\|=\left\|\frac{S\!A_w}{S\!B_w}\right\|\,\right)\,\!$
$\begin{matrix}\widehat{\sigma}_w\!\!\!&=&\!\!\!\arctan(\|\sec(\widehat{a}_w)\|\tan(\phi_w))=\arctan\!\left(\left\|\frac{\sin(\Delta\widehat{\sigma})}{S\!B_w}\right\|\tan(\phi_w)\right),\\&=&\!\!\!\!\!\!\arctan\!\left(\frac{\sqrt{{S\!A_w}^2+{S\!B_w}^2}}{\|S\!B_w\|}\tan(\phi_w)\right).\qquad\qquad\qquad\qquad\qquad\end{matrix}$

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

[edit] 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):

$\begin{matrix}{}_{.}\\\Delta\widehat{\sigma}\!\!&=&\!\!\!\!\widehat{\sigma}_d\;-\;\widehat{\sigma}_b\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\\&=&\!\!\!\!\!\!\!\!\!\arcsin\!\left(\sqrt{{S\!A}^2+{S\!B}^2}\,\right),\qquad\qquad\qquad\qquad\qquad\\&&\!\!\!\!\!\!\!\!\!{}^{\mathit{(can\,only\,find\,the\,first\,quadrant,\,i.e.,\;up\,to\,90^\circ)}}\qquad\qquad\qquad\\&=&\!\!\!\arccos(\sin(\phi_b)\sin(\phi_d)+\cos(\phi_b)\cos(\phi_d)\cos(\Delta\lambda)),\\&&\!\!\!\!\!\!\!\!\!\!\!{}^{\mathit{(not\,recommended\,for\,small\,angles,\;due\,to\,rounding\,error)}}\qquad\\&=&\!\!\!\!\!\!\!\arctan\!\left(\frac{\sin(\Delta\widehat{\sigma})}{\cos(\Delta\widehat{\sigma})}\right),\qquad\qquad\qquad\qquad\qquad\qquad\quad\\{}^{.}\end{matrix}\,\!$

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:

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

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

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