central angle: Definition from Answers.com

Angle AOB forms a central angle of circle O

A central angle is an angle which 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.

Contents

1 Coordinates
2 Calculation of TvL
3 Occupying great circle
4 Angular distance formulary
5 See also
6 References
7 External links

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 conjugate 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, $\Delta\widehat{\sigma}\,\!$ .

Calculation of TvL

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

$V_s,V_f,V_w,V_c:\mathrm{\;Standpoint,\ forepoint,\ working,\ coworking\ values};\,\!$

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

${}_{\color{white}.}\!\begin{pmatrix}\operatorname{sgn}(V)=|V|\!\cdot V^{-1};\quad\overrightarrow{\operatorname{sgn}}(V)=\operatorname{sgn}\big(\operatorname{sgn}(V)+\frac{1}{2}\big)\\{}_{(\,\operatorname{sgn}(0)=0;\qquad\overrightarrow{\operatorname{sgn}}(0)=+1\,)}\end{pmatrix}{}_{\color{white}.}\!\!\,\!$

$\Delta\lambda=\lambda_f-\lambda_s;\,\!$

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

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

$\begin{align}\phi_w=\phi_f;\; &\phi_c=\phi_s\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\\ &\widehat{\sigma}_f=\widehat{\sigma}_w\!\cdot\overrightarrow{\mbox{sgn}}(-S\!B_w)+\pi\!\cdot\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w)\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(-S\!B_w))\\ &\qquad\qquad\qquad\qquad\quad+2\pi\!\cdot\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w-\widehat{\sigma}_s));\end{align}\,\!$

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

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

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 $\widehat{\sigma}\,\!$ ,

$\begin{align}\widehat{\Alpha} &=\Big|\arcsin\big(\cos(\phi_w)\sin(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arccos\left(\frac{\sin(\phi_w)}{\sin(\widehat{\sigma}_w)}\right)\Big|,\\ &=\Big|\arctan\big(\cos(\widehat{\sigma}_w)\tan(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arctan\big(\sin(\widehat{\alpha}_w)\sin(\widehat{\sigma}_w)\cot(\phi_w)\big)\Big|.\end{align}\,\!$

Angular distance formulary

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

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

and, using half-angles,

$\begin{align}{}_{\color{white}.}\\ &=2\arcsin\!\left(\sqrt{\sin^2\!\left(\frac{\phi_f-\phi_s}{2}\right)+\cos(\phi_s)\cos(\phi_f)\sin^2\!\left(\frac{\Delta\lambda}{2}\right)}\,\right),\\ &=2\arccos\!\left(\sqrt{\cos^2\!\left(\frac{\phi_f-\phi_s}{2}\right)-\cos(\phi_s)\cos(\phi_f)\sin^2\!\left(\frac{\Delta\lambda}{2}\right)}\,\right),\\ &=2\arctan\!\left(\sqrt{\frac{\sin^2\left(\frac{\phi_f-\phi_s}{2}\right)+\cos(\phi_s)\cos(\phi_f)\sin^2\Big(\frac{\Delta\lambda}{2}\Big)}{\cos^2\left(\frac{\phi_f-\phi_s}{2}\right)-\cos(\phi_s)\cos(\phi_f)\sin^2\!\Big(\frac{\Delta\lambda}{2}\Big)}}\,\right).\\{}^{\color{white}.}\end{align}\,\!$

It can, as well, be found by means of finding the chord length via Cartesian subtraction^[1]:

$\begin{align} &\Delta{X}=\cos(\phi_f)\cos(\lambda_f) - \cos(\phi_s)\cos(\lambda_s);\\ &\Delta{Y}=\cos(\phi_f)\sin(\lambda_f) - \cos(\phi_s)\sin(\lambda_s);\\ &\Delta{Z}=\sin(\phi_f) - \sin(\phi_s);\\ &C_h=\sqrt{(\Delta{X})^2+(\Delta{Y})^2+(\Delta{Z})^2};\\ &\Delta\widehat{\sigma}=2\arcsin\left(\frac{C_h}{2}\right).\end{align}\,\!$

Also, by using Cartesian products rather than differences, the origin of the spherical cosine for sides becomes apparent:

$\begin{align} {\scriptstyle{\Pi}}X&=\cos(\phi_s)\cos(\phi_f)\cos(\lambda_s)\cos(\lambda_f);\\ {\scriptstyle{\Pi}}Y&=\cos(\phi_s)\cos(\phi_f)\sin(\lambda_s)\sin(\lambda_f);\\ {\scriptstyle{\Pi}}Z&=\sin(\phi_s)\sin(\phi_f);\\ \frac{{\scriptstyle{\Pi}}X\!\!+\!{\scriptstyle{\Pi}}Y}{\cos(\phi_s)\cos(\phi_f)}&=\cos(\lambda_s)\cos(\lambda_f)+\sin(\lambda_s)\sin(\lambda_f)=\cos(\Delta\lambda);\\ \Delta\widehat{\sigma}&=\arccos\Big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y+{\scriptstyle{\Pi}}Z\Big) =\arccos\Big({\scriptstyle{\Pi}}Z+\big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y\big)\Big),\\ &=\arccos\Big(\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)\Big).\end{align}\,\!$

There is also a logarithmical form:

${}_{\color{white}.}\;\mathbb{N}=\exp\left(\ln\!\left(\frac{\cos\left(\frac{\phi_f-\phi_s}{2}\right)}{\sin\left(\frac{\phi_s+\phi_f}{2}\right)}\right)-\ln\left(\tan\Big(\frac{|\Delta\lambda|}{2}\Big)\right)\right);\,\!$

${}_{\color{white}.}\;\mathbb{D}=\exp\left(\ln\!\left(\frac{\sin\left(\frac{|\phi_f-\phi_s|}{2}\right)}{\cos\left(\frac{\phi_s+\phi_f}{2}\right)}\right)-\ln\left(\tan\Big(\frac{|\Delta\lambda|}{2}\Big)\right)\right);\,\!$

${}_{\color{white}.}\quad\!\Delta\widehat{\sigma}=2\arctan\!\left(\,\left|\exp\left(\ln\!\left(\frac{\sin(\arctan(\mathbb{N}))}{\sin(\arctan(\mathbb{D}))}\right)+\ln\left(\tan\Big(\frac{|\phi_f-\phi_s|}{2}\Big)\right)\right)\right|\,\right).\,\!$

References

^ http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/dave2.html

External links

Central Angle of an Arc definition With interactive animation
Central Angle Theorem described With interactive animation
Inscribed and Central Angles in a Circle

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		American Heritage Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
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central angle

central angle

Central angle

Coordinates

Calculation of TvL

Occupying great circle

Angular distance formulary

See also

References

External links

central angle