Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated diameter with a normal coinciding with the bisector of $\theta$ when the planes are orthogonal (I think it is intuitively clear what this angle defines, even though the precise statement is a little long).

The lune on which the arc $AB$ is located has an area of $2r^2\theta_{\varphi=\pi/2}$, where $\theta_{\varphi=\pi/2}$ is the angle $\theta$ at $\varphi=\pi/2$. The angle $\theta$ is related to $AB$ by length$(AB) = r\theta$, but I would like to know the relation between $\theta$ and $\varphi$, once $\theta_{\varphi=\pi/2}$ has been fixed.

Picture

This is essentially so I can integrate some function from $\varphi=0$ to $\varphi=\pi$ to get the area of the lune. I need that to get an alternative definition, by integration, of the surface area of the $n$-sphere.

Edit: If it is not clear, the arcs from the north pole (topmost emphasized point) to $A$ and to $B$ are the same length, and $AB$ is a segment of a great circle (i.e. a geodesic).

share|improve this question
    
So $\theta_{\varphi=\pi/2}$ is the angle between the great circles containing $A$ and $B$? $\varphi$ is the angle between the pole and $A$ and $B$? –  robjohn May 9 '13 at 18:22
    
It might be simpler to define $\varphi$ as the angle $NC\frac{A+B}2$ where $N$ is the pole. –  Peter Taylor May 9 '13 at 18:24
    
@robjohn Yes, that's a better description –  JānisL May 9 '13 at 20:07
    
@robjohn Regarding your answer below, the side $\varphi$ should be $\varphi r$, and $\theta/2$ should be $\theta r/2$. I am assuming (though not 100% sure) that will change your relation accordingly. Is that correct? –  JānisL May 9 '13 at 20:20
    
@JimboBimbo: In spherical trigonometry, sides are measured by the angle they subtend at the center of the sphere. While it is true that the linear length of the side is $\varphi r$ where $r$ is the radius of the sphere, for use in spherical trig formulas, the side has angular length $\varphi$. –  robjohn May 9 '13 at 20:24
show 2 more comments

1 Answer

up vote 1 down vote accepted

Assuming that $\theta_{\varphi=\pi/2}$ is the angle between the great circle containing $N$ and $A$ and the great circle containing $N$ and $B$, and that $\varphi$ is the angle between the north pole and the great circle containing $A$ and $B$, then $\triangle N(\frac{A+B}{2})B$ is a right spherical triangle with the right angle at $\frac{A+B}{2}$. The basic relations for right spherical triangles yield $$ \tan\left(\frac{\theta_{\varphi=\pi/2}}{2}\right)\sin(\varphi)=\tan\left(\frac{\theta}{2}\right) $$ $\hspace{3.2cm}$enter image description here

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.