Saturday, 3 August 2019

Great circles

We know that a great circle is a line between two points on a sphere which is the intersection of a plane through the sphere's centre and the two points and the surface of the sphere. We also know that the shorter distance between the two points is the shortest distance between the two points if we are confined to the surface of the sphere.

Here we find the equation for a great circle and prove it is the same as the one given (rather ambiguously) in Wolfram Mathsworld and, loosely, the one by Professor Govindarajan. I also plotted some great circles to test the equations visually. That's the video. The reason for testing the equations so rigorously is that they did not seem to agree with Carroll's geodesic equation at (3.44). Eventually I showed that they did.

Polar coordinates

First we must deal with the angular coordinates ## \phi,\theta##. Some sources have ## \theta## as the longitude (azimuthal angle) and ## \phi## as the colatitude (angle from north pole) in other sources they are swapped around. I tried to follow the majority and did the former with ## \phi,\theta## in that order. Then Wikipedia told me that "in one system frequently encountered in physics ##(r,\theta,\phi)## gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books ##(r,\theta,\phi)## gives the radial distance, azimuthal angle, and polar angle. ... Other conventions are also used, so great care needs to be taken to check which one is being used." Not only are the meanings of ## \phi,\theta## swapped but also the order!! I have followed neither convention. Boo hoo. At Mathworld a table of seven variations is given. They use the same as me, but not in the same order. Carroll and Govindarajan follow the physicist convention. In future will follow the physicist convention but here and in the upcoming post on the parallel transport and geodesics, I stick to my own.

The great circle equations

Mine 
\begin{align}
\phi=\tan^{-1}{\left(\frac{A_z}{A_x\cos{\theta}+A_y\sin{\theta}}\right)}&\phantom {10000}\nonumber
\end{align}Wolfram Mathsworld
\begin{align}
\cos{\theta}\sin{\phi}\sin{c_2}+\sin{\theta}\sin{\phi}\cos{c_2}+c_1\cos{\phi}=0&\phantom {10000}\nonumber
\end{align}Professor Govindarajan's
\begin{align}
\cot{\phi}=A\cos{\left(\theta+\theta_0\right)}&\phantom {10000}\nonumber
\end{align}There are two or three constants in each equation.

Read the full details at Commentary 3.3 Great circle.pdf
The spreadsheet which generated the video is at Commentary 3.3 Parallel transport and geodesics.xlsm


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