On rereading the Wolfram article, their equation (9) is ##\phi=\delta=\pi / 2 - v##, so ##v##

According to www.mathworld.wolfram.com/GreatCircle.html (19) the geodesic equation on a sphere (great circle) is given below. It is derived from a somewhat specialised equation for a geodesic on a surface (http://mathworld.wolfram.com/Geodesic.html (30)), which itself is derived by considering a minimised line integral. Wolfram's (19) is given as

\begin{align}

a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{c}\mathrm{o}\mathrm{s} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(1) \\

\end{align}where ##a## is the radius of the sphere, ##c_1,c_2## are constants of integration, ##u,v## are respectively longitude and latitude. In the next equation it recasts that in Cartesian coordinates as\begin{align}

x{\mathrm{sin} c_2\ }+y{\mathrm{cos} c_2\ }-\frac{z}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(2) \\

\end{align}"which shows that the geodesic giving the shortest path between two points on the surface of the equation lies on a plane that passes through the two points in question and also through center of the sphere." (2) is indeed the equation of a plane which contains the origin, but it also implies that\begin{align}

x=a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ y=a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ z=\ a{\mathrm{c}\mathrm{o}\mathrm{s} v\ } & \phantom {10000}(3) \\

\end{align}This is very wrong. It would be correct if ##v## was the colatitude (angle measured from the pole). The colatitude is normally called ##\phi ## and ##\phi ={\pi }/{2}-v##, as they say in their #7. Alternatively one can swap all ##{\mathrm{sin} v\ },{\mathrm{cos} v\ }##. I guessed that the equation is therefore\begin{align}

a{\mathrm{cos} u\ }{\mathrm{cos} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{cos} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{sin} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(4) \\

\end{align}

The schematic shows great circles between cities. The right hand one shows the London-Peking great circle according Wolfram Mathworld. Perhaps this is what happened to the British Airways pilot who flew from London to Edinburgh instead of Düsseldorf in a month ago. (On the BBC here)

I have not been able to trace the source of the error in the Wolfram Mathworld proof. It may go as far back as their (7) where they might have intended to introduce ##\phi ##.

__is__the colatidude. I can only forgive myself because they earlier say ##\delta=90°-\phi##. It would be helpful if they wrote "where ##a## is the radius of the sphere, ##c_1,c_2## are constants of integration, ##u,v## are respectively longitude and**co**latitude" under their equation (19).According to www.mathworld.wolfram.com/GreatCircle.html (19) the geodesic equation on a sphere (great circle) is given below. It is derived from a somewhat specialised equation for a geodesic on a surface (http://mathworld.wolfram.com/Geodesic.html (30)), which itself is derived by considering a minimised line integral. Wolfram's (19) is given as

\begin{align}

a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{c}\mathrm{o}\mathrm{s} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(1) \\

\end{align}where ##a## is the radius of the sphere, ##c_1,c_2## are constants of integration, ##u,v## are respectively longitude and latitude. In the next equation it recasts that in Cartesian coordinates as\begin{align}

x{\mathrm{sin} c_2\ }+y{\mathrm{cos} c_2\ }-\frac{z}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(2) \\

\end{align}"which shows that the geodesic giving the shortest path between two points on the surface of the equation lies on a plane that passes through the two points in question and also through center of the sphere." (2) is indeed the equation of a plane which contains the origin, but it also implies that\begin{align}

x=a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ y=a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ z=\ a{\mathrm{c}\mathrm{o}\mathrm{s} v\ } & \phantom {10000}(3) \\

\end{align}This is very wrong. It would be correct if ##v## was the colatitude (angle measured from the pole). The colatitude is normally called ##\phi ## and ##\phi ={\pi }/{2}-v##, as they say in their #7. Alternatively one can swap all ##{\mathrm{sin} v\ },{\mathrm{cos} v\ }##. I guessed that the equation is therefore\begin{align}

a{\mathrm{cos} u\ }{\mathrm{cos} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{cos} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{sin} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(4) \\

\end{align}

**This is correct as I have proved by other means. (Here).**Numerically it can be shown with my great 3-D graph plotter (here and here). The incorrect equation is obviously not a great circle, whereas the correct one looks plausible;The schematic shows great circles between cities. The right hand one shows the London-Peking great circle according Wolfram Mathworld. Perhaps this is what happened to the British Airways pilot who flew from London to Edinburgh instead of Düsseldorf in a month ago. (On the BBC here)

I have not been able to trace the source of the error in the Wolfram Mathworld proof. It may go as far back as their (7) where they might have intended to introduce ##\phi ##.

**This error on Wolfram Mathworld caused me a lot of grief!**
## No comments:

## Post a Comment