Re: spherical geometry (was Re: navigation riddle)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Brian's elaborated explanation of his modified version of
the problem:

> > > Or how about this one:
> > > From a fixed point, head due South, continue in a straight line
> > > for one mile: turn due East and continue in that direction for one
> > > mile, then turn onto a due South heading and continue straight for
> > > one mile, to return to your start point.
> > >
> > >
> > > Brian Whatcott    Altus OK    Eureka!
> >
> > Brian, do you really want two southbound legs?  Also, What do you
> > mean by the phrase "in that direction for one mile" for the 2nd
> > leg?  Do you mean a) along a geodesically straight path, b) always
> > going due East, c) along a Euclideanly straight path, d) something
> > else?  Do you want the problem performed on the Earth's surface?
> >
> > David Bowman
>
>
>  A Southbound start, and a constant Easterly heading,
> and another Southbound turn on the Earth's surface,
> idealized to be spherical, of course.
>
> Brian Whatcott    Altus OK    Eureka!

Thanks for the clarification.  In this case it seems that again
there are an infinite number of solutions starting a fraction of a
mile from the South Pole.  The main trick is to realize that once
a person heads South towards the South Pole and continues in a
straight line past the pole one ends up heading North without any
change in direction.  Ignoring the tiny curvature effects of the
Earth's surface in the vicinity of the South Pole the solution
set seems to be that one starts about 1 - 1/(2*n*[pi]) miles from
the South pole where n is some positive integer.  After traveling
one mile in the same direction and crossing over the pole one
stops 1/(2*n*[pi]) miles from the pole on the other side.  One then
travels n times around the pole while traveling East ending up back
at the first turning point for this 2nd 1 mile long leg.  One then
travels again one mile back over the pole back to the initial
starting point.

David Bowman