From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Shortest Distances on Analytical Surfaces (Cylinder, ..)
Date: 24 Sep 1999 16:44:53 GMT
Newsgroups: sci.math
Keywords: geodesics on tori and other surfaces
In article <7s2gn3$n6j$1@newsread.f.de.uu.net>,
Achim Männer wrote:
>how can I calculate the shortest distance between two points on analytical
>surfaces?
The shortest distance between two points on a Riemannian manifold will
be along a path which is a _geodesic_, so your question (I suppose) is,
how do we find geodesics?
>On a sphere it is simple the arc length.
It is a consequence of being a geodesic that the curve will be parameterized
by arc length. That is, a geodesic (or more generally any path) is a
_function_ p : [0,1] -> M. Think of a path as being an itinerary, rather
than a road. But among all paths travelling a set of points, only
the ones traversed with constant speed can be geodesics.
>But in case of cylinder, cone
As others have remarked, in these cases the surfaces are locally isometric
with the flat plane, and so geodesics are the images of straight lines.
>or tori?
First off, let me remark that for surfaces in R^3, a curve is a geodesic
iff its acceleration vector is everywhere perpendicular to the surface.
(Evidently this is due to Gauss; see Spivak's "Comprehensive Introduction",
vol 2, p. 116) From this information alone one can rule out certain
simple suggestions for the geodesics -- for example, none are planar
except the ones which wrap the short way around the torus.
I don't know a nice description of the geodesics as curves in R^3,
but we can describe them as curves on the 2-dimensional surface using
local coordinates. Indeed, this approach can be used (in principle) for
any surface. The interested reader is referred to John Oprea's book
"Differential Geometry" for details, explicit computations, Maple code,
and some illustrations. Geodesics on surfaces are treated in section 5.2
(p. 156ff).
If we parameterize the torus with the function
x(u,v) = ( (R + r cos u) cos v, (R + r cos u) sin v, r sin u )
then a curve on the torus is given by describing u and v as functions
of another variable, t. This curve is a geodesic iff these functions
satisfy the geodesic equations
u'' + ((R + r cos u)/r )(sin u) (v')^2 = 0
v'' - 2 (r sin u )/(R + r cos u) u' v' = 0
With some machinations these reduce to the single equation
dv/du = c r sqrt( R + r cos u ) / sqrt( ( R + r cos u )^2 - c^2 )
which you can't solve symbolically but can sketch in the u-v plane,
and can then feed into x(u,v) to see the geodesics in R^3.
As a sample, the book shows the geodesic which leaves a point on the
top circle of the torus, in a direction tangent to that circle: it
wraps around the torus once the long way 'round before returning to
its starting point, but does not stay along the top circle at all -- rather,
it dips under the torus at "3 o'clock", back to the top at "6 o'clock",
back under at "9 o'clock", and returns to the top at "12-o'clock",
giving a curve with an isometry group equal to the symmetries of the square.
I have to confess I was not sufficiently patient to work out these
formulas and pictures on my own. This is not particularly easy work
for those not well versed in the terminology.
By the way, the geodesics need not be closed orbits, so there is no
hope that they can be described "analytically" e.g. as the intersections
of the torus with surfaces in some nice family. Certainly in the
category of algebraic varieties, we're stuck because that last
integral cannot be expressed analytically -- it's essentially a
question in the Galois theory of fields.
dave