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