Geodesics on Regular Constant Distance Surfaces

TL;DR

This paper characterizes when the projection between two convex surfaces preserves geodesics, showing it only occurs for concentric spheres or co-axial cylinders, and explores related open questions.

Contribution

It provides a necessary and sufficient condition for geodesic preservation under projection between convex surfaces, highlighting the special cases of spheres and cylinders.

Findings

Projection preserves geodesics only for concentric spheres or co-axial cylinders.

The main result fails for C^{1,1} surfaces, indicating optimality.

An example of geodesic-preserving projection for non-constant distance surfaces is given.

Abstract

Suppose that the surfaces K0 and Kr are the boundaries of two convex, complete, connected C^2 bodies in R^3. Assume further that the (Euclidean) distance between any point x in Kr and K0 is always r (r > 0). For x in Kr, let {\Pi}(x) denote the nearest point to x in K0. We show that the projection {\Pi} preserves geodesics in these surfaces if and only if both surfaces are concentric spheres or co-axial round cylinders. This is optimal in the sense that the main step to establish this result is false for C^{1,1} surfaces. Finally, we give a non-trivial example of a geodesic preserving projection of two C^2 non-constant distance surfaces. The question whether for any C^2 convex surface S0, there is a surface S whose projection to S0 preserves geodesics is open.