Geodesics on Regular Constant Distance Surfaces
Published In
Journal of Geometry
Document Type
Citation
Publication Date
12-1-2023
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.
Rights
© Springer Nature
Locate the Document
DOI
10.1007/s00022-023-00690-6
Persistent Identifier
https://archives.pdx.edu/ds/psu/40928
Publisher
Springer Nature
Citation Details
Veerman, J. J. P. (2023). Geodesics on regular constant distance surfaces. Journal of Geometry, 114(3).