Journal of Convex Analysis 27 (2020), No. 4, 1137--1156
Copyright Heldermann Verlag 2020
On Moving Chords in Constant Curvature 2-Manifolds
Juan Monterde
Dept. of Mathematics, University of Valencia, 46100 Burjassot-Valencia, Spain
juan.l.monterde@uv.es
David Rochera
Dept. of Mathematics, University of Valencia, 46100 Burjassot-Valencia, Spain
david.rochera@uv.es
An introduction to non-Euclidean geometry is given and a new approach to prove a generalization of Holditch's theorem in 2-dimensional constant curvature manifolds is presented. Moreover, the same procedure also makes possible to obtain generalized versions of Barbier's theorem for constant width curves and of Steiner's formulae for parallel curves. The main fact is that a general formula relating the geodesic curvatures of the involved curves is found, in such a way these results can be derived from there. Regularity of some curves generated geodesically from another is also studied.
Keywords: Holditch's theorem, Steiner's formula for parallel curves, Barbier's theorem, total geodesic curvature.
MSC: 53A35, 52A55.
[ Fulltext-pdf (970 KB)] for subscribers only.