Parallel and Transnormal Curves on Surfaces
The exterior parallelism of space curves in the sense of H.R. Farran and S.A. Robertson [J. London Math. Soc. (2) 35 (1987), 527-538] is studied in the case that the curves are located on a surface M in Euclidean 3-space. At first a local existence and uniqueness result is established under some genericity assumption for pairs of parallel curves passing through so-called EP-pairs on M. A surface for which every pair of points is an EP-Pair is shown to be an open part of a sphere or a plane. Continuous families of mutually parallel curves must consist of curvature lines. Cylinders, spheres and planes are the only surfaces which carry two such families. Finally, some special results are derived for parallel pairs and self-parallel resp. transnormal curves on surfaces of rotation and cylinders.
KeywordsConstant Width Normal Plane Continuous Family Normal Section Parallel Transfer
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