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Self-parallel and transnormal curves

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Abstract

The construction of examples of self-parallel curves from a closed central curve given by F. J. Craveiro de Carvalho and S. A. Robertson is extended to general closed regular curves without the assumption of non-vanishing curvature made there. Furthermore, every self-parallel curve in 3-space is shown to be of the type obtained by this construction, if the order of the self-parallel group is greater than 2. These considerations are used to present examples for transnormal curves in 4-space with arbitrarily high degree of transnormality, disproving a long-standing conjecture of M. C. Irwin.

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  1. Fachbereich Mathematik der Technischen Universität Berlin, Straße des 17. Juni 136, 1000, Berlin 12, Germany

    Bernd Wegner

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  1. Bernd Wegner
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Wegner, B. Self-parallel and transnormal curves. Geom Dedicata 38, 175–191 (1991). https://doi.org/10.1007/BF00181217

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  • DOI: https://doi.org/10.1007/BF00181217

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