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Results in Mathematics

, Volume 50, Issue 1–2, pp 109–124 | Cite as

Eigenvalue Equations in Curve Theory Part II: Evolutes and Involutes

  • Stephanie Müller
  • Angela Schwenk-Schellschmidt
  • Udo Simon
Article
  • 107 Downloads

Abstract.

We study plane Euclidean curves with positive curvature κ and spherical parametrization s.t. their radius of curvature κ−1 satisfies an eigenvalue equation. We investigate this class in detail in terms of evolutes and involutes and their geometric properties in relation to the eigenvalue equations considered. For a curve c in this class, the radius of curvature of its evolute satisfies the same eigenvalue equation like κ−1 of c, while the radii of curvature of its involutes satisfy a slightly modified eigenvalue equation. We give a complete local classification of all types of curves in this class.

Mathematics Subject Classification (2000).

53A04 34A30 34L15 53A17 

Keywords.

Euclidean plane curves curvature function support function evolutes involutes eigenvalue equations 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Stephanie Müller
    • 1
  • Angela Schwenk-Schellschmidt
    • 2
  • Udo Simon
    • 3
  1. 1.BerlinGermany
  2. 2.TFH BerlinBerlinGermany
  3. 3.Institut f. Mathematik MA 8-3TU BerlinBerlinGermany

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