Classical Curve Theory -
Solutions to Frenet's Equations

Ueber die Darstellung von Raumkurven aus ihren Invarianten (The Representation of Space Curves by their Invariants)

Toni Menninger, master thesis 1996

Mathematisches Institut der Bayerischen Julius-Maximilians-Universitaet Wuerzburg
Advisors: Johann Hartl, TU Munich, and Helmut Pabel, Wuerzburg

The thesis as pdf file (550 kB)

This thesis on curve theory explores a classical question: for which types of curves can the Frenet equations be explicitly solved, in other words: for which pairs of curvature κ and torsion τ functions is the parameterization of the corresponding curve known? The known answers are:

  • Plane curves -
    		• τ = 0
  • Helical curves -
    		• κ / τ = const.
  • Curves of constant precession1 -
    		• κ = ω cos(μ s),   τ = ω sin(μ s)   (ω, μ = const.)

    The thesis presents - for the first time to my knowledge - a generalization of curves of constant precession ("Kreisellinien", theorems 26 and 27) and the solution of the respective Frenet differential equations; they are characterized by the following relations:

    κ = ω cos(φ),   τ = ω sin(φ),   φ' / ω = const.   (φ(s) differentiable, ω(s) continuous)

    I also present a method to construct a series of curve classes with solutions for the Frenet equations (with plane curves, helical curves and generalized curves of constant precession as the first three classes in the series). Moreover, I attempt a complete and systematic overview of classical curve theory, including some results on moving frames (relationship between Frenet and Bishop Frame2), spherical curves and surface curves.


    Comments are welcome: toni_menninger at hotmail dot com

    Curves

    A curve of constant precession (Scofield 1995) with it's tangent (left), the arcs of which represent spherical helices.

    1 Paul D. Scofield: Curves of Constant Precession, The American Mathematical Monthly, Volume 102, Number 6, June-July 1995

    2 Bishop or Parallel Transport frames are characterized by the fact that the derivatives of both normal components are tangential. In other words, the moving frame twists only "as much as necessary", it comes as close to parallel translation as possible. Existence of a family of Bishop frames is guaranteed for any regular C2curve. For a discussion of applications and numerical algorithms see Hui Ma: Curve and Surface Framing for Scientific Visualization and Domain Dependent Navigation, 1996 (Ph.D. Dissertation)
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