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Spherical curves whose curvature depends on distance to a great circle

Ildefonso Castro, Ildefonso Castro-Infantes, Jesús Castro-Infantes

Abstract

Motivated by a problem posed by David A. Singer in 1999 and by the elastic spherical curves, we study the spherical curves whose curvature is expressed in terms of the distance to a great circle (or from a point). By introducing the notion of spherical angular momentum, we provide new characterizations of some well known curves, like the mentioned elastic curves, spherical catenaries, loxodromic-type spherical curves, the Viviani's curve, and the spherical Archimedean spirals curves. Furthermore, we show that they may be obtained as critical points of some energy curvature functionals. We also find out several new families of spherical curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions, and we are able to get arc length parametrizations of them.

Spherical curves whose curvature depends on distance to a great circle

Abstract

Motivated by a problem posed by David A. Singer in 1999 and by the elastic spherical curves, we study the spherical curves whose curvature is expressed in terms of the distance to a great circle (or from a point). By introducing the notion of spherical angular momentum, we provide new characterizations of some well known curves, like the mentioned elastic curves, spherical catenaries, loxodromic-type spherical curves, the Viviani's curve, and the spherical Archimedean spirals curves. Furthermore, we show that they may be obtained as critical points of some energy curvature functionals. We also find out several new families of spherical curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions, and we are able to get arc length parametrizations of them.

Paper Structure

This paper contains 14 sections, 9 theorems, 75 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Spherical curves such that $\kappa = \kappa (z)$ and the spherical angular momentum
  3. Elastic-type curves on the sphere
  4. A new characterization and a generalization of elastic curves
  5. Seiffert's spherical spirals
  6. Borderline spherical elastic curves
  7. Loxodromic-type curves on the sphere
  8. Case $0 <\mu <1$: spherical loxodromes
  9. Case $\mu =1$
  10. Case $\mu >1$
  11. Spherical catenaries
  12. New and classical spherical curves
  13. Spherical curves such that $\kappa(\varphi)=p \cos 2 \varphi/ \cos \varphi, \, 0<p<1$
  14. Viviani's curve and spherical Archimedean spirals

Key Result

Theorem \oldthetheorem

Let $\kappa=\kappa(z)$ be a continuous function. Then the problem of determining locally a spherical curve whose curvature is $\kappa(z)$ ---$z$ representing the (non constant) signed distance to the great circle $z\!=\!0$--- with spherical angular momentum $\mathcal{K} (z)$ satisfying anti K, is so

Figures (11)

  • Figure 1: Great circles $\mathbb{S}^2 \cap \{ \sqrt{1-c^2}\,y+c\,z=0 \}$: $\mathcal{K}\equiv c\in (-1,1)$.
  • Figure 2: Small circles: $\mathcal{K} (z)= k_0 \, z + c, \ k_0 >0$; $0\leq |c|<1$ (left), $c=\pm 1$ (center), $1<|c|<\sqrt{1+k_0^2}$ (right).
  • Figure 3: Seiffert's spirals ($p\approx 0$, $0<p<1$, $p\approx 1$, $p>1$): $\mathcal{K} (z)= p z^2 -p$, $p>0$.
  • Figure 4: Borderline elastic curves: $\mathcal{K} (z) = a z^2 -1, \, a>1/2$ (left: $1/2<a \leq 1$; right: $a\geq 1$).
  • Figure 5: Loxodromes: $\mathcal{K}(\varphi)=-\cos \alpha \cos\varphi$, $\alpha \in (0,\pi/2)$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem: $\kappa\!\equiv\! 0$
  • Example \oldthetheorem: $\kappa \equiv k_0 >0$
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • proof : Proof of Theorem \ref{['general elasticae']}
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • ...and 6 more