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Curves of Minimax Spirality

C. Yalçın Kaya, Lyle Noakes, Philip Schrader

TL;DR

This work formulates and analyzes the problem of finding curves that minimize the maximum rate of change of curvature (minimax spirality) for planar curves with fixed length, endpoints, and tangents, possibly under a curvature bound. By recasting the problem as an optimal-control problem with state constraints, the authors classify feasible optimal-structure curves as concatenations of Euler spirals, circular arcs, and straight segments, with distinct patterns depending on whether curvature bounds are active. They develop two numerical approaches—direct discretization and arc-parametrization—and illustrate the theory with three numerical examples, revealing both clean bang–bang structures and richer behaviors including boundary arcs and potential chattering. The results have practical implications for design of comfortable vehicle tracks and roads and offer a foundation for future multi-objective extensions that combine minimax spirality with length or other criteria.

Abstract

We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We consider the case when simple bounds (constraints) are also imposed on the curvature along the curve. The curvature at the endpoints may or may not be specified. We prove via optimal control theory that the optimal curve is some concatenation of Euler spiral arcs, circular arcs, and straight line segments. When the curvature is not constrained (or when the curvature constraint does not become active), an optimal curve is only made up of a concatenation of Euler spiral arcs, unless the oriented endpoints lie in a line segment or a circular arc of the prescribed length, in which case the whole curve is either a straight line segment or a circular arc segment, respectively. We propose numerical methods and illustrate these methods and the results by means of three example problems of finding such curves.

Curves of Minimax Spirality

TL;DR

This work formulates and analyzes the problem of finding curves that minimize the maximum rate of change of curvature (minimax spirality) for planar curves with fixed length, endpoints, and tangents, possibly under a curvature bound. By recasting the problem as an optimal-control problem with state constraints, the authors classify feasible optimal-structure curves as concatenations of Euler spirals, circular arcs, and straight segments, with distinct patterns depending on whether curvature bounds are active. They develop two numerical approaches—direct discretization and arc-parametrization—and illustrate the theory with three numerical examples, revealing both clean bang–bang structures and richer behaviors including boundary arcs and potential chattering. The results have practical implications for design of comfortable vehicle tracks and roads and offer a foundation for future multi-objective extensions that combine minimax spirality with length or other criteria.

Abstract

We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We consider the case when simple bounds (constraints) are also imposed on the curvature along the curve. The curvature at the endpoints may or may not be specified. We prove via optimal control theory that the optimal curve is some concatenation of Euler spiral arcs, circular arcs, and straight line segments. When the curvature is not constrained (or when the curvature constraint does not become active), an optimal curve is only made up of a concatenation of Euler spiral arcs, unless the oriented endpoints lie in a line segment or a circular arc of the prescribed length, in which case the whole curve is either a straight line segment or a circular arc segment, respectively. We propose numerical methods and illustrate these methods and the results by means of three example problems of finding such curves.
Paper Structure (17 sections, 6 theorems, 40 equations, 7 figures)

This paper contains 17 sections, 6 theorems, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related problems and relevance
  3. Contribution
  4. Reformulations and Maximum Principle
  5. Reformulations
  6. Maximum principle for Problem (OC)
  7. Classification of Critical Curves
  8. Unspecified curvatures at endpoints
  9. Numerical Methods and Experiments
  10. Numerical methods
  11. Direct discretization
  12. Arc parametrization
  13. Numerical experiments
  14. Example 1: unconstrained curvature
  15. Example 2: constrained curvature
  16. ...and 2 more sections

Key Result

Lemma 3.1

Suppose that the curvature constraints do not become active and that the optimal control $v(t)$ for Problem (OC) is singular, over an interval $[\zeta_1,\zeta_2]\subseteq[0,t_f]$. Then $\kappa(t) = 0$, i.e., $\theta(t)$ is constant, and it follows that $v(t) = 0$, for a.e. $t\in[\zeta_1,\zeta_2]$.

Figures (7)

  • Figure 1: Example 1---Critical curves of length $t_f = 2$ and of minimax spirality between the oriented points $(0,0,-\pi/3)$ and $(0.4,0.4,-\pi/6)$, and with $\kappa(0) = \kappa(2) = 0$. Each curve satisfies the maximum principle and therefore is critical. The curve in dark red has the smallest derivative of curvature, $b \approx 15.73$.
  • Figure 2: Example 1---"Best" critical solution with verification of the necessary optimality condition that $u(t) = -b\,\mathop{\mathrm{sgn}}\limits(\lambda_4(t))$. Note that $\kappa(t)$ is affine in $t$ along each spiral arc.
  • Figure 3: Example 2---Critical curve of length $t_f = 2$ and of minimax spirality between the oriented points $(0,0,-\pi/3)$ and $(0.4,0.4,-\pi/6)$, with constraint $|\kappa(t)| \le 5$. The three junctions/switching points are indicated by black dots. One gets $b\approx 19.01$.
  • Figure 4: Example 2---Numerical verification of the necessary optimality conditions.
  • Figure 5: Example 3a---Critical curve of length $t_f = 0.6$ and of minimax spirality between the oriented points $(0,0,\pi/3)$ and $(0.4,0.4,\pi/4)$ joining two circular arcs, with $\kappa(0) = 5$ and $\kappa(0.6) = 2$. Here one gets $b\approx 49.0$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 3.1: Singularity With Inactive Curvature Constraints
  • Lemma 3.2: Singularity With Active Curvature Constraints
  • Remark 5: Types of Solution Arcs
  • Lemma 3.3: Normality of Solutions
  • Lemma 3.4: Switching Function With Inactive Curvature Constraints
  • Lemma 3.5: Total Singularity
  • ...and 3 more