Existence d'une courbe \`a courbure positive maximisant le minimum du rayon de courbure -- "Observation num\'erique"

J\'er\^ome Bastien

arxiv: 1906.10010 · v1 · pith:LJJ6HMJNnew · submitted 2019-06-19 · 🧮 math.MG

Existence d'une courbe \`a courbure positive maximisant le minimum du rayon de courbure -- "Observation num\'erique"

J\'er\^ome Bastien This is my paper

Pith reviewed 2026-05-25 20:14 UTC · model grok-4.3

classification 🧮 math.MG

keywords curveminimumcourburecurvaturecurvesextremitiespositivealgebraic

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The pith

Existence proved for maximizer of min radius of curvature in positive-curvature curves with fixed ends; numerically observed to be circular arc plus line segment (Dubins case).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors consider curves in the plane that connect two given points with given tangent directions at the ends and that have positive curvature everywhere. For any such curve they measure the smallest radius of curvature that occurs along it. The goal is to choose the curve that makes this smallest radius as large as possible. They first give a mathematical argument that at least one curve achieving the largest possible value must exist. Using computer experiments they then observe that the optimal curve always appears to consist of a single circular arc joined smoothly to a straight-line segment (the segment sometimes shrinking to a point). This shape is already known in the literature on shortest paths with curvature bounds. The authors note that the observed curve can be used to improve the design of a mechanical component described in a patent.

Core claim

We first prove that there exists a curve of E which maximizes this minimum. Numerically, we observe then that this curve is equal to the unique curve of E composed of an arc of circle and a line segment, where appropriate reduced to a point.

Load-bearing premise

The numerical experiments correctly identify the global maximizer and do not miss other candidate shapes; this premise enters when the abstract states that the observed circle-plus-line curve is the maximizer without providing a supporting analytic proof.

read the original abstract

We consider the set E of curves with positive algebraic curvature, whose extremities and tangents in their extremities are given. For each of the curves of E, we define the minimum of the radius of curvature. We first prove that there exists a curve of E which maximizes this minimum. Numerically, we observe then that this curve is equal to the unique curve of E composed of an arc of circle and a line segment, where appropriate reduced to a point. This curve corresponds also to a particular case of Dubins's curve and will be used to improve the conception of a piece of a patent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard differential-geometry assumption that the set E of admissible curves is non-empty and that the minimum radius functional is well-defined and attains its supremum on a suitable closure of E.

axioms (1)

domain assumption The set E of curves with positive algebraic curvature, fixed extremities and fixed tangents at the extremities is non-empty and the minimum radius of curvature is a well-defined functional on E.
This defines the feasible set and the objective in the first sentence of the abstract.

pith-pipeline@v0.9.0 · 5635 in / 1314 out tokens · 46965 ms · 2026-05-25T20:14:43.533855+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We first prove that there exists a curve of E which maximizes this minimum. Numerically, we observe then that this curve is equal to the unique curve of E composed of an arc of circle and a line segment
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

En adaptant la démonstration de Dubins, on montre qu’il existe une courbe de E, maximisant le minimum du rayon de courbure

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.