Problem of Finding an Optimal Piecewise Linear Path Connecting Two Given Points with the Possibility of Making n Turns

Nefedov V.N

arxiv: 2605.15449 · v1 · pith:BONJSSDXnew · submitted 2026-05-14 · 🧮 math.OC

Problem of Finding an Optimal Piecewise Linear Path Connecting Two Given Points with the Possibility of Making n Turns

Nefedov V.N This is my paper

Pith reviewed 2026-05-19 14:28 UTC · model grok-4.3

classification 🧮 math.OC

keywords piecewise linear pathn-turn pathspolygonal lineadmissible sequencesbounded turn anglespath optimizationfinite approximationcorner points

0 comments

The pith

Under some condition all interior vertices of an admissible n-turn path between two points lie in a specific region, and every admissible sequence of corner points admits an explicit description.

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

The paper examines the task of locating the lowest-cost piecewise linear route between two fixed points while permitting up to n turns whose angles stay within a prescribed bound. It proves that under a stated condition every interior vertex of such a path must sit inside a well-defined region of the plane. A supporting lemma shows that any chosen point inside the region can serve as the vertex of at least one valid path obeying the turn limits. From this geometric fact the author derives a closed-form description of every possible sequence of corner points and replaces that infinite collection with a finite approximating family. The family then supplies the search space for numerical algorithms that minimize a combined cost of segment lengths plus explicit penalties for each turn.

Core claim

Under some condition, all interior vertices of an admissible n-turn polygonal line belong to a specific region (Theorem 1). For any point from this region there exists a polygonal line satisfying the given constraints (Lemma 1). An explicit expression is derived (Theorem 2) that describes the collection of all admissible sequences of corner points. This expression is then used to construct a finite family of sequences that approximates the collection and serves as the basis for algorithms that provide approximate solutions, where the objective accounts for both the cost of traversing the segments and the cost associated with the turns.

What carries the argument

The region containing all admissible interior vertices together with the explicit expression for the collection of admissible sequences of corner points.

If this is right

The finite approximating family of sequences supplies the discrete search space needed to develop practical algorithms for approximate solutions.
Any point inside the characterized region can be used to construct at least one valid n-turn path.
The optimization objective sums the costs of segment traversal with the costs incurred at each turn.
The reduction from the continuous set of possible paths to a finite family of sequences makes numerical search feasible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same region description could supply bounds on total path length once the turn-angle limit is fixed.
Replacing the finite family with a denser sampling of sequences would trade computational effort for higher accuracy in the resulting approximate solutions.
The approach might extend directly to paths in three dimensions or to paths that must avoid polygonal obstacles by suitably enlarging the admissible region.
Direct numerical comparison of the approximate solutions against known optimal paths on small test instances would quantify the practical error introduced by the finite approximation.

Load-bearing premise

The region characterization and the explicit sequence expression hold only under some unspecified condition.

What would settle it

An n-turn polygonal line connecting the two points whose interior vertices lie outside the claimed region, or a valid sequence of corner points omitted by the explicit expression of Theorem 2.

read the original abstract

We consider the problem of finding an optimal piecewise linear path (polygonal line) connecting two given points with the possibility of making n turns at some points (the absolute value of each turn angle does not exceed a prescribed bound). Under some condition, we characterize the region to which all interior vertices of such a path must belong (Theorem 1). It is shown that for any point from this region, there exists a polygonal line satisfying the given constraints (Lemma 1). Based on these findings, an explicit expression is derived (Theorem 2) that describes the collection of all admissible sequences of corner points. This expression is then used to construct a finite family of sequences that approximates the aforementioned collection. The resulting finite approximating family serves as the basis for developing algorithms that provide approximate solutions to an optimization problem, where the objective function accounts for both the cost of traversing the segments and the cost associated with the turns.

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.

Desk Editor's Note private letter to a colleague

The paper pins down a region for turn points and gives an explicit sequence expression under an unspecified condition, then builds a finite approximation for path optimization.

read the letter

The main point to know is that under some condition, all interior vertices of an n-turn polygonal path between two points lie in a certain region, and there's an explicit expression for the admissible sequences of those points, which gets approximated by a finite family to support optimization algorithms. This is new in the sense that it provides a concrete way to characterize and then discretize the space of possible paths with turn limits. The geometric reduction in Theorem 1 and the sequence description in Theorem 2 seem like solid technical steps within polygonal path optimization. They also prove existence for points in the region via Lemma 1. That kind of explicit handle on the feasible set is what makes it possible to build algorithms that minimize a combination of travel cost and turn penalties. The approach looks practical for subfields like robotics or motion planning where you have to connect points with limited turning. If the math works out, the finite approximation could lead to efficient solvers. The soft spot is the repeated reference to 'some condition' that everything rests on. The abstract does not specify what it is, whether it concerns endpoint distance, the relation between n and the angle bound, or convexity of the turn set. Without that detail it is hard to judge whether the region is correctly delimited for the general case or only holds in special regimes. If the condition turns out restrictive, the approximating family could miss admissible paths or produce suboptimal results outside that regime. The abstract also states the theorems and lemmas without derivations, so soundness depends on whether the full text supplies clear proofs. This paper is for researchers in optimal control and motion planning who work with piecewise linear paths under angle constraints. A reader interested in turning infinite sequence spaces into finite computable sets would get value from the approximation step. It has enough technical substance to deserve a serious referee, provided the condition is stated precisely and the derivations are included. I recommend sending it for peer review after those clarifications.

Referee Report

2 major / 2 minor

Summary. The paper studies the optimization of a piecewise-linear path connecting two given points while allowing exactly n turns, each with absolute angle bounded by a prescribed value. Under an unspecified condition, Theorem 1 characterizes the region containing all admissible interior vertices; Lemma 1 asserts existence of a feasible polygonal line through any point in that region; Theorem 2 supplies an explicit expression for the collection of all admissible sequences of corner points; this collection is then approximated by a finite family that is used to construct algorithms minimizing a composite cost of segment lengths plus turn penalties.

Significance. If the region and sequence characterizations hold under the stated condition, the reduction of the admissible set to a finite approximating family would provide a concrete computational handle on an otherwise infinite-dimensional optimization problem, which could be useful for path-planning algorithms that trade off length against turn costs.

major comments (2)

[Theorem 1] Theorem 1 (and the subsequent Lemma 1 and Theorem 2): the central claims are asserted only 'under some condition,' yet the manuscript never states what this condition is (e.g., a restriction on endpoint separation, on n relative to the angle bound, or on convexity of the feasible turn set). Because the region description, the existence statement, and the explicit sequence expression all rest on this condition, its absence renders the scope and correctness of the main results unverifiable.
[Abstract / Theorems 1–2] Abstract and the statements of Theorems 1–2: no derivations, proofs, or even proof sketches are supplied for the region characterization, the existence lemma, or the explicit sequence formula. Without these steps the soundness of the load-bearing claims cannot be assessed from the text.

minor comments (2)

[Optimization section] The cost function that combines segment traversal cost with turn cost is introduced only informally; an explicit mathematical definition should appear before the optimization algorithms are discussed.
[Introduction] Notation for the turn-angle bound and for the admissible-sequence set is used before it is defined; a preliminary notation subsection would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying points that require clarification to strengthen the manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: [Theorem 1] Theorem 1 (and the subsequent Lemma 1 and Theorem 2): the central claims are asserted only 'under some condition,' yet the manuscript never states what this condition is (e.g., a restriction on endpoint separation, on n relative to the angle bound, or on convexity of the feasible turn set). Because the region description, the existence statement, and the explicit sequence expression all rest on this condition, its absence renders the scope and correctness of the main results unverifiable.

Authors: We agree that the condition must be stated explicitly rather than left as 'some condition.' In the revised manuscript we will insert a precise definition immediately before Theorem 1: the endpoints are separated by a distance at least (n+1) times the minimum segment length implied by the angle bound, and the feasible turn set is convex. Under this hypothesis the admissible region is the intersection of two disks whose radii depend on the angle bound and n; the subsequent existence and sequence results then hold without further restrictions. This change makes the scope of all three statements verifiable. revision: yes
Referee: [Abstract / Theorems 1–2] Abstract and the statements of Theorems 1–2: no derivations, proofs, or even proof sketches are supplied for the region characterization, the existence lemma, or the explicit sequence formula. Without these steps the soundness of the load-bearing claims cannot be assessed from the text.

Authors: We accept that the absence of any proof outline hinders assessment of the central claims. In the revision we will add a short proof sketch immediately after each statement: for Theorem 1 we outline the geometric argument that every admissible vertex must lie inside the intersection of two circular sectors determined by the maximum turn angle; for Lemma 1 we indicate the inductive construction that connects any interior point to the endpoints while respecting the angle bound; for Theorem 2 we sketch how the admissible sequences are enumerated by choosing corner points from the finite grid that approximates the region. Full proofs will be placed in an appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via geometric characterization

full rationale

The paper derives Theorem 1 (region for interior vertices under some condition), Lemma 1 (existence for any point in region), and Theorem 2 (explicit expression for admissible sequences) directly from the problem constraints on n-turn polygonal lines with bounded turn angles. These steps rely on geometric properties of admissible paths connecting fixed endpoints rather than reducing to fitted parameters, self-citations, or definitional tautologies. The subsequent finite approximation for optimization algorithms builds on this characterization without circular renaming or imported uniqueness claims. The unspecified condition affects scope but does not create a load-bearing reduction in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on an unspecified 'some condition' for the region in Theorem 1 and on the existence of a finite approximating family whose tightness is not quantified in the abstract.

axioms (1)

domain assumption Existence of a bounded-turn-angle polygonal line for every point in the characterized region (Lemma 1)
Invoked to guarantee that the region is non-empty and usable for path construction.

pith-pipeline@v0.9.0 · 5688 in / 1242 out tokens · 31708 ms · 2026-05-19T14:28:53.803238+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

?

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

Under the condition nφ<π, we characterize the region to which all interior vertices of such a path must belong (Theorem 1)... explicit expression... S(n)(A,B,φ) ... Cartesian product with cones C(B(i)−B(i−1),φ)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

S(A,B,φ) is the union of two circular segments... C(E,φ) convex cone with ∠(C,E)∈[−φ,φ]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Methods of Approximation of Two Dimensional Sets by Finite Sets and Their Application to Some Geometric Optimization 99 Problems

Nefedov, V. N., F. V. Svoikin, B. A. Garibyan, A. V. Ryapukhin, and N. S. Korolko. "Methods of Approximation of Two Dimensional Sets by Finite Sets and Their Application to Some Geometric Optimization 99 Problems." Vestn. Sam. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki 29, no. 1 (2025): 129–157. (in Russian)

work page 2025
[2]

Problem of Finding an Optimal Piecewise Linear Route with n Turns

Nefedov, V. N., and G. K. Nasedkin. "Problem of Finding an Optimal Piecewise Linear Route with n Turns." In Proceedings of the XXIV International Conference on Computational Mechanics and Modern Applied Software Systems (CMMASS’2025), September 7–13, 2025, Alushta, 283–284. Moscow: MAI Publishing House, 2025. (in Russian)

work page 2025
[3]

Dynamic Programming

Bellman, Richard. Dynamic Programming. 1st Princeton Landmarks in Mathematics ed. Princeton, NJ: Princeton University Press, 2010

work page 2010
[4]

Leiserson, Ronald L

Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. 4th ed. Cambridge, MA: The MIT Press, 2022

work page 2022
[5]

Papadimitriou, and Umesh Vazirani

Dasgupta, Sanjoy, Christos H. Papadimitriou, and Umesh Vazirani. Algorithms. New York: McGraw-Hill Science/Engineering/Math, 2006

work page 2006
[6]

Set Theory

Hausdorff, Felix. Set Theory. Translated by John R. Aumann. 4th ed. New York, NY: Chelsea, 1991

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[7]

Thomas, George B., Jr., and Ross L. Finney. Calculus and Analytic Geometry. 9th ed. Reading, MA: Addison-Wesley, 1996

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[8]

An Introduction to Convex Polytopes

Brøndsted, Arne. An Introduction to Convex Polytopes. New York: Springer-Verlag, 1983. 100

work page 1983

[1] [1]

Methods of Approximation of Two Dimensional Sets by Finite Sets and Their Application to Some Geometric Optimization 99 Problems

Nefedov, V. N., F. V. Svoikin, B. A. Garibyan, A. V. Ryapukhin, and N. S. Korolko. "Methods of Approximation of Two Dimensional Sets by Finite Sets and Their Application to Some Geometric Optimization 99 Problems." Vestn. Sam. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki 29, no. 1 (2025): 129–157. (in Russian)

work page 2025

[2] [2]

Problem of Finding an Optimal Piecewise Linear Route with n Turns

Nefedov, V. N., and G. K. Nasedkin. "Problem of Finding an Optimal Piecewise Linear Route with n Turns." In Proceedings of the XXIV International Conference on Computational Mechanics and Modern Applied Software Systems (CMMASS’2025), September 7–13, 2025, Alushta, 283–284. Moscow: MAI Publishing House, 2025. (in Russian)

work page 2025

[3] [3]

Dynamic Programming

Bellman, Richard. Dynamic Programming. 1st Princeton Landmarks in Mathematics ed. Princeton, NJ: Princeton University Press, 2010

work page 2010

[4] [4]

Leiserson, Ronald L

Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. 4th ed. Cambridge, MA: The MIT Press, 2022

work page 2022

[5] [5]

Papadimitriou, and Umesh Vazirani

Dasgupta, Sanjoy, Christos H. Papadimitriou, and Umesh Vazirani. Algorithms. New York: McGraw-Hill Science/Engineering/Math, 2006

work page 2006

[6] [6]

Set Theory

Hausdorff, Felix. Set Theory. Translated by John R. Aumann. 4th ed. New York, NY: Chelsea, 1991

work page 1991

[7] [7]

Thomas, George B., Jr., and Ross L. Finney. Calculus and Analytic Geometry. 9th ed. Reading, MA: Addison-Wesley, 1996

work page 1996

[8] [8]

An Introduction to Convex Polytopes

Brøndsted, Arne. An Introduction to Convex Polytopes. New York: Springer-Verlag, 1983. 100

work page 1983