A proof of Jordan curve theorem based on the sweepline algorithm for trapezoidal decomposition of a polygon

Apurva Mudgal

Authors on Pith no claims yet

Pith reviewed 2026-05-07 12:08 UTC · model grok-4.3

classification 💻 cs.CG

keywords prooftheoremcurvejordanalgorithmdecompositionpolygonsweepline

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The pith

A proof of the Jordan curve theorem is presented by generalizing the sweepline algorithm for trapezoidal polygon decomposition and invoking Zorn's lemma, claimed as the first algorithmic proof using computational geometry.

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

The Jordan curve theorem states that any simple closed curve in the plane divides it into an interior region and an exterior region. This paper claims to prove the theorem by extending a standard computational geometry algorithm that sweeps a line across a polygon to create trapezoids. The generalization, combined with Zorn's lemma from set theory, is said to establish the separation property. The authors indicate the argument can be taught to undergraduates after basic discrete mathematics preparation.

Core claim

Load-bearing premise

That generalizing the sweepline algorithm for trapezoidal decomposition correctly encodes the topological separation property of a simple closed curve without unstated assumptions, and that Zorn's lemma applies appropriately in this algorithmic context.

read the original abstract

We prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a polygon. Our proof uses Zorn's lemma (or, equivalently the axiom of choice). Though several proofs have been given for the Jordan curve theorem by various authors, ours is the {\bf first algorithmic proof} of Jordan curve theorem using computational geometry. Further, with some preparation, the proof can be taught as part of an undergraduate discrete mathematics course, where till now the proof of this theorem was considered inaccessible.

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

This paper claims a new algorithmic proof of the Jordan curve theorem via generalized sweepline and Zorn's lemma on partial decompositions, but the upper-bound step for chains risks circularity by presupposing non-intersection.

read the letter

The core idea is to generalize the standard sweepline for trapezoidal decomposition of a simple polygon, build a poset of partial decompositions ordered by refinement, apply Zorn's lemma to reach a maximal element, and then read off the interior/exterior separation from that maximal decomposition. The abstract says this is the first such computational-geometry proof and that it can be taught in an undergrad discrete-math course after some setup. That framing is the main novelty on offer: a bridge from a routine CG algorithm to a classic topology result without heavy machinery beyond Zorn's lemma. If the details hold, it does give a fresh perspective that might feel more algorithmic than the usual topological arguments. The sweepline events themselves are handled in the usual way once the polygon is assumed simple, so that part is routine and well-trodden. The paper earns credit for trying to make the argument accessible rather than opaque. The soft spot sits in the Zorn application. For the poset to work, every chain needs an upper bound that is still a valid trapezoidal decomposition (no crossings, full coverage, proper adjacency). The construction of that bound, apparently via some form of union or limit, is not spelled out with an explicit check that the limit remains free of intersections and respects the sweep ordering. Because the local event handling already encodes the global non-intersection of the curve, the limit step can end up assuming the separation property it is meant to establish. That is the load-bearing part and the place where circularity could creep in. The rest of the argument does not appear to have similar gaps. This paper is for readers who follow alternative proofs of the Jordan curve theorem or who like seeing computational geometry tools applied outside their usual domain. A serious referee should look at it to verify whether the upper-bound construction actually closes without hidden assumptions; the claim is important enough to warrant that check even if revisions are needed. I would send it out for review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a simple polygon. It defines a poset of partial decompositions ordered by refinement, applies Zorn's lemma to obtain a maximal element, and argues that the properties of this maximal decomposition establish the separation of the plane into bounded interior and unbounded exterior regions by the curve. The authors position the result as the first algorithmic proof of the theorem in computational geometry and suggest it is accessible for undergraduate discrete mathematics courses after suitable preparation.

Significance. If the central argument holds, the paper would supply a novel bridge between computational geometry techniques and classical topology, offering a proof that relies on standard sweep-line primitives plus the axiom of choice rather than purely topological machinery. This could make the Jordan curve theorem more approachable in discrete-math settings, though the non-constructive use of Zorn's lemma departs from the constructive emphasis typical in computational geometry.

major comments (2)

[Zorn's lemma application and poset construction] In the section applying Zorn's lemma to the poset of partial trapezoidal decompositions, the argument that every chain possesses an upper bound (via union) lacks an explicit verification that the union remains a valid decomposition: it must be shown to introduce no new edge crossings, maintain consistent trapezoid adjacency, and cover the polygon without gaps or overlaps. The local ordering rules for sweep events presuppose global non-intersection, so the limit step risks assuming the separation property being proved.
[Poset of partial decompositions] The definition of the poset (ordered by refinement or inclusion) and the precise conditions for an element to be a valid partial decomposition are not stated with sufficient rigor to guarantee that a maximal element directly encodes the interior/exterior separation without additional unstated topological assumptions about the curve.

minor comments (2)

[Abstract] The abstract's claim that this is the 'first algorithmic proof using computational geometry' would benefit from a short comparison with existing proofs (e.g., those based on graph theory or other algorithmic approaches) to substantiate novelty.
[Algorithm description] Notation for sweep-line events and trapezoid adjacency could be illustrated with a small worked example or figure to improve readability for the intended undergraduate audience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough evaluation of our paper. The comments highlight areas where the presentation of the proof can be strengthened for clarity and rigor. We address each major comment below, and we will incorporate the suggested improvements in a revised version of the manuscript.

read point-by-point responses

Referee: In the section applying Zorn's lemma to the poset of partial trapezoidal decompositions, the argument that every chain possesses an upper bound (via union) lacks an explicit verification that the union remains a valid decomposition: it must be shown to introduce no new edge crossings, maintain consistent trapezoid adjacency, and cover the polygon without gaps or overlaps. The local ordering rules for sweep events presuppose global non-intersection, so the limit step risks assuming the separation property being proved.

Authors: We thank the referee for pointing out this potential gap in the exposition. Upon review, we agree that the proof that the union of a chain forms an upper bound requires more explicit verification to avoid any appearance of circularity. In the revised manuscript, we will insert a new lemma that demonstrates: (1) the union introduces no new edge crossings because any crossing would involve finitely many segments and thus be detected in some finite partial decomposition in the chain; (2) trapezoid adjacency is maintained by the local sweep rules applied consistently across the chain; and (3) coverage without gaps or overlaps follows from the fact that each partial decomposition covers its swept region completely, and the union covers the entire polygon as the sweep progresses to the end. The local ordering rules are applied only within each partial decomposition, where non-intersection is enforced by construction, not presupposing the global Jordan property. This addresses the concern without assuming the result. revision: yes
Referee: The definition of the poset (ordered by refinement or inclusion) and the precise conditions for an element to be a valid partial decomposition are not stated with sufficient rigor to guarantee that a maximal element directly encodes the interior/exterior separation without additional unstated topological assumptions about the curve.

Authors: The referee correctly identifies that the poset and validity conditions could benefit from greater formality. We will revise the manuscript to include a precise definition: Let P be a simple closed polygonal curve. A partial decomposition D is a finite collection of trapezoids such that their interiors are disjoint, their union is the region swept by the line up to the current position, boundaries align with segments of P or vertical lines, and no two edges cross within the swept area. The poset is ordered by refinement, where D1 ≤ D2 if D2 includes all trapezoids of D1 and further subdivides some. We will prove that any maximal element must correspond to a complete decomposition where the sweep line has traversed the entire curve, thereby partitioning the plane into the bounded interior (union of finite trapezoids) and unbounded exterior, using only the properties of the sweep without additional topological assumptions beyond the polygonal nature of the curve. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard Zorn application to poset of decompositions without reduction to inputs

full rationale

The paper presents a proof of the Jordan curve theorem via generalization of the sweepline algorithm for trapezoidal decomposition, invoking Zorn's lemma on a poset of partial decompositions. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear. The use of Zorn's lemma is an external axiom of choice application, not a tautological reduction. The sweepline events and decomposition steps are standard once a simple polygon is given, and the proof is self-contained against external benchmarks such as the axiom of choice and computational geometry primitives. No quoted step reduces the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on the axiom of choice (Zorn's lemma) as an external set-theoretic tool and on the correctness of the standard sweepline algorithm for trapezoidal decomposition as a starting point; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Zorn's lemma (axiom of choice)
Explicitly invoked in the abstract to support the proof.

pith-pipeline@v0.9.0 · 9076 in / 1182 out tokens · 105097 ms · 2026-05-07T12:08:54.054337+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A greedy maximal sweepline algorithm for a Jordan curve
cs.CG 2026-05 unverdicted novelty 4.0

A greedy sweepline algorithm for Jordan curves is proven maximal using König's lemma.