CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exterior-angle sum of a planar polygon

Statement

Let $P \subseteq R^{2}$ be a positively oriented simple Euclidean polygon: a simple closed piecewise linear regular plane curve with finitely many ordinary vertices, parametrized so that the bounded disk region it bounds lies on its left, and with matching unit tangents at the identified endpoint. Then its signed exterior angles $α_{1}, \dots, α_{m}$ , one at each vertex, satisfy $\sum_{j = 1}^{m} α_{j} = 2 π .$ Each $α_{j}$ is the principal signed turn in $(- π, π)$ from the incoming to the outgoing unit tangent; a straight vertex contributes $α_{j} = 0$ .

Facts & Assumptions

Given: A positively oriented simple Euclidean polygon $P$ with finitely many ordinary vertices, bounding a supplied disk region, with matching unit tangents at the identified endpoint and arclength parametrization on every side.

[F1]

A positively oriented simple closed piecewise $C^{2}$ regular plane curve that bounds a supplied disk region and has finitely many ordinary corners satisfies $\int_{γ} k_{p l a n e} d s + \sum_{j} α_{j} = 2 π$ (Hopf turning-tangent theorem with ordinary corners).

[F2]

Signed geodesic curvature is defined by $A_{γ} = k_{g} J T$ and $k_{g} = g (A_{γ}, J T)$ for the positive quarter-turn $J$ (Signed geodesic curvature).

Proof

technique · specialise Hopf turning to a curve whose smooth pieces are straight

1.1F2givenalgebra

Each side of $P$ is a straight segment and admits a unit-speed linear parametrization $s \mapsto p_{j} + s u_{j}$ with $∣ u_{j} ∣ = 1$ . Its ordinary second derivative vanishes, so the covariant acceleration of the ambient Euclidean connection vanishes, and [F2] gives $k_{p l a n e} = 0$ along every smooth piece.

1.2F1given

The polygon is a simple closed piecewise $C^{2}$ regular plane curve with finitely many ordinary corners, namely its vertices, and it bounds the supplied disk region with matching endpoint tangents. Hence [F1] applies to $P$ , and its signed exterior angles are the principal turns of [F1].

2.1F1step 1.1algebra∎

The curvature integral in [F1] vanishes by step 1.1, so $\sum_{j} α_{j} = 2 π$ . A straight vertex has incoming and outgoing tangents equal, so its principal turn is $0$ and contributes nothing; the identity is therefore a statement about the genuine turns only.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“Some Plane Geometry,” Corollary 9.6, printed pp. 156–161, records that the total signed curvature plus exterior angles of a positively oriented simple closed piecewise smooth plane curve equals $2 π$ ; Datar, Lectures on Riemannian Geometry, Lecture 1, §1.3, Theorem 1.3.2, proves the equivalent rotation-index statement. The specialization to straight sides uses only the library's definition of signed geodesic curvature of a straight unit-speed segment and the already authored Hopf turning theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)