Integral Geometry about the visual angle of a convex set

Agust\'i Revent\'os; Eduardo Gallego; Juli\`a Cuf\'i

arxiv: 1906.10374 · v1 · pith:LOGI5GILnew · submitted 2019-06-25 · 🧮 math.DG · math.MG

Integral Geometry about the visual angle of a convex set

Juli\`a Cuf\'i , Eduardo Gallego , Agust\'i Revent\'os This is my paper

Pith reviewed 2026-05-25 16:29 UTC · model grok-4.3

classification 🧮 math.DG math.MG

keywords visual angleconvex setintegral geometryCrofton formulaHurwitz formulaMasotti formulapairs of linesmotion-invariant measure

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The pith

The visual angle of a convex set equals an integral of densities with respect to the canonical measure on pairs of lines.

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

The paper interprets the integral formulas for the visual angle of convex sets, including those by Crofton, Hurwitz and Masotti, using integral geometry. It shows that these formulas can be seen as integrals of appropriate densities against the canonical measure in the space of pairs of lines. This view allows for new and simpler proofs of the formulas. A sympathetic reader would care because it unifies several classical results under a single measure-theoretic framework on line spaces.

Core claim

The integral formulas of Crofton, Hurwitz and Masotti for the visual angle admit an interpretation in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines, and new simpler proofs of them are given.

What carries the argument

The canonical motion-invariant measure on the space of pairs of lines, used to express the visual angle as an integral of a density.

If this is right

The formulas are instances of a general type of integral formulas in integral geometry.
Simpler proofs are obtained compared to earlier derivations.
Applies to a general type of integral formulas of the visual angle.

Where Pith is reading between the lines

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

This reinterpretation may suggest analogous measures for other geometric quantities like visual volume in 3D.
Sampling from the line-pair measure could provide Monte Carlo estimates for visual angles.
The approach might generalize to non-convex sets under suitable conditions.

Load-bearing premise

A canonical motion-invariant measure exists on the space of pairs of lines such that the visual angle of a convex set equals an integral of a suitable density against that measure.

What would settle it

For a unit disk, compute the visual angle directly as a function of distance and compare numerically to the value obtained by integrating the proposed density over pairs of lines; a discrepancy would disprove the equality.

Figures

Figures reproduced from arXiv: 1906.10374 by Agust\'i Revent\'os, Eduardo Gallego, Juli\`a Cuf\'i.

**Figure 1.** Figure 1: Direction of a line. With these new coordinates, proceeding as in [6], one has dG1 dG2 = |sin(α2 − α1)| dα1 dα2 dP. (12) We have used the fact that ϕ2 − ϕ1 = α2 − α1 + π where = (P, α1, α2) = 0, ±1, according to the position with respect to the origin of the lines G1, G2. As a consequence if f is a π-periodic function we have f(ϕ2 − ϕ1) dG1 dG2 = f(α2 − α1)|sin(α2 − α1)| dα1 dα2 dP. (13) 3.2 Integrals … view at source ↗

**Figure 2.** Figure 2: Visual angle of a convex set. Proof. For a given point P in the plane there are angles α(P), β(P) such that the pairs of lines G1, G2 through P that intersect the convex set K are those satisfying α(P) ≤ αi ≤ β(P), where αi = α(Gi). When P ∈ K we have α(P) = 0 and β(P) = π. We need to integrate the function f(α2 − α1)|sin(α2 − α1)| over [α, β] 2 with α = α(P) and β = β(P). In order to perform this integral… view at source ↗

**Figure 3.** Figure 3: Angles ω1 and ω2. Proposition 5. Let f be an anti π-periodic continuous function on R such that f(x) = f(−x) and H a C 2 function on [0, π] with H00(x) = f(x)·sin(x), x ∈ [0, π], and H(0) = H0 (0) = 0. Then Z Gi∩K6=∅ f(ϕ2 − ϕ1) dG1 dG2 = 2(2H(π/2) − H(π))F + 2 Z P /∈K (2H(ω1) + 2H(ω2) − H(ω)) dP. (15) Proof. In section 3.1 we have seen that ϕ2 − ϕ1 = α2 − α1 + π where = (P, α1, α2) = 0, ±1. Then Z Gi∩K… view at source ↗

read the original abstract

In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.

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 reinterprets Crofton-Hurwitz-Masotti formulas for visual angle as integrals over pairs of lines and supplies simpler proofs.

read the letter

The central contribution is a uniform measure-theoretic reading of the classical integral formulas for the visual angle of a convex set. The authors treat the visual angle as an integral of suitable densities against the canonical motion-invariant measure on the space of pairs of lines, and they supply new proofs that they describe as simpler than the originals by Crofton, Hurwitz, and Masotti. That framing is the main novelty on offer. It organizes several known results under one geometric measure and avoids some of the case-by-case arguments that appear in the older literature. If the derivations in the body are as direct as the abstract suggests, the paper performs a modest but useful housekeeping task inside integral geometry. The approach stays within the standard toolkit of the field, so the technical risk looks low. The main limitation is that the abstract gives no explicit construction of the measure or the densities, which makes it impossible to verify invariance or to compare proof length directly. Without those details the claim of simplicity remains plausible but untested. The citation pattern is conventional and does not raise red flags. This is a narrow technical note aimed at readers who already work with these formulas and want to see them recast in the language of pair measures. It will not interest people outside the subfield. The work is coherent enough on its own terms to merit referee attention rather than an immediate desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper reinterprets classical integral formulas for the visual angle of a convex set (due to Crofton, Hurwitz, and Masotti) from the viewpoint of integral geometry. It claims these formulas admit an interpretation as integrals of suitable densities against the canonical motion-invariant measure on the space of pairs of lines, and provides new simpler proofs of the formulas.

Significance. If the reinterpretation and proofs hold, the work offers a unified integral-geometric perspective on these classical results, potentially clarifying the role of motion-invariant measures on line pairs and simplifying access to the formulas for applications in convex geometry. Explicit construction of the measure and verification of invariance would strengthen the contribution.

major comments (1)

The central reinterpretation step invokes a 'canonical measure' on the space of pairs of lines with the property that the visual angle equals an integral of a density against it, but no explicit construction, uniqueness argument, or verification of motion-invariance is supplied in the abstract or visible text. This assumption is load-bearing for both the interpretation and the claimed simpler proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment. We address the major point below and agree that an explicit treatment of the measure will improve the manuscript.

read point-by-point responses

Referee: The central reinterpretation step invokes a 'canonical measure' on the space of pairs of lines with the property that the visual angle equals an integral of a density against it, but no explicit construction, uniqueness argument, or verification of motion-invariance is supplied in the abstract or visible text. This assumption is load-bearing for both the interpretation and the claimed simpler proofs.

Authors: The manuscript works throughout with the standard motion-invariant measure on the space of lines (and its natural extension to pairs), whose construction and invariance properties are classical in integral geometry. Nevertheless, we accept that an explicit construction, a short uniqueness argument (up to normalization), and a direct verification of motion-invariance are not spelled out in the visible text. In the revised version we will insert a short preliminary subsection that supplies precisely these elements, thereby making the reinterpretation and the subsequent proofs fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The visible abstract and context describe reinterpretation of classical Crofton/Hurwitz/Masotti formulas via integrals against a canonical motion-invariant measure on pairs of lines, plus new proofs. No equations, parameter fits, self-citations, uniqueness theorems, or ansatzes appear in the supplied text. Without explicit derivations or load-bearing steps that reduce by construction to inputs, the derivation chain cannot be shown to collapse. This matches the default expectation that most papers are non-circular when no reduction is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, invented entities or ad-hoc axioms; the work appears to rest on standard motion-invariant measures already present in integral geometry.

pith-pipeline@v0.9.0 · 5589 in / 970 out tokens · 33645 ms · 2026-05-25T16:29:57.352350+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

M. W. Crofton. On the theory of local probability. Phil. Trans. R. Soc. Lond., 158:181–199, 1868

work page
[2]

Cuf´ ı, E

J. Cuf´ ı, E. Gallego, and A. Revent´ os. On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set. Mathematika, 65:874–896, 2019. 17

work page 2019
[3]

I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products . Academic Press, New York-London-Toronto, Ont., 1980

work page 1980
[4]

A. Hurwitz. Sur quelques applications geometriques des s´ eries de Fourier. Annales scientiﬁques de l’ ´E.N.S., 3` eme s´ erie, 19:357–408, dec 1902

work page 1902
[5]

G. Masotti. La Geometria Integrale. Rend. Sem. Mat. Fis. Milano , 25:164–231 (1955), 1953–54

work page 1955
[6]

L. A. Santal´ o.Integral geometry and geometric probability. Cambridge Univer- sity Press, Cambridge, second edition, 2004. 18

work page 2004

[1] [1]

M. W. Crofton. On the theory of local probability. Phil. Trans. R. Soc. Lond., 158:181–199, 1868

work page

[2] [2]

Cuf´ ı, E

J. Cuf´ ı, E. Gallego, and A. Revent´ os. On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set. Mathematika, 65:874–896, 2019. 17

work page 2019

[3] [3]

I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products . Academic Press, New York-London-Toronto, Ont., 1980

work page 1980

[4] [4]

A. Hurwitz. Sur quelques applications geometriques des s´ eries de Fourier. Annales scientiﬁques de l’ ´E.N.S., 3` eme s´ erie, 19:357–408, dec 1902

work page 1902

[5] [5]

G. Masotti. La Geometria Integrale. Rend. Sem. Mat. Fis. Milano , 25:164–231 (1955), 1953–54

work page 1955

[6] [6]

L. A. Santal´ o.Integral geometry and geometric probability. Cambridge Univer- sity Press, Cambridge, second edition, 2004. 18

work page 2004