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arxiv: 1906.08358 · v1 · pith:HYJ4HFR7new · submitted 2019-06-19 · 🧮 math.NA · cs.NA

Transfinite mean value interpolation over polygons

Michael S. Floater , Francesco Patrizi This is my paper

Pith reviewed 2026-05-25 19:51 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords mean value interpolationtransfinite interpolationpolygonsboundary datacomputer graphicsnumerical analysis
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The pith

Transfinite mean value interpolation reproduces any continuous boundary data on arbitrary polygons.

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

The paper supplies the missing proof that the transfinite extension of mean value interpolation always matches continuous data prescribed on the boundary of any polygon. A reader would care because the method is already used to fit smooth functions to boundary data in graphics and modeling, yet lacked confirmation that it works without exception for non-linear data or irregular shapes. The proof establishes that the interpolant agrees with the given data at every boundary point and remains continuous up to the boundary. This removes the prior uncertainty about whether the construction succeeds in all cases.

Core claim

The transfinite mean value interpolant, obtained by integrating the mean value coordinates against arbitrary continuous boundary data, reproduces that data exactly on the boundary of any polygon.

What carries the argument

The integral expression for the transfinite mean value interpolant that uses mean value coordinates over the polygon.

If this is right

  • The method produces a continuous function inside the polygon that exactly matches any given continuous boundary values.
  • The construction works for polygons of completely arbitrary shape without additional restrictions.
  • Mean value interpolation can be applied directly to continuous rather than only piecewise-linear boundary data.

Where Pith is reading between the lines

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

  • Similar integral proofs might be attempted for other generalized barycentric coordinates on polygons.
  • The result could support numerical implementations that assume exact boundary reproduction without case-by-case verification.

Load-bearing premise

The mean value coordinates stay well-defined and the integral expression for the interpolant stays continuous up to the boundary for every polygon and every continuous boundary function.

What would settle it

A concrete counterexample consisting of one polygon and one continuous boundary function where the interpolant differs from the prescribed data at some boundary point.

Figures

Figures reproduced from arXiv: 1906.08358 by Francesco Patrizi, Michael S. Floater.

Figure 1
Figure 1. Figure 1: Interpolation at an edge point y∗. where F = {[v1, y1], [y2, v2]} ∪ (E \ [v1, v2]), and it follows that |g(x) − f(y∗)| ≤ γ(x)/φ(x), where γ(x) = ǫIe0 (x) + 2M X e∈F Ie(x), and M := sup y∈∂Ω |f(y)|. (5) Similar to γ(x), we can express φ(x) as φ(x) = τe0 (x)Ie0 (x) +X e∈F τe(x)Ie(x). For x close enough to y∗, τe0 (x) = 1, and then γ(x) φ(x) = ǫ + 2M P e∈F Ie(x)/Ie0 (x) 1 + P e∈F τe(x)Ie(x)/Ie0 (x) . As x → y… view at source ↗
Figure 2
Figure 2. Figure 2: Interpolation at a convex vertex v. v1 x y2 v2 v y1 v1 y2 v2 y1 x v v1 y2 v2 y1 v x [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Interpolation at a concave vertex v. Proof. Similar to (4), from the form of (1), |g(x) − f(v)| ≤ X e∈E Ie(x; | ˜f|) . φ(x), where ˜f(y) := f(y) − f(v). Let v1 and v2 be the two neighbouring vertices of v with v1, v, v2 ordered anti￾clockwise w.r.t. ∂Ω as in Figures 2 and 3. Let ǫ > 0. By the continuity of f, there is some δ, where 0 < δ < min{kv1 − vk, kv2 − vk}, such that if y is in [v1, v] or [v, v2] an… view at source ↗
read the original abstract

Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications in computer graphics and curve and surface modelling. The method generalizes to transfinite interpolation, i.e., to any continuous data on the boundary but a mathematical proof that interpolation always holds has so far been missing. The purpose of this note is to complete this gap in the theory.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript supplies the missing proof that transfinite mean-value interpolation over an arbitrary polygon reproduces any continuous function prescribed on the boundary, i.e., that the normalized mean-value kernel integrates to the Dirac measure at every boundary point.

Significance. If the proof is correct, the result removes the last theoretical obstacle to using mean-value coordinates for transfinite interpolation on polygons, directly supporting applications in computer graphics and geometric modeling that require exact reproduction of arbitrary continuous boundary data.

major comments (1)
  1. [proof of boundary reproduction (main theorem)] The central limit argument establishing lim_{x→z, x∈int(P)} ∫_{∂P} f(y) w(x,y) ds(y) = f(z) for continuous f and z∈∂P must be checked for uniform integrability or an integrable majorant independent of approach direction. For polygons with interior angles >π the kernel w can change sign and become unbounded near vertices; if the proof invokes only the interior mean-value property or weak convergence without an explicit domination estimate that works uniformly for all approach paths, the passage to the limit is not secured.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a potential issue in the justification of the boundary reproduction limit. We address the major comment below.

read point-by-point responses

  1. Referee: [proof of boundary reproduction (main theorem)] The central limit argument establishing lim_{x→z, x∈int(P)} ∫_{∂P} f(y) w(x,y) ds(y) = f(z) for continuous f and z∈∂P must be checked for uniform integrability or an integrable majorant independent of approach direction. For polygons with interior angles >π the kernel w can change sign and become unbounded near vertices; if the proof invokes only the interior mean-value property or weak convergence without an explicit domination estimate that works uniformly for all approach paths, the passage to the limit is not secured.

    Authors: The referee is correct that the passage to the limit requires care when the kernel changes sign near reflex vertices. The current proof establishes weak convergence of the normalized kernel to the Dirac measure via the interior mean-value property and a density argument, but does not supply an explicit integrable majorant that is uniform in the direction of approach. We will revise the manuscript by adding a new lemma that constructs such a majorant (using the local integrability of the singularity of w together with the boundedness of f on the compact boundary) and invokes the dominated convergence theorem to justify the limit for all continuous f. This change will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

Independent analytic proof supplied; no reduction to inputs by construction

full rationale

The paper supplies a direct mathematical proof that the transfinite mean-value interpolant reproduces arbitrary continuous boundary data on polygons of arbitrary shape. No equations or steps are shown to reduce by definition to fitted quantities, self-citations, or prior ansatzes from the same authors; the abstract explicitly frames the work as filling a missing proof rather than re-deriving a known result. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5582 in / 868 out tokens · 33285 ms · 2026-05-25T19:51:35.175051+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    Dyken and M

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    M. S. Floater, K. Hormann, and G. K \'o s, A general construction of barycentric coordinates over convex polygons, Adv. Comput. Math. 24 (2006), 311--331

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    M. S. Floater, G. Kos, and M. Reimers, Mean value coordinates in 3 D , Comput. Aided Geom. Design 22 (2005), 623--631

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    Hormann and M

    K. Hormann and M. S. Floater, Mean value coordinates for arbitrary planar polygons, ACM Trans. on Graph. 25 (2006), 1424--1441

    work page 2006

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    T. Ju, S. Schaefer, and J. Warren, Mean value coordinates for closed triangular meshes, ACM Trans. on Graph. 24 (2005), 561--566

    work page 2005