How many points in a point cloud is sufficient for accurate estimation of the curvature

R. Mirzaie

REVIEW 5 major objections 5 minor 14 references

How many points in a point cloud is sufficient for accurate estimation of the curvature

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a curve or surface of known length or area, the paper derives a sample-size bound that guarantees, with probability at least $p$, that every point has sampled neighbors close enough to estimate curvature — one circle radius for…

desk verdict The paper asks a real question about sample sizes for curvature estimation, but the main theorems are built on independence and geometry errors, and the surface estimator fails even on the unit sphere. read the letter →

arxiv 2506.06779 v2 pith:6HAWOGRY submitted 2025-06-07 math.DG

R. Mirzaie This is my paper · ORCID

classification math.DG MSC 53C2153C2362D05

keywords curvatureestimationpointcloudGaussianprincipalosculatingcirclemanifoldlearningsamplesizerandomsampling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how many random points must be sampled from a curve or surface before curvature can be estimated reliably at every point. It answers with explicit formulas: one for plane curves, based on guaranteeing two sample points within distance $\epsilon$ of each point that bracket it, and one for surfaces, based on guaranteeing four sample points inside the two principal cones at every point. If the formulas hold, the Gaussian curvature at a point is approximated by the product of the reciprocals of the two circumcircle radii determined by the point and each principal-direction pair. The practical upshot is a stopping rule: keep sampling until the point-cloud size exceeds the bound, and the desired confidence $p$ is reached. The paper itself notes in Remark 3.1 that no upper bound on the estimation error is possible for finite samples.

What carries the argument

The central object is the principal $(\theta,\epsilon)$-cone at a point: the exponential image of a standard sector of angle $\theta$ and radius $\epsilon$ centered on a principal direction $e_i$ in the tangent plane. The load-bearing identity is the relation between the number $m$ of sampled points and the probability that the four cones $C$, $-C$, $D$, $-D$ at every point in a covering each contain at least one sample point: $P \ge (1 - 4(1-\theta\epsilon^2/(2s))^m)^n$ with $n \le 12s/(\pi\epsilon^2)$. Inverting this inequality gives the sample-size formula in Theorem 2.14. For curves, the corresponding machinery is the $\epsilon$-triangle estimation: the radius $r(x,x',x'')$ of the circle through three points, computed by Heron's formula, with $1/r$ as the curvature estimate.

What would settle it

Take a cylinder of radius $R$ and a point $b$; sample $m$ points uniformly, and using the raw formula from Remark 2.13, pick two sampled points inside a narrow principal cone along the generator direction and compute the circumcircle radius $r$. Since the three points are nearly collinear, $1/r$ will not be near the principal curvature $0$ in that direction, and the discrepancy persists no matter how large $m$ is, showing that the claimed identification fails on this surface.

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Extended reading notes

Core claim

The central discovery is a probabilistic sample-complexity guarantee for curvature estimation from i.i.d. uniform point clouds. For a plane curve of length $l$, Theorem 2.7 shows that if $m > \frac{1}{2}\left(1 + \sqrt{1 + 8\frac{\log(1-p)-\log l+\log 2\epsilon}{\log(1-\epsilon^2/l^2)}}\right)$, then with probability at least $p$, every point $x$ of the curve has two sampled points $x'$, $x''$ with $[x',x,x'']$ and $\max\{d(x,x'), d(x,x'')\} < \epsilon$, so the reciprocal of the radius of the circle through the three points is an $\epsilon$-triangle estimation of $\kappa(x)$. For a surface of area $s$, Theorems 2.12 and 2.14 show that if $m \ge \frac{\log\big((1 - p^{\pi\epsilon^2/(12s)})/4\big)}{\log(1 - \theta\epsilon^2/(2s))}$, then with probability at least $p$, every point $b$ has points $x_{11}, x_{12}, x_{21}, x_{22}$ in the four principal $(\theta,\epsilon)$-cones at $b$, and the Gaussian curvature satisfies $\kappa(b) \simeq (r(x_{11},b,x_{12}))^{-1}(r(x_{21},b,x_{22}))^{-1}$. The proof is a covering argument: the surface is covered by geodesic balls whose number is bounded by $12s/(\pi\epsilon^2)$, and the probability that any one of the four cones at any ball center is empty is bounded by $4(1-\theta\epsilon^2/(2s))^m$, which is converted into the sample-size formula.

Load-bearing premise

The surface estimator rests on the unproved identification in Remark 2.13 that the circle through $b$ and two sampled points inside a principal cone has radius close to $1/\kappa_1(b)$, which can fail because the circumcircle depends on the full embedding, not just on the principal curvature in that direction.

Editorial extensions

If this is right

For curves, the formula in Theorem 2.7 gives a stopping criterion: once the point cloud contains more than the bound, every point is guaranteed to have two bracketing sample points within $\epsilon$, so the osculating-circle estimate is available everywhere with probability at least $p$.
For surfaces, the formula in Theorem 2.14 provides the four principal-cone points needed to estimate Gaussian curvature at every point, and the same four points yield the mean curvature as $(\kappa_1+\kappa_2)/2$.
Because the bounds depend only on the total length or area of the object, the same $m$ works simultaneously for all points of the curve or surface.
When $l$ or $s$ is unknown, the algorithms can plug in any established length or area estimator for the point cloud and then add points until $m$ exceeds the threshold.
The confidence $p$ can be chosen arbitrarily close to 1, with the formulas quantifying how much larger the sample size must become.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's genuinely new contribution is the probabilistic counting argument, not the curvature formula itself; the same four-cone covering argument would supply sample-size bounds for any local geometric query that needs a fixed number of points inside prescribed cones.
A provable replacement for the missing geometric step in Remark 2.13, giving an explicit bound on the angle between the broken geodesic and the principal normal section, would upgrade Theorem 2.14 from an existence guarantee for nearby points to an accuracy guarantee for the curvature estimate.
Because Remark 3.1 rules out universal error bounds for finite samples, the honest use of these formulas is as a confidence statement about the presence of nearby points, not about the size of the curvature error; future work could combine the sample-size bound with a surface regularity assumption to obtain a genuine error bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this one has a real problem, and the problem isn't the topic. The paper asks how many i.i.d. uniform points on a curve or surface are enough to guarantee, with probability p, that every point has nearby sample points to support a curvature estimate. That is a sensible question, and the attempted answer—counting arguments plus covering numbers—is a fair approach. The sample-size formulas in Theorem 2.7 and 2.14 look new, and Remark 3.1 is candid about the absence of error bounds. So there is something here worth acknowledging.

That said, the load-bearing arguments don't hold. Lemma 2.4 treats the C(m,2) pairs as independent to write P(Ei) = (1 - (li/l)^2)^{C(m,2)}. They aren't independent. The number of points in γi is binomial, so the chance that γi contains 0 or 1 point is (1-α)^m + mα(1-α)^{m-1}. That is a different formula, and it changes the sample size by a factor on the order of √(log n). The theorem understates what you need.

Theorem 2.7 has a simpler geometric slip. After finding xα and xβ in A1, on the 'bad' order case it introduces a point z with dγ(z,x)=ϵ and asks for two points in Bγ(z,ϵ/2). Those points satisfy dγ(x,w) ≤ 1.5ϵ, not < ϵ, so the promised ϵ-neighborhood guarantee is not established. Theorem 2.12's cone area is off by four: a standard cone of radius ϵ/2 and angle θ has area θϵ²/8, not θϵ²/2.

The surface result is worse. Remark 2.13 asserts that a broken geodesic through points in a principal (θ,ϵ)-cone approximates a principal curve, and that the circumcircle radius through such points gives 1/κ_i. No proof is given, and the claim is false generically. On the unit sphere at b=(0,0,1), you can pick ordered points in the two principal cones such that one circumcircle is a great circle (reciprocal radius 1) and the other lies in a plane at distance 1/√2 from the origin (reciprocal radius √2). The product tends √2, not the Gaussian curvature 1. So Theorem 2.14's estimator is wrong under its own hypotheses, independent of the probability lemmas.

I also can't check the numerics: no code or data, and the tables use p=0.1, which doesn't illustrate a 'close to one' confidence.

Who is this for? Someone building sample-size formulas for geometric estimation might read the curve part to see the covering strategy, but they would need to fix the concentration argument. The surface part should not be used. As an editor, I would desk reject this version; the main theorem is demonstrably false, and the flaws are elementary enough that referee time is better spent elsewhere.

Referee Report

5 major / 5 minor

Summary. The paper proposes an estimator for the curvature of plane curves and surfaces from i.i.d. uniform point clouds. For curves, Theorem 2.7 gives a sample-size threshold m such that, with probability at least p, every point x has two sample points x', x'' bracketing x on the curve within distance ε, so that the reciprocal of the circumradius of triangle x'x''x estimates κ(x). For surfaces, Theorem 2.14 claims that a similar threshold m ensures that every point b has four sample points that are (θ, ε)-principal ordered at b, and that the Gaussian curvature κ(b) is approximated by the product of the reciprocals of the two circumradii determined by b and those point pairs. Algorithms and numerical tables are included.

Significance. If the results were correct, the paper would provide explicit, parameter-free sample-size formulas for a practical curvature-estimation problem, with a probabilistic interpretation and no fitted parameters. The elementary counting approach is attractive, and the numerical tables suggest the author has working code. However, the central probabilistic and geometric arguments contain load-bearing errors: the independence assumption in Lemma 2.4 is false, the distance assertion in Theorem 2.7 is false, and the surface curvature identification in Remark 2.13 is unsupported and, in fact, false on a sphere. These errors invalidate the claimed guarantees, so the manuscript cannot currently be recommended for publication.

major comments (5)

[Lemma 2.4 and Remark 2.5] The proof treats the C(m,2) pairs of sample points as independent, writing P(E_i) = (1 - (l_i/l)^2)^{C(m,2)}. This is not justified: the events 'the pair {y_j,y_k} lies in γ_i' are dependent because pairs share sample points. The exact probability that γ_i contains fewer than two points is (1-α)^m + mα(1-α)^{m-1}, with α = l_i/l, which for small α is substantially larger than the expression used. Consequently the derived lower bound on m is an underestimate, and the sample-size formulas in Theorem 2.7 and Algorithm 3.1 are not valid as stated.
[Theorem 2.7] The proof contains a false distance assertion. After finding two sample points in B_γ(z, ε/2) with d_γ(z,x) = ε, the text states: 'It is clear that x''', x'''' ∈ B_γ(x, ε)'. This is false: a point in B_γ(z, ε/2) can have γ-distance from x as large as 3ε/2. Thus the constructed points need not satisfy max{d(x,x'), d(x,x'')} < ε, and the claimed ε-triangle estimation of κ(x) is not established.
[Remark 2.13 and Theorem 2.14] The load-bearing geometric identification is unproved and false. The paper asserts that the circumcircle through b and two points well ordered in a principal (θ, ε)-cone has radius close to 1/κ_1(b), and similarly for the second principal direction. No proof is given, and the claim fails on the unit sphere: take b=(0,0,1), e1=(1,0,0), e2=(0,1,0), x11=(δ,0,√(1-δ²)), x12=(-δ, δ², √(1-δ²-δ⁴)), x21=(0,δ,√(1-δ²)), x22=(0,-δ,√(1-δ²)). For small δ these points are well ordered in the principal cones, but the circle through x11, b, x12 lies in a plane at distance 1/√2 from the origin, so its reciprocal radius tends to √2, while the circle through x21, b, x22 is nearly a great circle with reciprocal radius tending to 1. The product tends to √2, not the Gaussian curvature 1. Thus Theorem 2.14 does not establish a curvature estimate even when the four ordered points exist.
[Theorem 2.12] The area estimate for the principal cones is too large by a factor of 4. A standard cone with total angle θ and radius ε/2 has area θ(ε/2)²/2 = θε²/8, not θε²/2 as written in the proof. This overestimates the probability that a sample point lies in each cone and changes the numerical constants in the sample-size formula. The theorem's conclusion is therefore not supported by the given computation.
[Remark 3.1 and Theorem 2.14] Even if the probabilistic estimates were repaired, Theorem 2.14 would not provide a quantitative curvature-estimation guarantee. The conclusion states κ(b) ≃ (r(x11,b,x12))^{-1}(r(x21,b,x22))^{-1} without any error bound, while Remark 3.1 explicitly concedes that for any finite sample set there exist submanifolds containing the sample with arbitrarily large curvature. The theorem at most controls the availability of nearby points; it does not control the accuracy of the proposed estimator.

minor comments (5)

[Throughout] There are numerous typos, including 'Gussian curvature', 'plan curve', 'Elementry differential geometry', and 'Accademic Press' in the references.
[Tables 1 and 2] The experiments set p = 0.1, which corresponds to a confidence of only 10%; since the text says p can be chosen arbitrarily close to one, the numerical illustration should use a standard high-confidence value such as p = 0.9 or 0.99, and should report the actual theoretical sample sizes used.
[Algorithm 3.2] The heuristic steps for finding the first and second principal directions (steps 7-8) are not derived from Theorem 2.14 and appear to rely on additional unstated assumptions about the geometry of the point cloud relative to the surface.
[Definition 2.9 and Remark 2.10] The term 'well ordered' is used to assert that points lie on a differentiable curve through x inside the cones, but the proof of Theorem 2.12 does not show that the sample points found in Ci, -Ci, Di, -Di actually lie on a common such curve, so the leap to 'well ordered' in Theorem 2.14 is not justified.
[Remark 2.8] The claim that the quantity under the square root is positive for all l ≥ ε is not clearly connected to the subsequent use of the formula; if the logarithmic expressions become undefined for some parameter choices, the algorithm should state restrictions on p and ε explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sample-size bounds are derived by elementary counting, and the curvature-identification step is an unproved geometric assumption rather than a self-referential reduction.

full rationale

The paper's central sample-size guarantees (Theorem 2.7 and Theorem 2.12) are derived from covering arguments, uniform i.i.d. sampling, and union-bound probability estimates; no parameter is fitted to data and no 'prediction' is obtained from the same quantity it is supposed to estimate. The curve-curvature step is the standard osculating-circle approximation, and the surface step rests on Remark 2.13, which asserts without proof that circumcircles through arbitrary points in principal cones approximate principal curvatures. That assertion is a mathematical soundness gap, not a circularity: the conclusion is not equivalent to the input by construction, and it is not justified by a self-citation or by a fitted parameter. There are no self-citations that carry the argument, no uniqueness theorem imported from the author's prior work, and no known result merely renamed. Thus the derivation chain is not circular, although Theorem 2.14's correctness depends on an unproved geometric identification.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claims rest on uniform sampling assumptions, known length or area, and an unproved identification of circumcircle radii with principal curvatures. The sample-size formulas depend on user-chosen parameters ϵ, p, and θ; no values are fitted to data. The main mathematical error is not an axiom but an incorrect probability computation in Lemma 2.4.

free parameters (3)

Resolution scale ϵ = 0.1 in experiments; otherwise user-chosen
Controls the radius of balls and cones and the triangle estimation accuracy. No data-driven selection is provided.
Target probability p = 0.1 in experiments
Confidence threshold in the sample-size formulas. The experiments use p=0.1, which is a low confidence level, but the theorems allow any p.
Cone angle θ = θ = ϵ/10 in the surface algorithm
Introduced in Theorem 2.12 and set arbitrarily in Algorithm 3.2. The choice is not derived from any accuracy requirement.

assumptions (5)

domain assumption Sample points are i.i.d. uniformly distributed with respect to the Riemannian volume measure induced from the ambient space.
Stated in Section 2; all probability bounds in the paper rely on this distributional assumption.
domain assumption Curves and surfaces are differentiable, have finite length or area, and have no boundary points.
Stated in Section 2; used for exponential maps, normal sections, and covering arguments.
ad hoc to paper The circumradius of a circle through a point b and two nearby points in a principal (θ,ϵ)-cone determines the corresponding principal curvature, and the product gives Gaussian curvature.
Remark 2.13 and Theorem 2.14 assert this without proof; this is the unsupported bridge from sample geometry to curvature.
standard math Standard facts about normal sections, principal curvatures, and exponential maps.
Remark 2.10 cites O'Neil [11]; these are standard inputs to the construction.
domain assumption For any compact surface and fixed radius, there is a cover by geodesic balls with multiplicity at most three.
Corollary 2.3(2) asserts this existence without proof; it is needed for the surface sample-size bound.

invented entities (1)

Principal (θ,r)-cone
purpose: Defines a neighborhood near a point in which sample points are considered close to a principal direction; used to select the four points for Gaussian curvature estimation.
Introduced in Definitions 2.9 through 2.11. It is a paper-specific construction with no independent validation outside the paper.

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Cite this review

Pith. "Pith review of How many points in a point cloud is sufficient for accurate estimation of the curvature." pith.science (2026). https://pith.science/paper/6HAWOGRY

@misc{pith2026250606779,
  author       = {Pith},
  title        = {Pith review of: How many points in a point cloud is sufficient for accurate estimation of the curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HAWOGRY}},
  note         = {Machine review of arXiv:2506.06779}
}

read the original abstract

We introduce an estimator for the curvature of curves and surfaces by using finite sample points drawn from sampling a probability distribution that has support on the curve or surface. First we give an algorithm for estimation of the curvature in a given point of a curve. Then, we extend it to estimate the Gaussian curvature of the surfaces. In the proposed algorithms, we use a relation between the number of selected points in the point cloud and the probability that a given point has a suffcient number of nearby points. This relation allows us to control the required number of points in the point cloud.

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

[1]

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Cazals, M

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Q. Charles, S. Hao, M. Kaichun, J. G. Leonidas, Deep learning on point sets for 3d classification and segmentation, Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition (CVPR), (2017). https://doi.org/10.48550/arXiv.1612.00593

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A. G. Ache, M. W. Warren, Ricci curvature and the manifold learning problem, Advances in Mathematics, 21 (2019), 14-66. Estimation of the curvature from random samples 15 https://doi.org/10.48550/arXiv.1410.3351

work page Pith review arXiv doi:10.48550/arxiv.1410.3351 2019

[2] [2]

P. J. Besl, R. Jain, Invariant surface characteristics for 3d object recognition in range images, Computer Vision, Graphics and Image Processing, 33 (1986), 33-80. https://doi.org/10.1016/0734-189X(86)90220-3

work page doi:10.1016/0734-189x(86)90220-3 1986

[3] [3]

Persistent homology detects curvature

P. Bubenik, M. Hull, D. Patel, B. Whittle, Persistent homology detects curva- ture, Inverse Problems, 36 025008(2), (2020). https://doi.org/10.48550/arXiv.1905.13196

work page Pith review arXiv doi:10.48550/arxiv.1905.13196 2020

[4] [4]

Cazals, M

F. Cazals, M. Pouget, Estimating differential quantities using polynomial fitting of osculating jets, Computer Aided Geometric Design, 22 (2005), 121-146. https://doi.org/10.1016/j.cagd.2004.09.004

work page doi:10.1016/j.cagd.2004.09.004 2005

[5] [5]

Charles, S

Q. Charles, S. Hao, M. Kaichun, J. G. Leonidas, Deep learning on point sets for 3d classification and segmentation, Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition (CVPR), (2017). https://doi.org/10.48550/arXiv.1612.00593

arXiv doi:10.48550/arxiv.1612.00593 2017

[6] [6]

J. Charton et al, Robust curvature estimation on surface triangulation, Iadis international conference computer graphics, visualization, computer vision and image processing, (2013), 43-45

work page 2013

[7] [7]

X. Chen, F. Schmitt, Intrinsic surface properties from surface triangulation, Eu- ropean Conference on Computer Vision, (1992), 739-743. https://doi.org/10.1007/3-540-55426-2-83

work page doi:10.1007/3-540-55426-2-83 1992

[8] [8]

PCPNET: Learning Local Shape Properties from Raw Point Clouds

P. Guerrero, Kleiman, M. Ovsjanikov, N. J. Mitra, Learning local shape proper- ties from raw point clouds, Computer Graphics Forum, 37 (2018), 75-85. https://doi.org/10.48550/arXiv.1710.04954

work page Pith review arXiv doi:10.48550/arxiv.1710.04954 2018