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REVIEW 2 major objections 4 minor 28 references

Supertoroid fitting of objects with holes for robotic grasping and scene generation

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Pith's one-line read A meridian radial distance allows supertoroids to be fitted to partial point clouds of objects with a single hole.

desk verdict Novel meridian radial distance for supertoroid fitting, but Eq. (12) has an inverted coefficient and exponent that invalidates the central derivation as written, and the empirical support is thin. read the letter →

arxiv 2412.04174 v3 pith:XZPZU6LL submitted 2024-12-05 eess.IV cs.RO

classification eess.IVcs.RO
keywords supertoroidsuperquadricspointcloudfittinggeometricmodelingroboticgraspingmeridianradialdistancegenus-oneobjectspartialview
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

This paper argues that supertoroids—rounded, degree-four surfaces that generalize toruses with separate squareness exponents for the cross-section and the loop—can serve as a fitting primitive for objects with one hole. It derives a meridian radial distance, Eq. (19), whose minimization recovers the six shape parameters and six pose parameters from a single-view, noisy point cloud. Previously, superquadrics could model a wide range of symmetric objects but not genus-one shapes such as handles, rings, or gears; the paper closes that gap by defining a distance that is both simple and geometric. If the claim holds, a robot could estimate the pose and shape of holed objects quickly, skip point-cloud completion, and use the returned parameters to plan grasps.

What carries the argument

The central object is the supertoroid, a supersurface of degree four obtained as the spherical product of a superellipse loop in the x-y plane and a superellipse cross-section in each plane containing the z-axis, parameterized by two shape exponents $\epsilon_1$ and $\epsilon_2$ and three sizes $a_1$, $a_2$, $a_3$ plus the hole-size parameter $a_4$. The carrying mechanism is the meridian radial distance of Eq. (19), built from the mean superellipse (the curve traced by the centers of the cross-sections) and the local cross-section superellipse; the two ratios $\beta_1$ and $\beta_2$ are each obtained from the inside-outside functions of those superellipses, so the distance has a closed form and a clear geometric meaning.

What would settle it

On a synthetic supertoroid with known parameters, generate a partial cloud that excludes the inner wall of the hole, add noise, and run the full two-stage fit. If the recovered hole-axis direction or fitted $a_4$ departs significantly from the ground truth while the meridian radial distance at the fitted parameters is smaller than at the true parameters, the objective is not a faithful fit for concealed-hole views.

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Extended reading notes

Core claim

The central claim is that the distance from a point to the supertoroid surface can be measured in the plane of the meridian through the point, using two scalar ratios. The projection of the point onto the x-y plane meets the mean superellipse at a vector $R_\pi$, and the major ratio $\beta_2$ locates that intersection; the vertical component then lies on a cross-section superellipse, whose inside-outside function fixes the minor ratio $\beta_1$. The resulting distance is $d_s = |(1-\beta_1)(p - \beta_2 p_\pi)|$, Eq. (19), and a least-squares fit of this distance yields the supertoroid parameters. Together with the derived tangent, normal, and curvature formulas, this gives a complete geometric model for grasp heuristics on holed objects.

Load-bearing premise

The two-stage pipeline assumes the first stage—fitting a mean superellipse to the point-cloud projection from three starting axis directions—recovers the hole direction; for narrow objects the authors had to adjust the initial guess for $a_3$ manually, and no failure analysis is given for views that hide the hole.

Editorial extensions

If this is right

  • Objects with a single hole—mug handles, rings, gears, pot handles—can be represented by one continuous parametric model rather than by a plural set of spheres or cylinders or by mesh completion.
  • Because the distance is a closed-form function of the 12 parameters, the fit runs in about 1–3 seconds on a CPU with downsampling, fast enough for a perception loop.
  • The same meridian radial distance covers superellipsoids in the limit $a_4 \to 0$, so a single fitting function spans genus-zero and genus-one symmetric shapes.
  • The recovered supertoroid parameters directly supply the object frame and the local differential geometry, allowing curvature-aware grasp selection from the same fit.

Reading between the lines

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

  • A testable extension is to apply the same two-ratio construction to other genus-one surfaces of revolution with non-elliptical profiles, such as square or asymmetric handles, by replacing the cross-section superellipse with a different closed generatrix.
  • The paper leaves open what happens when the single view does not reveal the hole's direction; a robust version would need a prior on the hole axis or a second view, which the present stage-1 alignment from three axis-aligned starts may not always provide.
  • If the meridian radial distance is an unbiased objective, it could also serve as a differentiable loss for learning-based shape parsers, replacing the signed-distance fields used in recent primitive abstraction pipelines.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper addresses geometric modeling of objects with holes by fitting supertoroids to point clouds. It derives a distance metric called the meridian radial distance, expressed in Eq. (19), whose minimization yields the intrinsic and extrinsic supertoroid parameters. The pipeline has two stages: first, the hole axis is estimated by fitting a mean superellipse to the projection of the point cloud; second, the full supertoroid is fit by minimizing the meridian radial distance. The paper also derives tangent vectors, normals, and normal curvatures for the supertoroid as a basis for grasp heuristics, and reports experiments on synthetic and real point clouds of holed objects such as gears, pots, bearings, and rolls. The central claimed contribution is a simple, fast, geometrically meaningful distance that extends superquadric fitting to genus-one objects, with code made publicly available.

Significance. If the derivation is corrected, the paper contributes a useful extension of superquadric fitting to objects with a single hole, which is relevant for robotic grasping and scene generation. The algebraic structure of the distance, the closed-form expressions for the differential geometry, and the public implementation are valuable assets. The claim that a single-view, partial, noisy point cloud can be fit without cloud completion is practically significant. However, the force of these contributions is currently weakened by an algebraic error in the core derivation and by an empirical evaluation that is largely qualitative and lacks baselines or quantitative error metrics.

major comments (2)
  1. [Section III, Eq. (12)] The relation in Eq. (12) is algebraically incorrect and is load-bearing for the central claim. From the parametric mean superellipse in Eq. (6), a point on it satisfies y/x = (a2/a1) tan^{ε2} ω_s. Requiring this point to lie on the ray through pπ gives tan ωπ = (a2/a1) tan^{ε2} ω_s, hence tan^{ε2} ω_s = (a1/a2) tan ωπ and tan ω_s = ((a1/a2) tan ωπ)^{1/ε2}. The paper instead prints tan^{1/ε2} ω_s = (a2/a1) tan ωπ, inverting both the coefficient and the exponent. Since Eq. (15) uses ω_s to define the local cross-section coordinates, and Eqs. (16)-(19) depend on those coordinates, the vector ps and distance ds in Eq. (19) do not, as written, correspond to a radial line from the cross-section center through p. The authors should correct Eq. (12), re-derive the subsequent expressions, and confirm explicitly that the released code implements the corrected relation; the current text does not establish the claimed meridian radial distance.
  2. [Section V-B, V-C, Table II] The empirical support for the claim of fast, accurate, and robust fitting is not yet convincing. The G/D/B counts in Table II are based on subjective qualitative criteria with no numerical thresholds; no quantitative fitting-error metric is reported, such as mean or median distance from the cloud to the fitted surface, chamfer distance, or parameter error against ground truth. No comparison is made to any baseline, such as a standard torus fit, a superquadric fit, or a signed-distance objective. The procedure also depends on several choices whose influence is not analyzed: the downsample sizes (150 and 1000 points), the post-hoc term added to maximize a4, and the manual adjustment of the a3 initial guess for narrow objects reported in Section V-C. Finally, the claim that stage 1 recovers the hole axis from a single partial view is not quantified; Section V-B states that with the correct alignment the fitting is fast and accurate, but the paper does not report how often or under which viewing conditions stage 1 produces that correct alignment, nor what happens when it fails.
minor comments (4)
  1. [Throughout] The manuscript contains several typographical and grammatical errors that should be fixed, including 'Polythechnic' in the affiliations, 'algorithsm' in Section I, 'this handlers rarely' in Section I, and 'usign' in Section IV.
  2. [Section II and III] The notation Fm and Fc is used in Eqs. (10) and (16) before it is defined; please define these as the left-hand sides of Eqs. (5) and (13), respectively, and remove the stray comma in Eq. (10).
  3. [Section V-A] Reference [16] is described as the method used to generate point clouds for supertoroids, but the cited paper addresses superellipse sampling; please clarify how the sampling is adapted to supertoroids, especially for partial views.
  4. [Section IV, Table I] The shorthand cω, sω, cη, sη in Table I and Eqs. (22)-(26) should be explicitly defined at first use, since the text otherwise uses the full trigonometric names.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the meridian radial distance is derived from the supertoroid's parametric and implicit equations, and the fitting optimizes genuine free parameters against point-cloud data.

full rationale

The derivation chain is self-contained. The supertoroid surface is defined in Eq. (2) and its implicit equation in Eq. (3); the mean superellipse (Eqs. (5)-(6)) and its radius (Eq. (7)) are direct consequences of that definition, not fitted quantities. The fitting distance is constructed geometrically: a point p is projected to the x-y plane, the major ratio beta2 is solved from the mean-superellipse inside-outside function (Eq. (10)), and the minor ratio beta1 is solved from the cross-section superellipse inside-outside function (Eq. (16)); Eq. (19) then combines these solved ratios. No parameter is fitted and then renamed a prediction; the only fitting is the least-squares minimization of the distance over the twelve intended shape and pose parameters in Section V-B. The one self-citation ([22]) appears in the introduction as a statement that the work extends prior grasping work; it is not used to justify the distance formula or the fitting result, so it is not load-bearing. Section V-B's caveats about initial guesses and the a3 adjustment are limitations of the two-stage optimizer, not circular inputs to the claimed derivation. A mathematical typo or algebraic slip in Eq. (12), if present, would be a correctness issue, not a circularity, since Eq. (12) is not assumed from the data or from a self-citation. Hence score 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central contribution is a derived distance function, not a new entity or fitted constant; the algorithm does introduce hand-tuned hyperparameters and domain assumptions about object symmetry.

free parameters (3)
  • downsample sizes = 150 (stage 1), 1000 (stage 2)
    Chosen by hand as a trade-off between speed and smoothness of cost; not justified by analysis.
  • a4 regularization weight = unspecified
    A term is added to maximize a4 during stage 2 for partial clouds; the weight is not given, making the objective not fully specified.
  • a3 initial guess adjustment for narrow objects = manual
    For 'roll' and 'long roll' the initial guess for a3 had to be adjusted; the adjustment rule is not formalized.
assumptions (3)
  • standard math Supertoroid parametric and implicit equations (Eq. 2, 3) as defined in Barr [23].
    The entire fitting is built on this surface model; it is accepted from prior literature.
  • domain assumption Objects of interest have a single hole and are symmetric enough to be represented by a supertoroid with a4 > 1.
    The method targets genus-one objects; a4 must exceed 1 for the hole to exist (stated in Section IV).
  • domain assumption The projection of each cloud point onto the x-y plane intersects the mean superellipse, allowing β2 to be computed via the inside-outside function.
    Used in Eq. (9)-(10); fails if the projected point lies at the origin or outside the mean superellipse's convex domain for general points.

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Cite this review

Pith. "Pith review of Supertoroid fitting of objects with holes for robotic grasping and scene generation." pith.science (2026). https://pith.science/paper/XZPZU6LL

@misc{pith2026241204174,
  author       = {Pith},
  title        = {Pith review of: Supertoroid fitting of objects with holes for robotic grasping and scene generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZPZU6LL}},
  note         = {Machine review of arXiv:2412.04174}
}
read the original abstract

One of the strategies to detect the pose and shape of unknown objects is their geometric modeling, consisting on fitting known geometric entities. Classical geometric modeling fits simple shapes such as spheres or cylinders, but often those don't cover the variety of shapes that can be encountered. For those situations, one solution is the use of superquadrics, which can adapt to a wider variety of shapes. One of the limitations of superquadrics is that they cannot model objects with holes, such as those with handles. This work aims to fit supersurfaces of degree four, in particular supertoroids, to objects with a single hole. Following the results of superquadrics, simple expressions for the major and minor radial distances are derived, which lead to the fitting of the intrinsic and extrinsic parameters of the supertoroid. The differential geometry of the surface is also studied as a function of these parameters. The result is a supergeometric modeling that can be used for symmetric objects with and without holes with a simple distance function for the fitting. The proposed algorithm expands considerably the amount of shapes that can be targeted for geometric modeling.

Figures

Figures reproduced from arXiv: 2412.04174 by the authors.

Figure 2
Figure 2. Different supertoroid shapes obtained with same [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. The angle ω on the x−y plane and the angle η for each cross-section. As in the case of the superellipsoid, the real positive exponent ϵ2 controls the shape in the canonical x−y plane and the real, positive exponent ϵ1 controls the shape in the perpen￾dicular planes containing the z axis. The coefficients ai control the dimensions in the x, y, z directions. We collect these intrinsic parameters in a vector vi = (a1, … view at source ↗
Figure 3
Figure 3. The mean superellipse can be seen as the mean value of the section [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: The vector ps of the closest point on the surface, according to the major and minor radial distance coefficients. The vertical projection plane is located at an angle ωπ from the canonical x-axis, tan ωπ = py px . (8) Considering the mean superellipse, we have the valu…
Figure 4
Figure 4. Figure 4: The vector to a point p is the sum of vector Rπ (intersection of the mean superellipse with the projection of p on plane x − y) and vector pR We define two ratios, the minor ratio β1 and the major ratio β2. The major ratio β2 is used to locate the intersecting point on…
Figure 6
Figure 6. Figure 6: The coordinate curves along parameters ω and η. The tangents (non-unit) along those curves are tω =    −a1ϵ2(a4 + cosϵ1 η) sin ω cosϵ2−1 ω a2ϵ2(a4 + cosϵ1 η) cos ω sinϵ2−1 ω 0    , (20) tη =    −a1ϵ1 sin η cos η ϵ1−1 cos ω ϵ2 −a2ϵ1 sin η cos η ϵ1−1 sin ω ϵ2 a3…
Figure 8
Figure 8. Figure 8: Tangent lines along coordinate curves (purple), and lines normal to [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 7
Figure 7. Figure 7: The tangent along the coordinate curves αω and αη shows similar behavior at the ω = 0 and η = 0 points, depending on the values of ϵ2 and ϵ1. Shown are the ϵ2 > 2 (cusp), ϵ2 = 2 (singular point), and ϵ2 < 2 B. Normal Vectors and Curvatures The tangents along the coordi…
Figure 9
Figure 9. Figure 9: A generated partial, noisy point cloud for a supertoroid (left) and a [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]
Figure 11
Figure 11. Figure 11: Result of the first optimization stage, showing the downsampled [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
Figure 14
Figure 14. Figure 14: Results of the point clouds of taller objects. The point clouds are [PITH_FULL_IMAGE:figures/full_fig_p007_14.png]
Figure 13
Figure 13. Figure 13: Result of three additional point cloud fittings, the supertoroid is [PITH_FULL_IMAGE:figures/full_fig_p007_13.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.