{"id":"a30fea1d-f452-43f7-a962-8f76cf98d706","arxiv_id":"2412.04174","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new meridian radial distance enables fast fitting of supertoroid shapes to partial, noisy point clouds of objects with a single hole.","lead":"This paper fits supertoroids, doughnut-like shapes with a hole, to partial point clouds of objects such as gears, handles, and rings, giving robots a compact 12-parameter model for pose and shape. The authors derive a fast distance formula for supertoroids and demonstrate it on real and synthetic single-view clouds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (12) appears to have the coefficient and exponent inverted; since Eqs. (15)-(19) build the fitting distance on it, the central claim that Eq. (19) is a valid meridian radial distance is not established as written.","rationale":"The paper's advertised contribution is a distance function that turns supertoroid fitting into a least-squares problem. That function is derived in Section III. The derivation is only valid if Eq. (12) is correct, because it determines the reference point R_pi on the mean superellipse and hence defines p_R and beta1. An independent re-derivation shows the printed relation is wrong in both the coefficient and the exponent. This is not a matter of empirical baselines or consensus; it is an internal algebraic inconsistency in the central derivation. I agree with the reader that the stage-1 initialization from partial single-view clouds is also under-analyzed, but that concern is secondary: even with a perfect initialization, the distance used in stage 2 must be correct. The available code makes the check easy, so a conditional verdict is appropriate: correct Eq. (12) (or show the implementation uses the correct relation) and add the quantitative evaluation the reader requested.","tokens_in":11166,"tokens_out":14872,"duration_ms":149755,"concrete_test":"Analytic check: take a1=2, a2=1, epsilon2=1 and a point whose projection has omega_pi=45 degrees. The ray intersects the mean ellipse (Eq. (6)) at parameter omega_s = arctan((a1/a2) tan omega_pi) = 63.4 degrees, whereas Eq. (12) gives omega_s = arctan((a2/a1) tan omega_pi) = 26.6 degrees; the latter point is not on the ray. Then run the released MATLAB fitting code on a noiseless synthetic supertoroid with the same a1, a2, epsilon2 initialized at the true pose: if the fitted cost is not approximately zero or the recovered parameters drift, the implemented distance inherits the error; if the fit is exact, Eq. (12) is a typographical error and the central algorithm survives with a corrected equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III's fitting distance rests on Eq. (12), which relates the polar angle omega_pi of p_pi to the superellipse parameter omega_s. From Eqs. (2) and (6), R_pi = beta2 p_pi and R_pi also equals (a1 a4 cos^{epsilon2} omega_s, a2 a4 sin^{epsilon2} omega_s); hence tan omega_pi = p_y/p_x = (a2/a1) tan^{epsilon2} omega_s. Solving gives tan omega_s = ((a1/a2) tan omega_pi)^{1/epsilon2}, equivalently tan^{epsilon2} omega_s = (a1/a2) tan omega_pi. The paper prints tan^{1/epsilon2} omega_s = (a2/a1) tan omega_pi. The coefficient a2/a1 and the exponent 1/epsilon2 are both transposed; for a1=a2 the printed relation implies tan omega_s = (tan omega_pi)^{epsilon2} rather than (tan omega_pi)^{1/epsilon2}. Since Eq. (15) uses omega_s to place the center of the cross-section superellipse and Eq. (16) computes beta1 in that frame, an incorrect omega_s moves the reference center off the ray through p, so the vector p_R and the distance d_s in Eq. (19) do not correspond to the claimed meridian radial distance. The central claim that Eq. (19) gives a simple, correct fitting metric for supertoroids is therefore not established by the paper as written; it needs either a corrected relation or an explicit demonstration that the implementation bypasses Eq. (12).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11549,"tokens_out":5859,"duration_ms":59519,"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":[{"comment":"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.","section":"Section III, Eq. (12)"},{"comment":"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.","section":"Section V-B, V-C, Table II"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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).","section":"Section II and III"},{"comment":"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.","section":"Section V-A"},{"comment":"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.","section":"Section IV, Table I"}],"recommendation":"major_revision","confidential_remarks":"The Eq. (12) error is central and must be addressed; it is likely a typographical transposition, but because it feeds directly into the fitting distance, the authors need to re-derive and re-run the experiments. If the implementation already uses the correct relation, the paper should say so explicitly. The empirical section needs quantitative metrics and at least one baseline before the robustness claims can be accepted. The code release and the differential-geometry expressions are positive aspects that make the revision tractable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main contribution is the meridian radial distance for supertoroids, a genuine extension of superquadric radial distance to genus-one surfaces. The two-stage fitting pipeline for partial clouds is a useful engineering step, and the paper ships code and data, which is good. The differential geometry section is standard but competently done.\n\nThere is a load-bearing problem in the derivation. Equation (12) relates the mean-superellipse angle ω_s to the projection angle ω_π. From the parametric equations, the correct relation is tan ω_π = (a2/a1) tan^{ε2} ω_s, which gives tan ω_s = ((a1/a2) tan ω_π)^{1/ε2}. The paper instead prints tan^{1/ε2} ω_s = (a2/a1) tan ω_π. Both the coefficient and the exponent are inverted. Since Eq. (15) uses ω_s to set the cross-section center and local coordinates, and Eq. (16) evaluates the inside-outside function there, an incorrect ω_s moves the reference point off the ray through p. The distance in Eq. (19) is then not the claimed meridian radial distance. As written, the central claim does not hold up. The experiments look plausible, so I suspect the implementation uses the correct relation, but the paper needs a corrected equation or an explicit statement that the code bypasses Eq. (12).\n\nThe experimental section is weak even apart from that. The G/D/B classification is subjective, there is no baseline comparison, and narrow objects require manual adjustment of the a3 initial guess. The post-hoc regularization to maximize a4 further weakens the evaluation. Timing numbers are provided, but no quantitative accuracy metrics.\n\nWho gets value from this paper? Researchers working on geometric primitive fitting for grasping, especially those using superquadrics, will find the meridian radial distance idea worth knowing once the error is fixed. As is, I would send it to peer review because the concept is novel and the error is likely fixable, but the referee should require the equation correction and a stronger empirical comparison.\n\nRecommendation: engage with the paper, but insist on resolving Eq. (12) before trusting the distance metric.","headline":"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.","tokens_in":12044,"tokens_out":7066,"would_cite":false,"duration_ms":59835,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A meridian radial distance allows supertoroids to be fitted to partial point clouds of objects with a single hole.","keywords":["supertoroid","superquadrics","point cloud fitting","geometric modeling","robotic grasping","meridian radial distance","genus-one objects","partial view fitting"],"falsifier":"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.","tokens_in":11027,"feed_emoji":"🤖","tokens_out":6284,"duration_ms":58442,"temperature":0.7,"pith_summary":"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.","feed_headline":"One distance formula fits holed objects for robotic grasping","feed_subtitle":"The supertoroid's meridian radial distance recovers shape and pose from a single noisy view of handles and rings.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines superquadrics including supertoroids and provides the inside-outside function used to build the ratios.","marker":"[23]"},{"why":"Supplies the radial Euclidean distance for superquadrics that the meridian radial distance modifies for the toroidal case.","marker":"[24]"},{"why":"Sampling algorithm used to generate synthetic supertoroid point clouds for the fitting experiments.","marker":"[16]"},{"why":"Earlier superquadric-based grasping pipeline by the same group that the supertoroid fitting extends to holed objects.","marker":"[22]"},{"why":"Standard reference for superquadric segmentation and recovery that the fitting procedure follows.","marker":"[9]"},{"why":"Cloud-completion alternative for household objects that the partial-cloud fit is designed to avoid.","marker":"[19]"}],"fun_headline_variants":["One distance formula fits supertoroids to holed objects","Supertoroid fit gives holed objects a graspable model","Pose and shape of rings and handles from one supertoroid","Holed objects: one distance formula for robotic grasping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One distance formula fits supertoroids to holed objects","Supertoroid fit gives holed objects a graspable model","Pose and shape of rings and handles from one supertoroid","Holed objects: one distance formula for robotic grasping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3139,"prompt_tokens":911,"completion_tokens":2228,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":2158}},"tokens_in":527,"tokens_out":2228,"duration_ms":17395,"temperature":1.0,"reasoning_tokens":2158,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:39:38.547530+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Superquadrics and angle-preserving transformations,","cited_arxiv_id":null,"evidence_quote":"Defines superquadrics including supertoroids and provides the inside-outside function used to build the ratios."},{"cited_title":"From uncertainty to visual exploration,","cited_arxiv_id":null,"evidence_quote":"Supplies the radial Euclidean distance for superquadrics that the meridian radial distance modifies for the toroidal case."},{"cited_title":"Equal-distance sampling of superellipse models,","cited_arxiv_id":null,"evidence_quote":"Sampling algorithm used to generate synthetic supertoroid point clouds for the fitting experiments."},{"cited_title":"Grasping unknown objects in clutter by superquadric representation,","cited_arxiv_id":null,"evidence_quote":"Earlier superquadric-based grasping pipeline by the same group that the supertoroid fitting extends to holed objects."},{"cited_title":"Jakli ˇc, A","cited_arxiv_id":null,"evidence_quote":"Standard reference for superquadric segmentation and recovery that the fitting procedure follows."},{"cited_title":"Exploiting symmetries and extrusions for grasping household objects,","cited_arxiv_id":null,"evidence_quote":"Cloud-completion alternative for household objects that the partial-cloud fit is designed to avoid."}],"review_version":1}