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REVIEW 3 major objections 6 minor 54 references

OffsetCrust: Variable-Radius Offset Approximation with Power Diagrams

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The variable-radius offset of a surface can be computed as the separator between two kinds of weighted sites in a power diagram, with displaced points placed along a radius-dependent direction.

desk verdict A practical variable-radius offset method with a clean core idea and strong experiments; Theorem 1 needs a real proof and the manifoldness claim is too strong, but it deserves serious peer review. read the letter →

arxiv 2507.10924 v2 pith:HTOGS4YF submitted 2025-07-15 cs.GR

Zihan Zhao , Pengfei Wang , Minfeng Xu , Shuangmin Chen , Shiqing Xin , Changhe Tu , Wenping Wang This is my paper
classification cs.GR
keywords variable-radiusoffsetpowerdiagramsurfacecrust-basedreconstructionmedialaxistransformsharpfeaturepreservationtrianglemeshgeneralizeddistancefunction
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 claims that the variable-radius offset of a surface---the envelope of balls whose radii vary from point to point---can be computed directly as a subset of power-diagram facets. One set of sites is points $p$ sampled on the base surface, weighted by $R(p)^2$; the other is the same points displaced a small distance $\epsilon$ along a radius-dependent direction, weighted by $(R(p)-\epsilon)^2$. Because the offset point $p^{\mathrm{off}}_R$ of a contributing base point satisfies $\|p^{\mathrm{off}}_R-p\|^2-R(p)^2=\|p^{\mathrm{off}}_R-p^{\epsilon}_R\|^2-(R(p)-\epsilon)^2=0$, it lies exactly on the bisector of the matched pair, and occluded points cannot contribute because they are dominated by another site's power distance. If the claim is right, variable-radius offsetting on triangle meshes becomes a sampling-and-tessellation problem with a small least-squares correction for facet misalignment, and the same construction recovers boundary surfaces from medial axis transforms.

What carries the argument

The central object is the power diagram of two weighted point sets: base samples $p$ with weight $R(p)^2$ and displaced samples $p^{\epsilon}_R=p+\epsilon\,n^{\mathrm{off}}_R(p)$ with weight $(R(p)-\epsilon)^2$. The offset surface is extracted as the separator between the two classes of power cells, and the load-bearing identity is equation (4): a point on the offset satisfies equal power distances to the two members of a matched pair exactly when it satisfies the generalized offset equation. Theorem 1 supplies the displacement direction that makes this identity track the true envelope, while the regularized least-squares step (equations (14)--(15)) fits each power-diagram vertex to the tangent-plane constraints $(v-p_i)\cdot n^{\mathrm{off}}_i=R(p_i)$ and thereby suppresses the misaligned facets that generic crust sampling produces.

What would settle it

Take a surface with an analytically known variable-radius offset, such as a sphere with $R(x)=1+a x$, compute the true contact direction $n^{\mathrm{off}}_R$ at several points from the envelope, and compare it with the rotated normal given by Theorem 1; a discrepancy beyond sampling and $\epsilon$ discretization error, or a failure of the power-diagram facet boundary to converge to $\phi(x)=0$ as samples increase, would refute the central claim.

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Extended reading notes

Core claim

The central discovery is that the generalized distance function $\phi(x)=\min_{p\in S}(\|x-p\|-R(p))$ can be represented by a power diagram whose sites come in matched pairs. For any base point $p$ that contributes to the offset, the identity $\|p^{\mathrm{off}}_R-p\|^2-R^2(p)=\|p^{\mathrm{off}}_R-p^{\epsilon}_R\|^2-(R(p)-\epsilon)^2=0$ places the offset point on the power bisector of $p$ and $p^{\epsilon}_R$, and the paper argues that a point whose offset is occluded by another ball lies inside the power cell of that dominant site, so it is automatically excluded from the extracted separator. The companion result is Theorem 1: the displacement direction $n^{\mathrm{off}}_R(p)$ is the unit surface normal rotated by $\alpha=\arcsin(\|\nabla R(p)\|)$ around the axis $(\nabla R(p)/\|\nabla R(p)\|)\times n_p$, which reduces to the usual normal when $R$ is constant. Misaligned facets caused by pairs $(p_i, p^{\epsilon}_j)$ with $i\neq j$ are then repaired by moving each power-diagram vertex to the regularized least-squares position that makes $(v-p_i)\cdot n^{\mathrm{off}}_i-R(p_i)$ close to zero.

Load-bearing premise

The load-bearing premise is the offset-direction formula in Theorem 1: the displaced point must lie along the normal rotated by $\arcsin(\|\nabla R(p)\|)$, which requires $R$ to be differentiable with $\|\nabla R\|\le 1$ and the normal field to be smooth; on a triangle mesh the method bridges gradient discontinuities only by heuristic interpolation, so if the formula misplaces displaced sites the power-diagram separator will not match the true offset.

Editorial extensions

If this is right

  • Variable-radius offsets on triangle meshes can be computed without parametric offsetting or explicit self-intersection trimming: the power-diagram separator is manifold in the non-degenerate configurations the paper examines, and occluded samples automatically fall inside other cells.
  • Constant-radius offsets are the special case $\alpha=0$, so the same implementation covers a whole family of offsetting problems that previously required separate distance-field pipelines.
  • Reconstructing a surface from a medial axis transform becomes a variable-radius offset of the MAT, since the MAT radius function satisfies $\|\nabla R\|\le 1$ by construction.
  • Sharp features are treated by a one-base-many-displaced sampling around edges and vertices, and by restricting the allowed displacement directions the method can also produce feature-preserved or topology-changing offsets, such as in dilation-erosion closing operations.

Reading between the lines

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

  • Extension: the same weighted-site construction should work for radius functions defined on point clouds or implicit surfaces, provided one can estimate a gradient and a normal field; the paper demonstrates only triangle-mesh inputs.
  • Testable consequence: as sampling density grows and $\epsilon\to 0$, the least-squares fix should make the extracted facets converge to the true zero level set; an experiment varying only sample count would show whether residual error comes from sampling or from the offset-direction approximation.
  • The method's sharp-feature fidelity depends on the heuristic spherical interpolation between face-based displacement directions, so varying the dihedral-angle threshold at fixed sample count would isolate the contribution of Theorem 1 from the contribution of the sampling heuristic.
  • The MAT application suggests a stress test: feed in a medial axis with a radius field that locally violates $\|\nabla R\|\le 1$, such as near a crease, and check whether the unresolvable misalignments predicted by the paper appear exactly where the constraint fails.
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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

3 major / 6 minor

Summary. The paper proposes OffsetCrust, a crust-based method for approximating variable-radius offset surfaces of triangle meshes. Given a radius function R on a base surface S, it samples base points p with power weight R(p)^2 and displaced points p_R^epsilon = p + epsilon n_R^off(p) with weight (R(p)-epsilon)^2, then extracts the offset as the subset of power diagram facets separating base and displaced sites. The displacement direction n_R^off is claimed to be the surface normal rotated by arcsin(||grad R||) around an axis determined by grad R and n_p (Theorem 1). A least-squares refinement moves power diagram vertices to reduce misaligned facets. The method is validated quantitatively on a Dupin cyclide, qualitatively on CAD and freeform models, and on 7,609 Thingi10K meshes for the constant-radius case; applications include expressive surface design, channel surfaces, and MAT reconstruction.

Significance. If the central identity is correct, the paper offers a novel and efficient way to compute variable-radius offsets without explicit self-intersection handling, with a clean geometric interpretation via power diagrams. Strengths include the simple formulation, the large-scale constant-radius evaluation on Thingi10K, and the public code release. The method also enables interesting extensions such as feature-preserving offsets by restricting displacement directions. However, the load-bearing derivation of the displacement direction is only sketched, and some strong topological claims are contradicted by the paper's own appendix statistics; these issues need to be fixed before the contribution is fully established.

major comments (3)
  1. [§3.2, Theorem 1 / Figure 3] The displacement direction n_R^off is the single most load-bearing quantity in the paper: every displaced site, every power facet, and the misalignment objective in Eq. (14) depend on it. Yet the proof of Theorem 1 is only a verbal description of a right triangle; it does not derive the triangle from the stationarity condition of Eq. (1), does not state whether n_p is the inward or outward normal, and does not handle the piecewise-linear case where grad R is per-face and undefined on edges and vertices. The correct smooth-surface identity is that the tangent component of n_R^off equals -grad_S R(p), i.e., n_R^off = sqrt(1-||grad R||^2) n_p - grad R(p) for an outward offset, with the sign of the normal component reversed for an inward offset. The rotation formula in Theorem 1 is consistent with this identity, but the text never establishes it. I ask the authors to replace the diagram-based argument with a derivation from Eq. (1) and to state explicitly how the edge/vertex Slerp and 1vN heuristics in §4.1 relate to this continuous formula.
  2. [§3.2 'Manifold Output...' vs Appendix E, Table 3] The subsection titled 'Manifold Output Without the Need for Explicit Self-Intersection Handling' claims that non-manifoldness cannot occur. This is contradicted by the authors' own Appendix E statistics: for delta = -2% with the optimized pipeline, Table 3 reports mean non-manifold edge proportion 0.002% (max 0.376%), mean non-manifold vertex proportion 0.023% (max 0.398%), and mean non-coplanar intersecting face proportion 11.759% (max 66.176%). Even the raw output has up to 5.128% intersecting faces for delta = -2%. The claim should be corrected to state that non-manifold and self-intersecting artifacts are rare but not eliminated by construction, especially after the misalignment optimization.
  3. [§5.1-5.2] The quantitative validation of the central variable-radius claim is thin. The only quantitative variable-radius evaluation is on the Dupin cyclide (§5.1.1), a single smooth analytical surface. The Thingi10K large-scale evaluation in §5.2 is for constant-radius offsets (delta = ±2%) only, and the qualitative comparisons in §5.1.2 do not report error metrics against a ground truth. Since the method's novelty is precisely the variable-radius case, I would like to see at least a few quantitative variable-radius tests on triangle meshes, for example by comparing against a high-resolution sampled version of Eq. (1) or against offset spheres/cones/slabs for models with known radius functions.
minor comments (6)
  1. [§3.2, Theorem 1] The theorem statement uses grad R(p) without specifying that it is the surface (tangential) gradient; in the discrete setting R is piecewise linear per triangle, so grad R is not defined on edges and vertices. Please make this distinction explicit.
  2. [§5.2] The sentence 'the NC metric exhibits significant improvement, improving by approximately an order of magnitude' is imprecise because NC values are percentages near 100%; please report the actual before/after values or clarify what 'order of magnitude' refers to.
  3. [Table 1] The formatting of bold entries is confusing because both the best and second-best scores are marked in bold; some cells also appear to run together. Please reformat the table so the best and second-best scores are visually distinguishable.
  4. [§4.1] Several inset figures are referenced without numbers, which hurts readability. Please number the insets for the edge-type, vertex-type, and cylinder-based sampling strategies.
  5. [Appendix A] The notation d_pow(x, p_i) is used in Eqs. (19)-(26) without being redefined in the appendix; please add a pointer to Eq. (6) so the appendix is self-contained.
  6. [References] Reference [31] is cited as 'Qun et al.' but the paper is by Qun and Rokne; please correct the citation style.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the offset is defined externally by Eq. (1), the power-diagram weights are chosen so that bisectors coincide with tangent planes at true offset points, the displacement direction comes from ∇R via Theorem 1 rather than from the output, and accuracy is checked against analytical and distance-field ground truth.

full rationale

The derivation chain is self-contained. The offset surface is fixed by Eq. (1), φ(x) = min_{p∈S}( ‖x−p‖ − R(p) ) = 0, which never references the power diagram. The weighted-site construction is an engineering identity, not a prediction: Eq. (3)-(4) ('‖p_off_R − p‖ − R(p) = ‖p_off_R − p_eps_R‖ − (R(p) − ε) = 0', 'or equivalently' the squared form) hold because the weights R(p)² and (R(p) − ε)² are chosen so that the bisector plane between p and p_eps_R passes through p_off_R and is perpendicular to n_off_R(p); the bisector is the tangent plane of the true offset only if n_off_R(p) is the true displacement direction. That direction is not fitted: Theorem 1 (rotating the unit normal n_p by α = arcsin(‖∇R(p)‖) around ∇R/‖∇R‖ × n_p) uses only R and the surface geometry. Its proof in Section 3.2 is a sketch (a right triangle with hypotenuse ‖pq‖ and leg |R(p) − R(q)|) and is under-specified for piecewise-linear meshes, but an under-proved identity is a rigor gap, not circularity; the paper's own Section 7 explicitly concedes the arcsin domain restriction ‖∇R‖ ≤ 1 and the heuristic design of the face/edge/vertex sampling strategies. Validation is external: the Dupin cyclide analytical ground truth (Section 5.1.1), projection-distance error against the original surface for constant offsets (Section 5), comparisons with PointDC, which builds the distance field directly from Eq. (1) (Section 5.1.2), and MAT reconstruction measured against the original boundary surfaces (Section 6.3). The misalignment elimination (Eq. 14-15) minimizes (v − p_i)·n_off_i − d_i with d_i := R(p_i), i.e., the input radius, not a measured target, and the reported Normal Consistency gains are computed against external normals, so the fine-tuning is not a disguised fit to the expected answer. Self-citations (PCO [14], Q-MAT [52]) serve only as a comparison baseline and as a generator of MAT test inputs; neither justifies a load-bearing premise. No uniqueness theorem is imported from the authors' prior work, and the weighted-site ansatz is derived in-paper from Eq. (4) rather than adopted via citation. Accordingly, no specific circular reduction can be quoted, and the score is 0.

Assumptions & free parameters 7 free parameters · 6 assumptions · 1 invented entities

No constants are fitted to the offset ground truth; all free parameters are implementation defaults tuned via ablations. The central correctness depends on standard power diagram geometry, smoothness of R with ||grad R|| <= 1, watertight input meshes, and the (sketched) envelope argument behind Theorem 1.

free parameters (7)
  • epsilon (displacement distance) = 1e-6
    Displacement distance in Eq. 2; theory allows any small positive value, and Appendix A shows smaller epsilon localizes misalignments; value is set by hand.
  • lambda (regularization weight) = 0.01
    Regularization weight in Eq. 15; ensures the linear system is invertible; chosen small and empirically.
  • rho (clearance parameter) = 5% of average edge length
    Sets radius of protected spheres and cylinders around vertices and edges; ablation in Appendix B shows small offsets prefer smaller rho; set by hand.
  • blue-noise sample count = 70,000
    Default number of base samples; denser sampling improves accuracy (Appendix B) but increases runtime; chosen empirically.
  • vertex sphere discretization = 642 vertices
    Resolution of 1vN spherical sampling around vertices; more vertices improve large-offset accuracy but cost more; chosen empirically.
  • dihedral angle threshold = 0.2 rad
    Optional feature-line detection threshold; Appendix B Table 2 shows it balances quality and runtime; chosen empirically.
  • TetWild preprocessing edge length = 0.5
    Used to remesh inputs; affects triangle quality and feature detection but is not a fitting constant.
assumptions (6)
  • domain assumption Input meshes are watertight, free of self-intersections, and have good triangle quality.
    Section 4.1 lists these as explicit requirements; the sampling, feature detection, and normal estimation assume consistent, non-degenerate input.
  • domain assumption The radius function R is smooth with ||grad R(p)|| <= 1 everywhere.
    Required by Theorem 1 because alpha = arcsin(||grad R||) must be defined; Section 3.2 and Section 7 state that violating it leads to unresolvable misalignments.
  • domain assumption The power diagram is generic: no more than three sites determine a vertex in 2D and four in 3D.
    Section 3.2 assumes this to claim manifold output; CGAL is said to handle degeneracies, but the manifoldness argument is only for the generic case.
  • domain assumption If a point p does not contribute to the offset, there exists another point q whose ball strictly contains p_R^off.
    Used in Section 3.2 to argue non-contributing points cannot appear in the crust; requires strict containment and continuous envelope behavior.
  • domain assumption The proof of Theorem 1 treats pq, |R(p)-R(q)|, and the tangent-plane leg as a right triangle, which holds to first order only for smooth surfaces and smooth R.
    No error estimate is given for piecewise-linear R on meshes; the method relies on this first-order envelope geometry.
  • standard math Power diagram properties and weighted distance geometry from Aurenhammer (1987).
    Used throughout Section 3 to identify offset facets as power diagram bisectors.
invented entities (1)
  • paired power diagram sites (base point p with weight R(p)^2 and displaced point p_R^eps with weight (R(p)-eps)^2)
    purpose: Construct a power diagram whose separating facets approximate the variable-radius offset surface.
    The two-point site pair is an algorithmic construction introduced by the paper; it has no direct external observable, though the predicted offset meshes are testable.

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Cite this review

Pith. "Pith review of OffsetCrust: Variable-Radius Offset Approximation with Power Diagrams." pith.science (2026). https://pith.science/paper/HTOGS4YF

@misc{pith2026250710924,
  author       = {Pith},
  title        = {Pith review of: OffsetCrust: Variable-Radius Offset Approximation with Power Diagrams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTOGS4YF}},
  note         = {Machine review of arXiv:2507.10924}
}
abstract

Offset surfaces, defined as the Minkowski sum of a base surface and a rolling ball, play a crucial role in geometry processing, with applications ranging from coverage motion planning to brush modeling. While considerable progress has been made in computing constant-radius offset surfaces, computing variable-radius offset surfaces remains a challenging problem. In this paper, we present OffsetCrust, a novel framework that efficiently addresses the variable-radius offsetting problem by computing a power diagram. Let $R$ denote the radius function defined on the base surface $S$. The power diagram is constructed from contributing sites, consisting of carefully sampled base points on $S$ and their corresponding off-surface points, displaced along $R$-dependent directions. In the constant-radius case only, these displacement directions align exactly with the surface normals of $S$. Moreover, our method mitigates the misalignment issues commonly seen in crust-based approaches through a lightweight fine-tuning procedure. We validate the accuracy and efficiency of OffsetCrust through extensive experiments, and demonstrate its practical utility in applications such as reconstructing original boundary surfaces from medial axis transform (MAT) representations.

Figures

Figures reproduced from arXiv: 2507.10924 by the authors.

Figure 1
Figure 1. Offsetting a solid “L”. (a) Constant-radius offset: for a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Insight into why the power diagram helps determine the variable-radius offset. (a) A point [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Proof to Theorem 1. Let ϵ > 0 be sufficiently small, and define: p ϵ R := p + ϵ · n off R (p). (2) Then, the three points p, p ϵ R, and p off R are colinear. We thus obtain: ∥p off R − p∥ − R(p) = ∥p off R − p ϵ R∥ − (R(p) − ϵ) = 0, (3) or equivalently: ∥p off R − p∥ 2 − R2 (p) = ∥p off R − p ϵ R∥ 2 − (R(p) − ϵ) 2 = 0. (4) This motivates the construction of two groups of points and the use of a power diagram to dete… view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: In a 2D power diagram, a vertex typically has a [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: Larger radii are specified at four positions on the [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 8
Figure 8. Figure 8: Inward offset of a cube with a square hole, visual [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: A gallery of variable-radius offset surfaces computed by our [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: Construction of a Dupin cyclide surface using 1vN [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: Offsetting the chair with offset radii defined at the vertices. Two types of features are compared: detailed [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: Qualitative comparisons of constant-radius offsets. In the outward offset [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]
Figure 15
Figure 15. Figure 15: presents boxplots comparing the performance metrics of DC and our method (with and without mis￾alignment elimination). For CD and HD, our method is comparable to DC in median and interquartile range. No￾tably, our method achieves smaller minimum and maximum values, an…
Figure 13
Figure 13. Figure 13: Quantitative results for constant-radius offsets with [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 14
Figure 14. Figure 14: Runtime performance of constant-radius offsets [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]
Figure 16
Figure 16. Figure 16: By assigning varying offset distances (visualized using different colors) to the vertices of the input Centaur model, [PITH_FULL_IMAGE:figures/full_fig_p010_16.png]
Figure 17
Figure 17. Figure 17: Construction of diverse knots from a parameterized central curve, defined as [PITH_FULL_IMAGE:figures/full_fig_p010_17.png]
Figure 18
Figure 18. Figure 18: Surface reconstruction from medial axis transforms (MATs) generated using variable-radius offsets computed by [PITH_FULL_IMAGE:figures/full_fig_p011_18.png]
Figure 19
Figure 19. Figure 19: Variable-radius offset of a discrete sphere [PITH_FULL_IMAGE:figures/full_fig_p011_19.png]
Figure 20
Figure 20. Figure 20: demonstrates the relationship between the regions for four random sites. Misaligned facets are almost in￾evitable in any sampling configuration when implementing OffsetCrust [PITH_FULL_IMAGE:figures/full_fig_p013_20.png]
Figure 21
Figure 21. Figure 21: illustrates the configuration of sites used in OffsetCrust. While 0 < ϵ < R(p) is theoretically effective, as analyzed in Section 3, selecting a small ϵ > 0 improves [PITH_FULL_IMAGE:figures/full_fig_p013_21.png]
Figure 22
Figure 22. Figure 22: With smaller ϵ, the misalignments are more localized and thus easier to optimize. In contrast, larger ϵ leads to more complex misalignment adjacencies, making optimiza￾tion more difficult. stability when addressing misaligned facets. This is because the ϵ-balls have r…
Figure 23
Figure 23. Figure 23: For the ablation study on the block model with sharp features, we use a high-quality mesh with [PITH_FULL_IMAGE:figures/full_fig_p014_23.png]
Figure 24
Figure 24. Figure 24: Different numbers of blue noise samples for the [PITH_FULL_IMAGE:figures/full_fig_p014_24.png]
Figure 25
Figure 25. Figure 25: Ablation study on the Fandisk model with dihedral angle threshold for filtering. [PITH_FULL_IMAGE:figures/full_fig_p015_25.png]
Figure 26
Figure 26. Figure 26: Two types of undesirable behaviors affect the quality [PITH_FULL_IMAGE:figures/full_fig_p015_26.png]
Figure 27
Figure 27. Figure 27: 2D illustration of Figure 26. When the point [PITH_FULL_IMAGE:figures/full_fig_p015_27.png]
Figure 28
Figure 28. Figure 28: Qualitative results of the constant-radius offset of CAD models at various offset distances. The behaviors of sharp [PITH_FULL_IMAGE:figures/full_fig_p017_28.png]
Figure 29
Figure 29. Figure 29: Qualitative results of the constant-radius offset of high-genus models at various offset distances. Topology changes [PITH_FULL_IMAGE:figures/full_fig_p017_29.png]
Figure 30
Figure 30. Figure 30: Qualitative results of the constant-radius offset of real scan models at various offset distances. The details are [PITH_FULL_IMAGE:figures/full_fig_p017_30.png]
Figure 31
Figure 31. Figure 31: Given the letter “A” (left), OffsetCrust can produce either a feature-preserved offset (middle) or a rounded￾feature offset (right) by controlling the displacement di￾rections when generating edge-around and vertex-around displaced points [PITH_FULL_IMAGE:figures/ful…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.