Single Edge Collapse Quad-Dominant Mesh Reduction

Julian Knodt

REVIEW 4 major objections 5 minor 2 references

Single Edge Collapse Quad-Dominant Mesh Reduction

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Single edge collapse can keep most quads while decimating meshes.

desk verdict A genuinely practical quad-preserving decimation method with clear wins in topology retention, but the abstract overstates the geometric-quality claim and the key ablation shows a real trade-off. read the letter →

arxiv 2411.16874 v1 pith:5Z6G3HTK submitted 2024-11-25 cs.GR

Julian Knodt This is my paper · ORCID

classification cs.GR

keywords quadmeshreductionquadricerrormetricsedgecollapselevelofdetailskinneddecimationsymmetrypreservationtopologyattribute

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 claims that the standard single-edge-collapse decimation used for triangle meshes can also preserve most quads of a quad-dominant input without sacrificing geometric fidelity. Its method adds a dihedral-angle-weighted quadric for every edge, groups edges with nearly equal error into equivalence classes, and resolves collapses within each class using a recency rule that collapses opposing quad edges in sequence. The same per-edge weights can steer simplification toward symmetry planes and joint regions, and treating joint influences as ordinary attributes is reported to outperform prior articulated simplification on animated meshes. If correct, this offers a simpler bridge between triangle-oriented decimation pipelines and the artist-made quad meshes used in games and animation.

What carries the argument

The engine of the method is the per-edge quadric $Q_{\text{edge}}$ in Eq. (2), a plane quadric lying orthogonal to a face's normal, weighted by dihedral angle and edge length, which keeps collapsed vertices from drifting in the face's tangent plane. Around it sit Eq. (4), approximate equality $|a-b|<\epsilon_{\text{abs}}$ that turns the total ordering of edge costs into a partial ordering; a 'recency' counter that, after a collapse, raises the priority of edges on the opposite side of incident quads, causing quad-chords to collapse in sequence; and Eq. (5), $QEM(e,v)=(Q_{e_0}+Q_{e_1})(v)-(Q_{e_0}(e_0)+Q_{e_1}(e_1))$, which measures introduced rather than accumulated error and lets quadrics stay fixed at initialization.

What would settle it

Take a set of quad-dominant meshes whose opposing quad edges have substantially different quadric errors, decimate each with Eq. 5 and with true memoryless simplification, and compare Chamfer and Hausdorff distances; if the Eq. 5 outputs are consistently worse than memoryless simplification, the claim of no geometric degradation fails.

Watch

Extended reading notes

Core claim

The central discovery is that the obstacle to quad preservation is not the edge-collapse operator itself but the overly strict ordering it imposes. Edges on a quad often have nearly identical quadric errors, and a total ordering dictated by floating-point QEM values destroys topology by collapsing equivalent edges in effectively random order. The paper introduces three coordinated changes: per-edge quadrics weighted by dihedral angle; grouping edges with approximately equal quadric error and ordering within a group by recency so that opposing quad edges are collapsed in sequence; and a modified QEM error that subtracts current endpoint errors, replacing memoryless simplification. The paper claims these changes preserve most input quads at equal or better Chamfer and Hausdorff error across 67 static and 19 animated meshes, and that the same per-edge weighting preserves symmetry and joint influences.

Load-bearing premise

The method's promise of equal geometric quality rests on the claim that the modified error in Eq. 5 replaces memoryless simplification without formal proof or multi-mesh evidence, so if that error measure distorts true geometric cost, the 'without degrading geometric quality' claim collapses.

Editorial extensions

If this is right

A decimation tool can now accept hybrid quad/triangle meshes and still retain a large fraction of quads, closing the gap between triangle decimators and pure-quad decimators.
For a quad-dominant character mesh reduced to 25% of its triangle count, the paper reports a median quad-retention ratio of about 0.76 versus about 0.44 for the QEM baseline.
Joint influences can be preserved by ordinary attribute quadrics, so skinning-aware decimation does not need poses or alternating optimization; the paper reports lower Chamfer and Hausdorff error over the first 50 animation frames on most of its 19 animated models.
Symmetry preservation becomes a soft, per-edge weight rather than a hard constraint, allowing artists to keep radial or reflectional symmetry without a separate symmetry-aware algorithm.

Reading between the lines

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

Increasing $\epsilon_{\text{abs}}$ is effectively a user-facing dial that trades slight geometric error for more quad preservation; LOD tooling could expose it directly instead of hiding it as an implementation constant.
The recency heuristic may generalize beyond quads: any regular substructure whose edges have nearly equal cost, such as edge loops or lattice sheets, could be preserved by defining analogous opposing-edge recency.
Because the animated meshes in the dataset were triangle-only GLB files, the paper does not demonstrate quad preservation on skinned quad meshes; a direct test on quad-dominant skinned inputs would close that gap.
Per-edge quadric weights are a general paintbrush, so the same mechanism used for symmetry and joints could preserve user-painted regions, creases, or material boundaries.

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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

You should read this paper if you care about quad-dominant mesh decimation. The core idea is simple and convincing: keep single-edge QEM but fix three things that destroy quads — ill-conditioned tangent planes, arbitrary ordering of near-equal edge costs, and memoryless simplification discarding input constraints. The per-edge dihedral-weighted quadric, the approximate-equality equivalence classes with recency ordering, and the Eq. 5 error re-formulation are new as a combination, and they work. On 67 static and 19 animated meshes, quad retention goes up substantially (median output-quad ratio about 0.95 vs 0.67 at 50% reduction), and the visual results, especially in Figs. 7 and 13, show real topology preservation that QEM and MeshLab just don't give you. The joint-influence experiment is also a nice, honest piece of work: directly testing Hoppe's attribute quadrics against Landreneau-Schaefer and showing it wins on most animated models fills a gap in the literature. The soft spots are real, but they are in the claims, not in the method. The abstract says "consistently outperforms prior work with lower Chamfer and Hausdorff distance," and Table 1 contradicts that for static Hausdorff at 50%: your average Hausdorff is 7.692e-3 versus 6.081e-3 for QEM and 4.977e-3 for MeshLab. Chamfer is consistently better, and Hausdorff is better at 25% and 10%, but the 50% case alone kills "consistently." The bigger issue is the phrase "without degrading geometric quality." Your own ablation in Fig. 13 shows that disabling recency drops Hausdorff from 3.457e-3 to 1.051e-3 — a 3.3x penalty. You call that "slightly" lower quality in the text; it is not slight, and it is exactly the kind of trade-off the abstract denies. Also, Eq. 5 is asserted to serve the same purpose as memoryless simplification without proof, and the only ablation is one mesh. That is a weak foundation for a load-bearing design choice. No error bars, no per-model win counts for static meshes, and no code or dataset makes the average numbers harder to interpret. None of this sinks the paper. The method is new, practical, and clearly valuable; the experiments are substantial. But the author needs to rewrite the abstract to say "usually" or "on average" where appropriate, report per-model counts and significance for static Hausdorff, and either prove or more carefully test Eq. 5. This is a strong candidate for peer review after those fixes. I would send it out, and I would cite it once the claims match the data.

Referee Report

4 major / 5 minor

Summary. The paper proposes modifications to quadric-error-metric (QEM) mesh simplification so that single edge collapses preserve most input quads on quad-dominant meshes. The three main ingredients are per-edge dihedral-angle-weighted quadrics (Eq. 2), a partial ordering of edges with approximately equal QEM costs (Eq. 4) combined with a recency-based ordering that implicitly collapses quad chords, and a modified QEM error (Eq. 5) that subtracts the current endpoint quadric errors to mimic memoryless simplification without actually recomputing quadrics. The authors also extend per-edge weights to soft symmetry preservation and joint-influence preservation, and they test Hoppe's attribute-preserving QEM for joint influences. Experiments are carried out on 67 static and 19 animated Sketchfab meshes, comparing against an internal QEM implementation and MeshLab's quadric decimation. The paper reports substantially higher quad-preservation ratios (median output-quad ratio about 0.949 vs 0.674 at 50% target) and mostly lower average Chamfer distances, but at the 50% static target the average Hausdorff distance is higher for the proposed method than for both baselines.

Significance. If the claims are properly qualified, this is a practically valuable contribution: it shows that a simple modification of the standard single-edge-collapse QEM pipeline can retain a large fraction of quad topology, which is relevant for artist-authored LODs and skinned-mesh workflows. The test on 67 static and 19 animated meshes is a genuinely large corpus, and the quad-preservation gains are consistent and visually clear. The paper also performs a useful service by directly testing Hoppe's 1999 attribute-preserving QEM for joint influences rather than assuming it is inadequate. The main weakness is that the central claim of preserving quads 'without degrading geometric quality' is not supported by the paper's own ablation, and the abstract overstates the geometric results. These issues are local to the claims and evaluation rather than to the core algorithmic idea, so the contribution is salvageable with a revision.

major comments (4)

[Abstract and Section 5, Table 1] The abstract states that the approach 'consistently outperforms prior work with lower Chamfer and Hausdorff distance,' but Table 1 shows that at the 50% static target the average Hausdorff distance of the proposed method (7.692e-3) is higher than both the internal QEM baseline (6.081e-3) and MeshLab (4.977e-3). The more cautious wording in Section 5 ('comparable or better') is justified, but the abstract, the introduction, and the contribution list in Section 4 ('higher geometric similarity than prior work') are not. Please correct the overstatement and qualify the 'without degrading geometric quality' claim accordingly.
[Section 6.1, Fig. 13] The recency ordering is the central mechanism for quad preservation, yet the ablation on a single mesh shows a large geometric cost: with recency enabled, Hausdorff distance increases from 1.051e-3 to 3.457e-3 (a factor of 3.3) and Chamfer distance from 1.249e-4 to 1.830e-4. The text says the geometric quality 'suffers slightly,' which misrepresents this degradation. Because the paper's central claim is that quads can be preserved 'without degrading geometric quality,' this ablation directly undermines that claim. The paper should either soften the claim to 'comparable geometric quality in most cases, with a trade-off quantified here,' or provide dataset-wide statistics on the geometric cost of recency (e.g., per-model paired comparisons with and without recency).
[Section 3.2, Eq. (5)] The paper asserts that the modified QEM error in Eq. (5) 'serves the same purpose as memoryless simplification, and is equally performant both in runtime and efficacy,' but no experiment directly compares Eq. (5) against memoryless simplification. The ablation in Fig. 14 compares the new error to the original QEM, both without memoryless simplification, on a single mesh. This is insufficient to support the equivalence claim, which is load-bearing because the method deliberately avoids memoryless simplification to keep the equivalent-edge ordering stable. Please either add a direct comparison to memoryless simplification or rephrase the claim as a hypothesis and temper the surrounding justification.
[Section 4 and Section 5 (static evaluation)] The static-mesh evaluation reports only averages and medians (Table 1) without per-model win counts, variance estimates, or paired significance tests. Given the large spread visible in the per-model tables (Tables 4-6), the claim of 'consistently' better or even 'comparable' geometric quality is not statistically supported. For example, the 50% Hausdorff average is driven by several models with large Hausdorff increases (e.g., 'Spiral Staircase' and 'Speeder Bike' in Table 4). Please add per-model counts of wins/losses/ties and a paired test (e.g., Wilcoxon signed-rank) over the dataset for Chamfer and Hausdorff at each target ratio, and discuss the trade-off between quad preservation and geometric error in those terms.

minor comments (5)

[Section 2] There is a typo in the first paragraph of Related Work: 'gemoetry processing' should be 'geometry processing.'
[Table 2] In the animated-mesh results table, the row label '25%△' appears twice; the second occurrence should presumably be '10%△' to match the three target ratios described in the text.
[Section 3.1] The phrase 'This quadric is weighed according to the dihedral angle' uses 'weighed' where 'weighted' is the standard term; the same inconsistency appears elsewhere (e.g., 'weighed by edge length' in Fig. 15).
[Algorithm 2] In the pseudocode, the line 'matched(𝑒)+ = 1' is unusual notation; it should be written as 'matched(e) += 1' for clarity, and similarly for 'matched(𝑒)+ = 2'.
[Section 5, QEM baseline description] The internal QEM baseline is described only briefly ('battle-tested implementation of [Hoppe 1999] and [Landreneau and Schaefer 2009]'), and it is not publicly available. Please provide more detail on the exact QEM variant, attribute handling, and any post-processing used for quad recovery, so that the comparison is reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quad-preservation mechanism is evaluated against external geometric metrics and independent baselines, and the paper's evidentiary weaknesses are overclaims rather than circular derivations.

full rationale

The paper's central claim is that single edge collapse can preserve most input quads without degrading geometric quality. The mechanism for this is a set of constructive modifications to QEM: per-edge dihedral-angle weighted quadrics (Eq. 2), approximate-equality grouping and recency ordering (Eq. 4 and Alg. 1), and a modified collapse cost (Eq. 5). None of these components is defined in terms of the evaluation metrics. Quad preservation is measured by the ratio of output quads to triangles relative to the input ratio, and geometric quality is measured by Chamfer and Hausdorff distance against the original mesh, both of which are external to the algorithm's objective. Eq. 5 defines collapse cost as the difference between the new vertex's quadric error and the current endpoint errors; this is a new cost model, not a metric that is later reported as a 'prediction.' The comparison baselines are the author's internal QEM implementation and MeshLab; although the internal baseline is not independently vetted, this is an experimental-control concern, not a circularity. There are no self-citations, no imported uniqueness theorems, and no fitted parameter that is renamed as a prediction. The paper does contain internal contradictions: Table 1 shows higher average Hausdorff distance for 'Ours' than both QEM and MeshLab at the 50% triangle target, and the Fig. 13 recency ablation shows a roughly 3.3x Hausdorff penalty for quad preservation, while the text calls the quality difference 'slight.' These are evidence-quality problems that undermine the strength of the claim 'without degrading geometric quality,' but they do not make the derivation circular. The derivation chain is self-contained: the method's components are justified by ablations and comparisons against external error measures, not by restating the output as the input.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The method introduces several hand-chosen thresholds and weights (epsilon_abs, lambda_joint, lambda_sym) and relies on unproven modeling choices: the dihedral-angle weighting formula, the modified QEM error equation, and the equivalence-class ordering. These do not constitute circularity but are the principal sources of tuning risk.

free parameters (5)

epsilon_abs = 5e-6
Threshold for approximate equality of quadric errors (Eq. 4). Controls the quads-vs-geometry trade-off; set manually for all main experiments and ablated in Sec. 6.5.
lambda_joint = 1
Weight for joint-influence difference when scaling per-edge quadrics (Eq. 2). Set manually; ablated in Sec. 6.4.
lambda_sym = 0
Weight for symmetry matching in per-edge quadrics; disabled in main experiments, used in symmetry-preservation demos. Chosen manually, no systematic tuning described.
max joint influences stored per vertex = 16
Hard cap for stack-allocated quadrics; if more than 16, smallest weight dropped. Not varied or justified.
post-decimation joint count = 4
After decimation, top 4 joint influences are kept and normalized; arbitrary choice, not derived.

assumptions (7)

standard math Quadric error metrics for plane and attribute preservation (Garland-Heckbert 1997; Hoppe 1999) are valid and sufficient as a base.
The method builds directly on these prior quadrics without modifying the fundamental plane-quadric minimization. Invoked throughout Section 3.
standard math Linear least squares solves the attribute functional fit (Eq. 7).
Used to compute per-vertex attribute functions for joint influences and other attributes; standard linear algebra is assumed.
domain assumption Artist-created quad-dominant meshes contain regular quad elements, and preserving quad topology and edge loops is a worthwhile objective.
The entire goal and evaluation metric (quad ratio preservation) presuppose this value judgment. Stated in Introduction and Discussion.
domain assumption Joint influences can be treated as linearly interpolated attributes and Hoppe's attribute quadrics preserve them adequately.
Section 'Skinned Mesh Reduction' argues prior work gave no evidence against this; the paper then tests it empirically rather than proving it.
ad hoc to paper The per-edge dihedral-angle-weighted quadric (Eq. 2) with edge-length-squared scaling correctly resolves the tangent-space degeneracy of coplanar faces.
The weighting formula w = (1/pi) arccos(...) * ||e0-e1||^2 is introduced heuristically; ablation (Fig. 15) is on a single mesh.
ad hoc to paper Approximate equality defined by Eq. 4 (|a-b| < epsilon_abs) is a valid basis for treating edges as incomparable, and recency ordering preserves quads without harming geometric quality.
This is the core ordering heuristic; it depends on the hand-set epsilon_abs and on the unproven assertion that ordering equal-cost edges by quad-chord continuity does not degrade geometry.
ad hoc to paper The modified QEM error of Eq. 5 serves the same purpose as memoryless simplification while keeping quadrics fixed at initialization.
Asserted in Section 3.2; no theoretical derivation, and the only ablation (Fig. 14) is on a single mesh.

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Cite this review

Pith. "Pith review of Single Edge Collapse Quad-Dominant Mesh Reduction." pith.science (2026). https://pith.science/paper/5Z6G3HTK

@misc{pith2026241116874,
  author       = {Pith},
  title        = {Pith review of: Single Edge Collapse Quad-Dominant Mesh Reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5Z6G3HTK}},
  note         = {Machine review of arXiv:2411.16874}
}

read the original abstract

Mesh reduction using quadric error metrics is the industry standard for producing level-of-detail (LOD) geometry for meshes. Although industry tools produce visually excellent LODs, mesh topology is often ruined during decimation. This is because tools focus on triangle simplification and preserving rendered appearance, whereas artists often produce quad dominant meshes with clean edge topology. Artist created manual LODs preserve both appearance and quad topology. Furthermore, most existing tools for quad decimation only accept pure quad meshes and cannot handle any triangles. The gap between quad and triangular mesh decimation is because they are built on fundamentally different operations, triangle simplification uses single edge collapses, whereas quad decimation requires that entire sets of edges be collapsed atomically. In this work, we demonstrate that single edge collapse can be used to preserve most input quads without degrading geometric quality. Single edge collapse quad preservation is made possible by introducing dihedral-angle weighted quadrics for every edges, allowing optimization to evenly space edges while preserving features. It is further enabled by explicitly ordering edge collapses with nearly equivalent quadric error that preserves quad topology. In addition to quad preservation, we demonstrate that by introducing weights for quadrics on certain edges, our framework can be used to preserve symmetry and joint influences. To demonstrate our approach is suitable for skinned mesh decimation (a key use case of quad meshes), we show that QEM with attributes can preserve joint influences better than prior work. On both static and animated meshes, our approach consistently outperforms prior work with lower Chamfer and Hausdorff distance, while preserving more quad topology.

Figures

Figures reproduced from arXiv: 2411.16874 by the authors.

**Figure 1.** Our approach is able to better preserve input topology and quads than QEM [Garland and Heckbert, 1997, Hoppe, 1999] for hybrid triangle/quad [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** To measure the symmetry of each edge, we compute a matching of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Cube mesh simplified with our approach. All edges in the input cube [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (14 more)

**Figure 6.** Figure 6: Before edge collapses, all edges have 0 recency (marked in black). [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: Our approach compared to an implementation of [Hoppe, 1999] (labeled QEM) and [Cignoni et al [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Decimation of a hybrid quad mesh with many components at multiple resolutions. Our approach preserves most quads without sacrificing geometric [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Visualization of quad preservation against geometric quality for [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 11.** Figure 11: Our approach identifies and preserves radial symmetry. Left: input [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 10.** Figure 10: As compared to normal QEM, our approach is able to preserve more [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 13.** Figure 13: We ablate our approach with and without recency on a complex [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

**Figure 14.** Figure 14: Ablation of the new QEM formulation from Eq. 5 against the original [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

**Figure 16.** Figure 16: Ablation of our approach with varying 𝜆joint. As 𝜆joint increases, the area around joints is more strongly preserved, at the cost of higher decimation in other areas. Despite the rest pose chamfer distance increasing, we see that at 𝜆joint = 1 there is an improvement …

**Figure 17.** Figure 17: We ablate the effect of varying 𝜖abs from Eq. 4. As 𝜖abs increases, there is stronger quad preservation, but a slight increase in the distance from the original mesh. cbfelipegall. cost, and demonstrate how these modifications can be extended to preserve symmetry and …

**Figure 20.** Figure 20: ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2024 [PITH_FULL_IMAGE:figures/full_fig_p013_20.png]

**Figure 18.** Figure 18: We show additional examples of symmetry preservation on some example meshes. By increasing the weight of edges along planes of symmetry, our [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]

**Figure 19.** Figure 19: Comparison of geometric distance for each of 19 animated models for our approach and “QEM” which implements [Landreneau and Schaefer, 2009]. [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]

**Figure 20.** Figure 20: Visual comparison of our approach against QEM, which implements [Landreneau and Schaefer, 2009], on two animated meshes. For each mesh, the [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]

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Reference graph

Works this paper leans on

2 extracted references

[2018]

ACM Trans

Tetrahedral Meshing in the Wild. ACM Trans. Graph. 37, 4, Article 60 (July 2018), 14 pages. https://doi.org/10.1145/3197517.3201353 IEEE. 1985. IEEE standard for binary floating-point arithmetic . Institute of Electrical and Electronics Engineers, New York. Note: Standard 754–1985. Wenzel Jakob, Marco Tarini, Daniele Panozzo, and Olga Sorkine-Hornung. 201...

arXiv 2009
[2021]

ACM Trans

Reliable feature-line driven quad-remeshing. ACM Trans. Graph. 40, 4, Article 155 (July 2021), 17 pages. https://doi.org/10.1145/3450626.3459941 Bay Raitt and Greg Minter. 1998. Digital Sculpture Techniques (How to Ap- ply the Principles of Traditional Sculpture to make Stunning 3D Charac- ters). https://web.archive.org/web/20160218133259/https://www.ualb...

arXiv 1997

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [2018]

ACM Trans

Tetrahedral Meshing in the Wild. ACM Trans. Graph. 37, 4, Article 60 (July 2018), 14 pages. https://doi.org/10.1145/3197517.3201353 IEEE. 1985. IEEE standard for binary floating-point arithmetic . Institute of Electrical and Electronics Engineers, New York. Note: Standard 754–1985. Wenzel Jakob, Marco Tarini, Daniele Panozzo, and Olga Sorkine-Hornung. 201...

arXiv 2009

[2] [2021]

ACM Trans

Reliable feature-line driven quad-remeshing. ACM Trans. Graph. 40, 4, Article 155 (July 2021), 17 pages. https://doi.org/10.1145/3450626.3459941 Bay Raitt and Greg Minter. 1998. Digital Sculpture Techniques (How to Ap- ply the Principles of Traditional Sculpture to make Stunning 3D Charac- ters). https://web.archive.org/web/20160218133259/https://www.ualb...

arXiv 1997