pith. sign in

arxiv: 2605.16305 · v1 · pith:J7VXL57Nnew · submitted 2026-04-26 · 💻 cs.GR · cs.NA· math.CV· math.NA

Conformal tubular parameterization and toroidal bending of tube-like surfaces

Shunyu Yao , Gary P. T. Choi This is my paper

Pith reviewed 2026-05-20 23:57 UTC · model grok-4.3

classification 💻 cs.GR cs.NAmath.CVmath.NA
keywords conformal parameterizationtubular surfacestoroidal mappingquasi-conformal correctionsurface parameterizationvascular surfacesgeometry processing
0
0 comments X

The pith

Tube-like surfaces can be conformally mapped to 3D tubular and toroidal domains via cut-based lifting and localized seam correction.

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

The paper develops a conformal parameterization method specifically for open tube-like surfaces that have two boundary components. It starts by cutting the input mesh, computing a disk-to-rectangle conformal map, and lifting the result into a three-dimensional tubular domain. A localized quasi-conformal correction step then reduces distortion near the cut seam while leaving distant regions unchanged. For meshes with noisy or irregular boundaries, a free-boundary variant uses extension and smoothing to impose constraints on artificial outer rings. The framework also supplies two conformal maps that bend the tubular result into toroidal shapes while keeping the underlying tube topology intact. A reader would care because many real surfaces in medicine and engineering, such as blood vessels or pipes, lose their natural longitudinal and circumferential structure when forced into flat domains.

Core claim

The central claim is that a fixed-boundary tubular parameterization is obtained by cutting the mesh and lifting a disk-to-rectangle conformal map to three dimensions, after which a localized quasi-conformal correction on an annular domain mitigates seam artifacts, a free-boundary variant handles noisy boundaries through extension and cycle-Laplacian smoothing, and two conformal toroidal bending maps transform the result while preserving topology, yielding low-distortion maps suitable for downstream tasks on synthetic tubes and real vascular surfaces.

What carries the argument

The central mechanism is the disk-to-rectangle conformal map that is cut from the tube-like surface and lifted into a three-dimensional tubular domain, combined with an annular quasi-conformal correction that targets only the seam region.

If this is right

  • Low-distortion parameterizations are produced for both synthetic tube meshes and real vascular surfaces.
  • Seam-induced artifacts are effectively reduced by the annular correction step.
  • Robustness improves when input boundaries are noisy or irregular through the free-boundary extension approach.
  • Flexible tubular and toroidal target domains become available for downstream surface processing tasks.
  • Topology of the original tube is preserved under the toroidal bending maps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The resulting maps could support more accurate finite-element simulations on tubular medical structures by keeping circumferential directions aligned.
  • Engineering applications such as stress analysis on cylindrical shells might benefit from the topology-preserving domains without requiring additional periodic cuts.
  • Integration with existing texture-mapping pipelines could become simpler because the 3D tubular output already encodes natural longitudinal and angular coordinates.

Load-bearing premise

The input surfaces are open tube-like meshes with exactly two boundary components that admit a suitable cut allowing a disk-to-rectangle conformal map to be computed and lifted to a 3D tubular domain without introducing uncorrectable global distortion.

What would settle it

A concrete falsifier would be a tube-like surface on which the lifted parameterization shows global distortion metrics that remain high after the localized quasi-conformal correction is applied, or on which the derived toroidal bending maps fail to remain conformal.

Figures

Figures reproduced from arXiv: 2605.16305 by Gary P. T. Choi, Shunyu Yao.

Figure 1
Figure 1. Figure 1: Overview of the proposed conformal tubular parameterization and toroidal bending framework. Top: an input tube-like surface is mapped to a tubular parameter domain and can then be conformally bent into toroidal geometries. Bottom: the localized seam correction reduces the distortion concentrated near the cut seam, as illustrated by the distortion heat map. In this work, we propose a flexible conformal para… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustration of distortion correction by quasi-conformal composition. The map f transforms an infinitesimal circle into an ellipse with principal stretches |fz|(1 ± |µf |). Choose a quasi-conformal map g with µg ≡ µf−1 cancels the anisotropy so that the composition satisfies µg◦f = 0. Computation of quasi-conformal maps By the Measurable Riemann Mapping The￾orem [33], a quasi-conformal map exists… view at source ↗
Figure 3
Figure 3. Figure 3: Pipeline of the initial tubular parameterization. Starting from an open surface with two boundary loops (colored in blue and green), we cut the mesh along a shortest cut path (colored in red), compute a disk harmonic map ϕ, conformally transform it to a rectangle via ψL∗ , and then lift it to a 3D tube via α. Initial tubular parameterization Our initial tubular parameterization builds on the rectangular co… view at source ↗
Figure 4
Figure 4. Figure 4: Quasi-conformal correction for the initial tubular parameterization. The initial tubular parameterization η is first transferred to the annulus A via β −1 . A strip neighborhood (colored in yellow) of the cut seam is then corrected by a localized LBS map ζ, while the rest of the annulus is kept fixed. Finally, the corrected annulus parameterization is mapped back to the tube via β. where NV (i) denotes the… view at source ↗
Figure 5
Figure 5. Figure 5: Pipeline of the free-boundary tubular parameterization. Boundary loops are extended and smoothed to form an augmented mesh Mext (extended part is colored in orange), the fixed-boundary parameterization Φfix is applied on Mext and the result is finally restricted back to the original mesh M. Raw extension of boundary loops Let the two boundary loops be ∂M0 and ∂M1. For each boundary loop, we iteratively app… view at source ↗
Figure 6
Figure 6. Figure 6: Raw boundary extension on a boundary loop. At each boundary vertex, an outward direction di is estimated from local tangent ti , normal ni and inward ui . where ni is an area-weighted incident face normal vector. Since bei may point inward or outward depending on local orientation, we correct its sign by bi = ( −bei , if bei · ui > 0, bei , otherwise. We then combine the lateral with normal components to i… view at source ↗
Figure 7
Figure 7. Figure 7: Cycle-Laplacian smoothing of the raw extension. The smoothed ring balances data fidelity, anchor regularization, and cycle smoothness, thereby suppressing high-frequency zigzags. where ω is a weighting coefficient. This is a convex quadratic problem, and hence imposing the first-order optimality condition ∇Esmooth = 0 yields a sparse symmetric linear system (I + ωLcycle) x = x raw , where the i-th row of x… view at source ↗
Figure 8
Figure 8. Figure 8: Examples of conformal toroidal bending using the two different bend￾ing maps. The results are obtained by full wrapping along the major circle Ψmaj and full wrapping along the minor circle Ψmin, respectively. where E = θ 2 u + (R + cos θ) 2ϕ 2 u , F = θuθz + (R + cos θ) 2ϕuϕz, G = θ 2 z + (R + cos θ) 2ϕ 2 z . To construct a conformal bending map, we impose E = G and F = 0 [38]. A tractable choice is to ass… view at source ↗
Figure 9
Figure 9. Figure 9: Representative input surfaces and corresponding mapping results from the synthetic and real datasets. Here, we illustrate several examples from a real vascular structure dataset (left) and a synthetic dataset (right), together with their parameterization results under the proposed framework. ble 1). This unified experiment simultaneously serves as an ablation of the correction step and as a sensitivity ana… view at source ↗
read the original abstract

Tube-like surfaces are widely encountered in geometry processing, engineering structures, and medical anatomy, yet their intrinsic longitudinal and circumferential topology is not well preserved by conventional planar annular or rectangular parameterization domains. In this work, we propose a conformal parameterization framework for open tube-like surfaces with two boundary components. The proposed method first constructs a fixed-boundary tubular parameterization by cutting the input mesh, computing a disk-to-rectangle conformal map, and lifting the result to a three-dimensional tubular domain. To reduce residual distortion introduced near the cut seam, we further introduce a localized quasi-conformal correction scheme formulated on an annular domain, which improves conformality while leaving regions away from the seam unchanged. To handle noisy or irregular input boundaries, we also develop a free-boundary variant based on boundary extension and cycle-Laplacian smoothing, allowing the prescribed boundary constraints to be imposed on artificial outer rings rather than directly on the original surface. Finally, we derive two conformal toroidal bending maps that transform the tubular parameterization into toroidal geometries while preserving the underlying tube topology. Experiments on synthetic tube meshes and real vascular surfaces demonstrate that the proposed framework produces low-distortion parameterizations, effectively mitigates seam-induced artifacts, improves robustness for boundary-noisy inputs, and provides flexible tubular and toroidal target domains for downstream surface processing tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a conformal parameterization framework for open tube-like surfaces with two boundary components. It first constructs a fixed-boundary tubular parameterization by cutting the mesh, computing a disk-to-rectangle conformal map, and lifting the result to a 3D tubular domain. A localized quasi-conformal correction on an annular domain is then applied near the seam to mitigate artifacts while leaving distant regions unchanged. A free-boundary variant uses boundary extension and cycle-Laplacian smoothing for noisy inputs. Finally, two conformal toroidal bending maps are derived to produce toroidal target domains. Experiments on synthetic tube meshes and real vascular surfaces are reported to show low distortion, seam mitigation, and robustness.

Significance. If the low-distortion claims hold with quantitative support, the work would provide a practical tool for topology-preserving parameterization of tube-like surfaces in geometry processing and medical applications such as vascular modeling, where standard planar domains fail to respect longitudinal and circumferential structure.

major comments (2)
  1. [§3] §3 (Fixed-boundary tubular parameterization): the localized quasi-conformal correction is formulated only on an annular domain near the seam and explicitly leaves regions away from the seam unchanged. The low global distortion claim therefore depends on the initial disk-to-rectangle conformal map already having low distortion for the chosen cut; no analysis or experiments quantifying residual distortion as a function of cut placement are provided, making this assumption load-bearing for both fixed- and free-boundary variants.
  2. [Experiments] Experiments section: the reported results claim low distortion and effective mitigation of seam artifacts on synthetic and vascular meshes, yet no quantitative metrics (e.g., conformal distortion energy, angle or area error tables, or comparisons against baselines) or sensitivity analysis to cut choice appear in the evaluation, weakening support for the central claims.
minor comments (2)
  1. [§3.2] Clarify the precise definition and construction of the annular domain used for the quasi-conformal correction step, including how the seam is identified and mapped.
  2. [§4] The toroidal bending maps are stated to be conformal and topology-preserving; include a brief derivation sketch or reference to the underlying complex analysis to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of our claims regarding distortion and evaluation. We address each major comment below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [§3] §3 (Fixed-boundary tubular parameterization): the localized quasi-conformal correction is formulated only on an annular domain near the seam and explicitly leaves regions away from the seam unchanged. The low global distortion claim therefore depends on the initial disk-to-rectangle conformal map already having low distortion for the chosen cut; no analysis or experiments quantifying residual distortion as a function of cut placement are provided, making this assumption load-bearing for both fixed- and free-boundary variants.

    Authors: We agree that the global distortion properties rely in part on the quality of the initial disk-to-rectangle conformal map away from the seam. The localized correction is intentionally restricted to the annular region to preserve the conformal property elsewhere, but we acknowledge that explicit quantification of residual distortion versus cut placement would better support the robustness claim. In the revised manuscript we will add a sensitivity study on synthetic tube meshes using several representative cut placements, reporting both local and global distortion measures to demonstrate that the framework remains effective for standard cut choices. revision: yes

  2. Referee: [Experiments] Experiments section: the reported results claim low distortion and effective mitigation of seam artifacts on synthetic and vascular meshes, yet no quantitative metrics (e.g., conformal distortion energy, angle or area error tables, or comparisons against baselines) or sensitivity analysis to cut choice appear in the evaluation, weakening support for the central claims.

    Authors: The current evaluation relies primarily on visual inspection of distortion and seam artifacts. To provide stronger quantitative evidence, the revised version will include tables reporting conformal distortion energy, maximum angle and area distortion, and comparisons against baseline methods such as direct disk-to-rectangle conformal mapping without the localized correction. The sensitivity analysis to cut placement requested in the previous comment will also be incorporated into the experiments section. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's pipeline applies standard disk-to-rectangle conformal mapping after a cut, lifts the result to a 3D tubular domain, applies localized quasi-conformal correction on an annular domain, and derives toroidal bending maps that preserve topology. These steps rely on established conformal and quasi-conformal techniques with explicit correction formulations that do not define output distortion or low-distortion claims in terms of the method's own fitted parameters or inputs. No load-bearing self-citations, self-definitional reductions, or ansatzes smuggled via prior work are present; the central claims rest on independent experimental validation rather than tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework relies on existence of conformal maps for simply connected domains and standard mesh processing operations; no new physical entities or heavily fitted parameters are introduced in the abstract description.

axioms (2)
  • standard math Conformal maps exist and can be computed between disk-like domains and rectangles for suitable meshes.
    Invoked in the initial disk-to-rectangle conformal map step described in the abstract.
  • domain assumption Quasi-conformal maps can be localized to annular domains to correct seam distortion without affecting distant regions.
    Central to the correction scheme; treated as a workable property of the formulation.

pith-pipeline@v0.9.0 · 5764 in / 1495 out tokens · 65066 ms · 2026-05-20T23:57:17.220119+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Surface parameterization: a tutorial and survey,

    M. S. Floater and K. Hormann, “Surface parameterization: a tutorial and survey,” in Advances in Multiresolution for Geometric Modelling(N. A. Dodgson, M. S. Floater, and M. A. Sabin, eds.), (Berlin, Heidelberg), pp. 157–186, Springer Berlin Heidel- berg, 2005

    work page 2005

  2. [2]

    Mesh parameterization methods and their ap- plications,

    A. Sheffer, E. Praun, and K. Rose, “Mesh parameterization methods and their ap- plications,”Found. Trends Comput. Graph. Vis., vol. 2, no. 2, pp. 105–171, 2006

    work page 2006

  3. [3]

    How to draw a graph,

    W. T. Tutte, “How to draw a graph,”Proc. London Math. Soc., vol. s3-13, no. 1, pp. 743–767, 1963

    work page 1963

  4. [4]

    Parametrization and smooth approximation of surface triangula- tions,

    M. S. Floater, “Parametrization and smooth approximation of surface triangula- tions,”Comput. Aided Geom. Des., vol. 14, no. 3, pp. 231–250, 1997

    work page 1997

  5. [5]

    MIPS: An efficient global parametrization method,

    K. Hormann and G. Greiner, “MIPS: An efficient global parametrization method,” inCurve and Surface Design: Saint-Malo 1999(P.-J. Laurent, P. Sablonni` ere, and L. L. Schumaker, eds.), pp. 153–162, Nashville, TN: Vanderbilt University Press, 2000

    work page 1999

  6. [6]

    Intrinsic parameterizations of surface meshes,

    M. Desbrun, M. Meyer, and P. Alliez, “Intrinsic parameterizations of surface meshes,”Comput. Graph. Forum, vol. 21, no. 3, pp. 209–218, 2002

    work page 2002

  7. [7]

    Least squares conformal maps for automatic texture atlas generation,

    B. L´ evy, S. Petitjean, N. Ray, and J. Maillot, “Least squares conformal maps for automatic texture atlas generation,”ACM Trans. Graph., vol. 21, no. 3, pp. 362–371, 2002

    work page 2002

  8. [8]

    ABF++: fast and robust angle based flattening,

    A. Sheffer, B. L´ evy, M. Mogilnitsky, and A. Bogomyakov, “ABF++: fast and robust angle based flattening,”ACM Trans. Graph., vol. 24, no. 2, p. 311–330, 2005

    work page 2005

  9. [9]

    A local/global approach to mesh parameterization,

    L. Liu, L. Zhang, Y. Xu, C. Gotsman, and S. J. Gortler, “A local/global approach to mesh parameterization,” inProceedings of the Symposium on Geometry Processing, SGP ’08, (Goslar, DEU), pp. 1495–1504, Eurographics Association, 2008

    work page 2008

  10. [10]

    Locally injective parametrization with arbitrary fixed boundaries,

    O. Weber and D. Zorin, “Locally injective parametrization with arbitrary fixed boundaries,”ACM Trans. Graph., vol. 33, no. 4, 2014

    work page 2014

  11. [11]

    Scalable locally injective mappings,

    M. Rabinovich, R. Poranne, D. Panozzo, and O. Sorkine-Hornung, “Scalable locally injective mappings,”ACM Trans. Graph., vol. 36, no. 4, p. 1, 2017. 25

    work page 2017

  12. [12]

    Con- formal surface parameterization for texture mapping,

    S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle, “Con- formal surface parameterization for texture mapping,”IEEE Trans. Vis. Comput. Graph., vol. 6, no. 2, pp. 181–189, 2000

    work page 2000

  13. [13]

    Opti- mized conformal surface registration with shape-based landmark matching,

    L. M. Lui, S. Thiruvenkadam, Y. Wang, P. M. Thompson, and T. F. Chan, “Opti- mized conformal surface registration with shape-based landmark matching,”SIAM J. Imaging Sci., vol. 3, no. 1, pp. 52–78, 2010

    work page 2010

  14. [14]

    Conformal virtual colon flatten- ing,

    W. Hong, X. Gu, F. Qiu, M. Jin, and A. Kaufman, “Conformal virtual colon flatten- ing,” inProceedings of the 2006 ACM Symposium on Solid and Physical Modeling, SPM ’06, (New York, NY, USA), pp. 85–93, Association for Computing Machinery, 2006

    work page 2006

  15. [15]

    Nondistorting flat- tening maps and the 3-D visualization of colon CT images,

    S. Haker, S. B. Angenent, A. R. Tannenbaum, and R. Kikinis, “Nondistorting flat- tening maps and the 3-D visualization of colon CT images,”IEEE Trans. Med. Imaging, vol. 19, pp. 665–670, 2000

    work page 2000

  16. [16]

    Genus zero surface con- formal mapping and its application to brain surface mapping,

    X. Gu, Y. Wang, T. Chan, P. Thompson, and S.-T. Yau, “Genus zero surface con- formal mapping and its application to brain surface mapping,”IEEE Trans. Med. Imaging, vol. 23, no. 8, pp. 949–958, 2004

    work page 2004

  17. [17]

    Brain surface conformal parameterization using Riemann sur- face structure,

    Y. Wang, L. M. Lui, X. Gu, K. M. Hayashi, T. F. Chan, A. W. Toga, P. M. Thomp- son, and S.-T. Yau, “Brain surface conformal parameterization using Riemann sur- face structure,”IEEE Trans. Med. Imaging, vol. 26, no. 6, pp. 853–865, 2007

    work page 2007

  18. [18]

    Boundary first flattening,

    R. Sawhney and K. Crane, “Boundary first flattening,”ACM Trans. Graph., vol. 37, no. 1, 2017

    work page 2017

  19. [19]

    Parallelizable global conformal parameterization of simply-connected surfaces via partial welding,

    G. P. T. Choi, Y. Leung-Liu, X. Gu, and L. M. Lui, “Parallelizable global conformal parameterization of simply-connected surfaces via partial welding,”SIAM J. Imaging Sci., vol. 13, no. 3, pp. 1049–1083, 2020

    work page 2020

  20. [20]

    A constructive algorithm for disk conformal parame- terizations,

    W.-H. Liao and M.-H. Yueh, “A constructive algorithm for disk conformal parame- terizations,”J. Sci. Comput., vol. 92, no. 2, p. 40, 2022

    work page 2022

  21. [21]

    Hemispheroidal parameterization and harmonic de- composition of simply connected open surfaces,

    G. P. T. Choi and M. Shaqfa, “Hemispheroidal parameterization and harmonic de- composition of simply connected open surfaces,”J. Comput. Appl. Math., vol. 461, p. 116455, 2025

    work page 2025

  22. [22]

    Spherical parameterization balancing angle and area distortions,

    S. Nadeem, Z. Su, W. Zeng, A. Kaufman, and X. Gu, “Spherical parameterization balancing angle and area distortions,”IEEE Trans. Vis. Comput. Graph., vol. 23, no. 6, pp. 1663–1676, 2016

    work page 2016

  23. [23]

    Advanced hierarchical spherical parameterizations,

    X. Hu, X.-M. Fu, and L. Liu, “Advanced hierarchical spherical parameterizations,” IEEE Trans. Vis. Comput. Graph., vol. 24, no. 6, pp. 1930–1941, 2017

    work page 1930

  24. [24]

    A novel local/global approach to spher- ical parameterization,

    Z. Wang, Z. Luo, J. Zhang, and E. Saucan, “A novel local/global approach to spher- ical parameterization,”J. Comput. Appl. Math., vol. 329, pp. 294–306, 2018. 26

    work page 2018

  25. [25]

    Convergence of Dirichlet energy minimization for spherical conformal parameterizations,

    W.-H. Liao, T.-M. Huang, W.-W. Lin, and M.-H. Yueh, “Convergence of Dirichlet energy minimization for spherical conformal parameterizations,”J. Sci. Comput., vol. 98, no. 1, p. 29, 2024

    work page 2024

  26. [26]

    Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces,

    G. P. T. Choi, “Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces,”J. Comput. Appl. Math., vol. 447, p. 115888, 2024

    work page 2024

  27. [27]

    Riemannian gradient descent for spherical area-preserving mappings,

    M. Sutti and M.-H. Yueh, “Riemannian gradient descent for spherical area-preserving mappings,”AIMS Math., vol. 9, no. 7, p. 19414, 2024

    work page 2024

  28. [28]

    Spherical density-equalizing map for genus-0 closed surfaces,

    Z. Lyu, L. M. Lui, and G. P. T. Choi, “Spherical density-equalizing map for genus-0 closed surfaces,”SIAM J. Imaging Sci., vol. 17, no. 4, pp. 2110–2141, 2024

    work page 2024

  29. [29]

    A new efficient algorithm for volume-preserving parameterizations of genus-one 3-manifolds,

    M.-H. Yueh, T. Li, W.-W. Lin, and S.-T. Yau, “A new efficient algorithm for volume-preserving parameterizations of genus-one 3-manifolds,”SIAM J. Imaging Sci., vol. 13, no. 3, pp. 1536–1564, 2020

    work page 2020

  30. [30]

    Toroidal density-equalizing map for genus-one surfaces,

    S. Yao and G. P. T. Choi, “Toroidal density-equalizing map for genus-one surfaces,” J. Comput. Appl. Math., vol. 472, p. 116844, 2026

    work page 2026

  31. [31]

    Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory,

    G. P. T. Choi, “Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory,”J. Sci. Comput., vol. 87, no. 3, p. 70, 2021

    work page 2021

  32. [32]

    TEMPO: Feature-endowed Teichm¨ uller extremal mappings of point clouds,

    T. W. Meng, G. P.-T. Choi, and L. M. Lui, “TEMPO: Feature-endowed Teichm¨ uller extremal mappings of point clouds,”SIAM J. Imaging Sci., vol. 9, no. 4, pp. 1922– 1962, 2016

    work page 1922

  33. [33]

    Riemann’s mapping theorem for variable metrics,

    L. Ahlfors and L. Bers, “Riemann’s mapping theorem for variable metrics,”Ann. Math., vol. 72, no. 2, pp. 385–404, 1960

    work page 1960

  34. [34]

    Texture map and video compression using Beltrami representation,

    L. M. Lui, K. C. Lam, T. W. Wong, and X. Gu, “Texture map and video compression using Beltrami representation,”SIAM J. Imaging Sci., vol. 6, no. 4, pp. 1880–1902, 2013

    work page 1902

  35. [35]

    A note on two problems in connexion with graphs,

    E. W. Dijkstra, “A note on two problems in connexion with graphs,”Numer. Math., vol. 1, no. 1, p. 269–271, 1959

    work page 1959

  36. [36]

    Setting the boundary free: a composite ap- proach to surface parameterization,

    R. Zayer, C. R¨ ossl, and H.-P. Seidel, “Setting the boundary free: a composite ap- proach to surface parameterization,” inProceedings of the Third Eurographics Sym- posium on Geometry Processing, SGP ’05, (Goslar, DEU), p. 91–es, Eurographics Association, 2005

    work page 2005

  37. [37]

    A signal processing approach to fair surface design,

    G. Taubin, “A signal processing approach to fair surface design,” inProceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’95, (New York, NY, USA), pp. 351–358, Association for Computing Machinery, 1995

    work page 1995

  38. [38]

    Global Conformal Surface Parameterization,

    X. Gu and S.-T. Yau, “Global Conformal Surface Parameterization,” inEurographics Symposium on Geometry Processing(L. Kobbelt, P. Schroeder, and H. Hoppe, eds.), The Eurographics Association, 2003. 27

    work page 2003

  39. [39]

    Superfluid vortex dynamics on a torus and other toroidal surfaces of revolution,

    N.-E. Guenther, P. Massignan, and A. L. Fetter, “Superfluid vortex dynamics on a torus and other toroidal surfaces of revolution,”Phys. Rev. A, vol. 101, p. 053606, 2020

    work page 2020

  40. [40]

    The vascular model repository: A public resource of medical imaging data and blood flow simulation results,

    N. M. Wilson, A. K. Ortiz, and A. B. Johnson, “The vascular model repository: A public resource of medical imaging data and blood flow simulation results,”J. Med. Devices, vol. 7, no. 4, p. 040923, 2013. 28

    work page 2013