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Documents authored by Shewchuk, Jonathan Richard

Document
DOI: 10.4230/LIPIcs.SoCG.2026.90
Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells

Authors: Jonathan Richard Shewchuk

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The restricted Delaunay triangulation of a closed surface Σ and a finite point set V ⊂ Σ is a subcomplex of the Delaunay tetrahedralization of V whose triangles approximate Σ. It is well known that if V is a sufficiently dense sample of a smooth Σ, then the union of the restricted Delaunay triangles is homeomorphic to Σ. We show that an ε-sample with ε ≤ 0.3245 suffices. By comparison, Dey proves it for a 0.18-sample; our improved sampling bound reduces the number of sample points required by a factor of 3.25. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of 21. The first step of our homeomorphism proof is particularly interesting: we show that for a 0.44-sample, the restricted Voronoi cell of each site v ∈ V is homeomorphic to a disk, and the orthogonal projection of the cell onto T_vΣ (the plane tangent to Σ at v) is star-shaped.
Cite as

Jonathan Richard Shewchuk. Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 90:1-90:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shewchuk:LIPIcs.SoCG.2026.90,
  author =	{Shewchuk, Jonathan Richard},
  title =	{{Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{90:1--90:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.90},
  URN =		{urn:nbn:de:0030-drops-258961},
  doi =		{10.4230/LIPIcs.SoCG.2026.90},
  annote =	{Keywords: Restricted Delaunay triangulation, restricted Voronoi diagram, surface sampling, surface mesh generation, surface reconstruction, \epsilon-sample, homeomorphism}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.91
The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks

Authors: Jonathan Richard Shewchuk and Sagnik Bhattacharya

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A linear neural network computes a linear transformation of its input vector. Given a fully-connected linear network, the set of all weight vectors for which the network computes the same linear transformation is an algebraic variety in weight space, called a fiber under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into manifolds of various dimensions called strata. We characterize how these strata are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element. Our main result describes the relationship between this partial order and the ranks of certain matrices in the network. Each stratum represents a different pattern of information flow through the network, expressed as a barcode. Connections among the strata are best understood through simple transformations of the barcodes called barcode moves.
Cite as

Jonathan Richard Shewchuk and Sagnik Bhattacharya. The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 91:1-91:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shewchuk_et_al:LIPIcs.SoCG.2026.91,
  author =	{Shewchuk, Jonathan Richard and Bhattacharya, Sagnik},
  title =	{{The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{91:1--91:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.91},
  URN =		{urn:nbn:de:0030-drops-258971},
  doi =		{10.4230/LIPIcs.SoCG.2026.91},
  annote =	{Keywords: Linear neural network, real algebraic variety, stratification, multilinear algebra, product of matrices, persistence barcode, real algebraic geometry, discrete geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.49
Restricted Constrained Delaunay Triangulations

Authors: Marc Khoury and Jonathan Richard Shewchuk

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We introduce the restricted constrained Delaunay triangulation (restricted CDT), a generalization of both the restricted Delaunay triangulation and the constrained Delaunay triangulation. The restricted CDT is a triangulation of a surface whose edges include a set of user-specified constraining segments. We define the restricted CDT to be the dual of a restricted Voronoi diagram defined on a surface that we have extended by topological surgery. We prove several properties of restricted CDTs, including sampling conditions under which the restricted CDT contains every constraining segment and is homeomorphic to the underlying surface.
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Marc Khoury and Jonathan Richard Shewchuk. Restricted Constrained Delaunay Triangulations. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{khoury_et_al:LIPIcs.SoCG.2021.49,
  author =	{Khoury, Marc and Shewchuk, Jonathan Richard},
  title =	{{Restricted Constrained Delaunay Triangulations}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{49:1--49:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.49},
  URN =		{urn:nbn:de:0030-drops-138481},
  doi =		{10.4230/LIPIcs.SoCG.2021.49},
  annote =	{Keywords: restricted Delaunay triangulation, constrained Delaunay triangulation, surface meshing, surface reconstruction, topological surgery, portals}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.47
Fixed Points of the Restricted Delaunay Triangulation Operator

Authors: Marc Khoury and Jonathan Richard Shewchuk

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
The restricted Delaunay triangulation can be conceived as an operator that takes as input a k-manifold (typically smooth) embedded in R^d and a set of points sampled with sufficient density on that manifold, and produces as output a k-dimensional triangulation of the manifold, the input points serving as its vertices. What happens if we feed that triangulation back into the operator, replacing the original manifold, while retaining the same set of input points? If k = 2 and the sample points are sufficiently dense, we obtain another triangulation of the manifold. Iterating this process, we soon reach an iteration for which the input and output triangulations are the same. We call this triangulation a fixed point of the restricted Delaunay triangulation operator. With this observation, and a new test for distinguishing "critical points" near the manifold from those near its medial axis, we develop a provably good surface reconstruction algorithm for R^3 with unusually modest sampling requirements. We develop a similar algorithm for constructing a simplicial complex that models a 2-manifold embedded in a high-dimensional space R^d, also with modest sampling requirements (especially compared to algorithms that depend on sliver exudation). The latter algorithm builds a non-manifold representation similar to the flow complex, but made solely of Delaunay simplices. The algorithm avoids the curse of dimensionality: its running time is polynomial, not exponential, in d.
Cite as

Marc Khoury and Jonathan Richard Shewchuk. Fixed Points of the Restricted Delaunay Triangulation Operator. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{khoury_et_al:LIPIcs.SoCG.2016.47,
  author =	{Khoury, Marc and Shewchuk, Jonathan Richard},
  title =	{{Fixed Points of the Restricted Delaunay Triangulation Operator}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{47:1--47:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.47},
  URN =		{urn:nbn:de:0030-drops-59396},
  doi =		{10.4230/LIPIcs.SoCG.2016.47},
  annote =	{Keywords: restricted Delaunay triangulation, fixed point, manifold reconstruction, surface reconstruction, computational geometry}
}
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