Vietoris--Rips Shadow for Euclidean Graph Reconstruction

Atish Mitra; Rafal Komendarczyk; Sushovan Majhi

REVIEW 3 major objections 4 minor 16 references

Vietoris--Rips Shadow for Euclidean Graph Reconstruction

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A Vietoris–Rips complex built from the epsilon-path metric recovers the homotopy type of a Euclidean graph from any sufficiently close sample, and in the plane its shadow is both homotopy equivalent and Hausdorff-close to the graph.

desk verdict Useful extension of graph reconstruction results, but Theorem 3.1's surjectivity proof rests on an unproved equality that looks false as stated, and Theorem 5.9 inherits the gap. read the letter →

arxiv 2506.01603 v3 pith:FWH2RNO6 submitted 2025-06-02 math.AT cs.CG

Rafal Komendarczyk , Sushovan Majhi , Atish Mitra This is my paper · ORCID

classification math.ATcs.CG MSC 55U1055P1005C10

keywords Vietoris–Ripscomplexgraphreconstructionshadowepsilon-pathmetriclarge-scaledistortionhomotopyequivalenceHausdorffdistancetopologicaldataanalysis

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 proves that the Vietoris–Rips complex of a noisy Euclidean sample, built with a path-based metric rather than the plain Euclidean distance, recovers the full homotopy type of an underlying embedded graph, provided the sample is Hausdorff-close and the Rips scale is chosen below a quarter of the graph's systole. In the plane, it goes further: the geometric shadow of that complex is itself homotopy equivalent to the graph and stays Hausdorff-close to it. The payoff is a quantitative recipe for choosing the sampling density and Rips scale from two graph invariants, the shortest loop length and the large-scale distortion, with no reliance on reach or curvature bounds that vanish at graph vertices.

What carries the argument

The load-bearing construction is the $\varepsilon$-path metric $d^\varepsilon_S$, which measures distance between sample points by the infimum length of chains whose consecutive points are within Euclidean distance $\varepsilon$. Because short Euclidean jumps erase the spurious cycles that ordinary Euclidean Rips complexes create near graph vertices, this metric lets the Rips complex track the graph's intrinsic geometry. The proof is carried by three supporting mechanisms: the large-scale distortion $\delta^\varepsilon_\beta(G)$ (the worst ratio between intrinsic distance in $G$ and in its $\varepsilon$-thickening, taken over pairs at distance at least $\beta$), a Hausmann-type theorem identifying the length-metric Rips complex of $G$ with $G$ itself at scales below $\ell(G)/3$ via circumcenters, and, for the planar shadow statement, the shadow radius $\Delta(G)$ together with a lifting condition on intersecting edges that guarantees the shadow projection induces a $\pi_1$-epimorphism.

What would settle it

Take $G$ to be a circle of circumference larger than 4 and pick a sample $S$ of points slightly offset from the circle so that the nearest-point round trip $\Phi(\Psi(s))$ never equals $s$ but $d_H(G,S)$ is below the theorem's $\tfrac12 \xi\varepsilon$ threshold. If the resulting $R^\varepsilon_\beta(S)$ has a fundamental group different from $\mathbb{Z}$, or if the map induced on $\pi_1$ by the shadow fails to be surjective, the theorem's conclusion fails; the equality used in the surjectivity proof can be checked directly on the vertices of any triangulation of a sphere mapped into $S$.

Watch

Extended reading notes

Core claim

The central result is Theorem 3.1: for a compact connected metric graph $G \subset \mathbb{R}^N$, any $\xi \in (0,\frac14)$, any $\beta < \ell(G)/4$, and any $\varepsilon \leq \beta/3$ with large-scale distortion $\delta^\varepsilon_\beta(G) \leq \frac{1+2\xi}{1+\xi}$, every sample $S$ with $d_H(G,S) < \tfrac12 \xi\varepsilon$ satisfies $R^\varepsilon_\beta(S) \simeq G$. The proof builds simplicial maps between the length-metric Rips complex of $G$ and the $\varepsilon$-path Rips complex of $S$, uses circumcenters to show the maps are contiguous to inclusions, and invokes Whitehead's theorem. The companion geometric result, Theorem 5.9, restricts to planar graphs with strictly positive angles between incident edges and shows that the shadow $\mathcal{S}(R^\varepsilon_\beta(S))$ is homotopy equivalent to $G$ and lies within Hausdorff distance $\beta + \tfrac12 \xi\varepsilon$ of $G$.

Load-bearing premise

The surjectivity argument assumes that for every sample point $s$ used in a sphere triangulation, the nearest-point projection from $S$ to $G$ followed by the nearest-point projection from $G$ back to $S$ returns $s$ itself; the proof's displacement bounds only guarantee these points are close, not equal.

Editorial extensions

If this is right

For any compact connected Euclidean graph, the homotopy type is recoverable from a noisy sample whenever the sampling noise is a small fraction of the chosen $\varepsilon$, with $\varepsilon$ and $\beta$ fixed by the systole and the large-scale distortion alone.
Graphs with cusps or zero vertex angles, which have vanishing $\mu$-reach and lie outside classical reach-based sampling theory, are still covered by the homotopy reconstruction.
In the plane, the reconstructed object is not just an abstract complex but an embedded polyhedral shadow that is simultaneously homotopy equivalent and Hausdorff-close to the original graph.
The $\varepsilon$-path metric removes the spurious small loops that the Euclidean Vietoris–Rips complex inevitably creates near vertices, so the correct homotopy type appears at much simpler scales.
Because the shadow approach reproduces the planar Euclidean Vietoris–Rips $\pi_0$ and $\pi_1$ isomorphism results in the path-metric setting, it supplies an embedded geometric reconstruction rather than a purely homological one.

Reading between the lines

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

If the hidden premise that the nearest-point maps $\Phi$ and $\Psi$ are mutual inverses on the vertices of sphere triangulations fails in some sample, the surjectivity half of Theorem 3.1 would need a density or separation condition on $S$; the stated Hausdorff bound alone may be insufficient.
The planar geometric reconstruction suggests a practical pipeline: compute the $\varepsilon$-path Rips complex, take its shadow, and then simplify the shadow's medial axis to obtain a one-dimensional proxy, which the paper notes is itself a geometric graph in the planar case.
A natural next stress test is three-dimensional graphs, where the shadow projection is known not to behave as well; the paper's $\pi_1$ lifting condition would need a new mechanism because the planar Jordan-curve arguments in Proposition 5.5 do not transfer.
The large-scale distortion is monotone in $\varepsilon$ and tends to 1 as $\varepsilon$ goes to 0, so the theorem's sampling condition is always satisfiable in principle; this also suggests that in practice one could estimate $\delta^\varepsilon_\beta(G)$ from the sample itself and adapt $\varepsilon$ until the inequality holds.

Share X Bluesky LinkedIn Reddit HN

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

First thing you should know: the homotopy reconstruction theorem (3.1) is not proven as written. The gap is in the surjectivity half. They define eg on the barycentric subdivision using circumcenters, then try to apply Lemma A.2 with f = g and the second map = Φ∘eg. Lemma A.2 requires g(v) = (Φ∘eg)(v) for every vertex v of K. For a vertex, eg(v) = Ψ(g(v)), so the condition is Φ(Ψ(g(v))) = g(v). The proof asserts this equality without argument. The maps Φ and Ψ are only defined by displacement bounds; they are nearest-point-like correspondences between G and S, and composing them will generically move points. The Hausdorff condition gives ||Φ(Ψ(s)) − s|| < ξε, not equality. And the mixed-distance estimate for the simplex condition (2) also needs that equality. So the Whitehead step doesn't close. Since Theorem 5.9 imports Theorem 3.1, the geometric reconstruction theorem inherits the gap.

Now the credit. The paper is not a repackaging. The extension of the Chambers–de Silva–Erickson–Ghrist π1-shadow result from the Euclidean metric to the epsilon-path metric is a genuine step, and the move from global distortion to large-scale distortion is a real improvement for graphs with cusp-like features. The shadow radius Δ(G) is a new sampling parameter with real geometric content, and the quantitative bounds tying ε and β to the systole and distortion are exactly what a practitioner would want. The writing is generally careful, the appendix contains substantial proofs (Proposition 5.2 is serious), and the references to prior work look honest.

Soft spots beyond the main gap: Proposition 5.5's Case B invokes the Jordan curve theorem loosely—the epsilon-path pieces don't necessarily form a simple closed curve, and the crossing claim needs more care. Corollary 5.10 is stated with the proof deferred to an unnamed convexity argument, which is an incomplete proof, not a referee's gift. These are minor relative to the Theorem 3.1 issue.

Should a serious editor send this to referees? Yes. The results are new, the approach is plausible, and the gap, while load-bearing, may be repairable by strengthening the hypotheses or restructuring the surjectivity argument. But as it stands, the central result is not established, and I would not cite Theorem 3.1 or Theorem 5.9 in their current form.

Referee Report

3 major / 4 minor

Summary. The paper studies reconstruction of Euclidean metric graphs from Hausdorff-close samples using Vietoris–Rips complexes built with the ε-path metric. Theorem 3.1 claims that the abstract Vietoris–Rips complex R^ε_β(S) is homotopy equivalent to a compact connected graph G under quantitative conditions involving β, ε, the systole ℓ(G), and the large-scale distortion δ^ε_β(G). Theorem 5.9 claims that for planar graphs satisfying regularity conditions (A1)–(A4), the shadow S(R^ε_β(S)) is homotopy equivalent and Hausdorff-close to G, with explicit scale bounds. Section 4 develops a general lifting condition for the shadow projection to induce a π_1-epimorphism, and the appendix contains supporting geometric propositions.

Significance. If the main theorems are correct, the paper gives a useful extension of the Chambers–de Silva–Erickson–Ghrist shadow theorem from the Euclidean metric to path-based metrics and offers quantitative sampling conditions based on the systole and large-scale distortion rather than on global distortion or µ-reach. The statements are explicit and do not involve fitted constants, which is a genuine strength. However, the proof of Theorem 3.1 currently rests on an unjustified equality, and several steps in the geometric reconstruction section are not fully justified, so the contribution is not yet established.

major comments (3)

[Section 3, Theorem 3.1 proof, Surjectivity] The line "Since g(v_i)=(Φ∘eg)(v_i)" is asserted without proof. From the definition of eg on vertices, eg(v_i)=Ψ(g(v_i)), so the claimed equality is Φ(Ψ(g(v_i)))=g(v_i). The inequalities used to define Φ and Ψ only imply ‖Φ(Ψ(s))−s‖<ξε, which does not force equality. This equality is necessary for condition (1) of Lemma A.2 and for replacing the vertices in condition (2); without it, the set {g(v_0),...,g(v_l),(Φ∘eg)(bσ_l)} is not shown to be a simplex. Consequently the surjectivity of π_*(Φ) is not established, and since Theorem 5.9 invokes Theorem 3.1, the main homotopy reconstruction claim is unsupported. A repair would require selecting Φ and Ψ so that Φ∘Ψ fixes the finite set g(K^(0)), but this selection is not part of the current proof.
[Appendix, Proposition 5.5] The statement of Proposition 5.5 is incomplete: it uses points a,b∈G that are never defined, presumably the images of A,B under the nearest-point map Ψ. In the proof of Case B, the claim "The Jordan curve theorem implies... that the image of AB transversely intersects Q" is not justified. If the ε-path Q is disjoint from the path P∪AB, the fact that the segment CD intersects AB does not imply that C and D lie in different components of the complement of the Jordan curve P∪AB, because the curve need not be convex; hence Q need not cross AB. This step is essential for applying the shadow radius property, and therefore for Lemma 5.7 and Theorem 5.9.
[Section 5, Lemma 5.8] The simplicial map Ψ' on the shadow complex is not shown to be well-defined. For a shadow vertex P that is the transverse intersection of several pairs of edges, the definition Ψ'(P):=Ψ(A) depends on the arbitrarily chosen edge [AB], and no argument is given that the value is independent of this choice or that the resulting vertex map sends shadow edges to simplices of R^L_{6β}(G). In particular, for a shadow edge [U,V] with U,V∈S, the assertion that "there is nothing to show" assumes that [U,V] is an edge of R^ε_β(S), which is not a consequence of [U,V] being a shadow edge. This affects the injectivity half of the π_1-isomorphism claim used in Theorem 5.9.

minor comments (4)

[Appendix, Proposition 2.5 proof] The geodesic convexity of the metric ball B_G(c(A),r) for r<ℓ(G)/3 is stated without proof; a brief justification would help, since the statement is not immediate in graphs containing cycles.
[Section 4, Theorem 4.4 proof] The homotopy notation in the proof of Theorem 4.4, such as "AED≃(AE)D≃...", is nonstandard and the homotopies are not explicitly defined, which makes the argument difficult to verify.
[Appendix, Proposition 5.2 proof] The proof refers to items "[13(ii)]" and "[12(i)]" that do not correspond to numbered equations in the current manuscript; these references should be replaced by explicit equations or clarifications.
[Section 5, Corollary 5.10] The proof of Corollary 5.10 is only a sketch; in particular, the pointwise displacement bound for the linear extension of Ψ' to the shadow is not demonstrated, and the claim that the maps are the nearest-point correspondences requires more detail.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and no prediction reduces to a fitted input or to a self-citation chain.

full rationale

The paper's central results are derived from clearly stated geometric parameters of the hidden graph (systole, large-scale distortion, shadow radius) and from the Hausdorff distance between the graph and the sample. No parameter is fitted to the target reconstruction, and no equation identifies an output with an input by construction. The ε-path metric and large-scale distortion are taken from the authors' earlier work, but those notions are introduced with explicit definitions and the key comparison and distortion estimates are either proved in the appendix or are parameter-free external facts; thus the self-citations are not load-bearing in a circular way. The one serious concern in the surjectivity proof of Theorem 3.1 is the asserted equality g(v_i) = (Φ∘eg)(v_i), which is not justified by the displacement bounds. This is a genuine mathematical gap or hidden assumption about the behavior of nearest-point maps, but it is not a circularity: it does not make the theorem's conclusion equivalent to its assumptions by definition, and it involves no fitted parameter renamed as a prediction. Accordingly, the appropriate verdict is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper introduces no fitted numerical parameters. The quantities beta, epsilon, and xi are variables constrained by explicit inequalities, and ell(G), delta^epsilon_beta(G), Theta, and Delta(G) are determined by the graph G. The axioms are standard background results plus domain assumptions about the sampling condition and the epsilon-path metric, many borrowed from prior published work. The shadow radius and shadow complex are new mathematical constructs but are defined from the graph and the complex, not postulated ad hoc.

assumptions (6)

domain assumption Compactness and finiteness: G is compact with finitely many edges, so the systole ell(G) is positive.
Remark 2.3; used throughout to guarantee positive ell(G) and well-defined circumcenters.
domain assumption The epsilon-path metric d^epsilon_S is a metric on S under the sampling condition d_H(G,S) < xi epsilon / 2.
Definition 2.7 and Proposition A.1; needed for R^epsilon_beta(S) to be a Rips complex of a metric space.
domain assumption Large-scale distortion delta^epsilon_beta(G) is monotone in epsilon and anti-monotone in beta, and tends to 1 as epsilon tends to 0.
Properties stated in Section 2.5, taken from Komendarczyk-Majhi-Tran [11]; used to guarantee the existence of epsilon satisfying the theorem's bounds.
domain assumption Comparison estimate: for a,b in G and A,B in S with ||a-A||, ||b-B|| < xi epsilon / 2, one has ||A-B|| <= d^epsilon_S(A,B) <= d_G(a,b) + xi epsilon / (1 - xi).
Proposition 2.8, cited from Majhi [13]; a key bound in the simplicial map chain.
ad hoc to paper For planar graphs with properties (A1)-(A4), the shadow radius is positive: Delta(G) > 0.
Proposition 5.2 defines a new regularity constant; the proof relies on C^1 regularity and minimum incident angle, but the quantity itself is newly introduced for this paper.
standard math Standard algebraic topology tools: Whitehead theorem, Simplicial Approximation Theorem, Nerve lemma, Jordan curve theorem.
Used in proofs of Theorem 2.6, Theorem 3.1, and Proposition 5.5.

invented entities (2)

Shadow radius Delta(G)
purpose: New geometric parameter of a planar graph quantifying the scale below which points near a segment are geodesically close to its endpoints; used to set the admissible reconstruction scale beta.
Definition 5.1. Purely mathematical construct; its positivity is proved from assumptions (A1)-(A4), so it is not empirical.
Shadow complex SC(K)
purpose: A planar triangulation of the shadow of a complex K used to construct simplicial maps for injectivity of the shadow projection.
Definition 4.1 and Remark 4.2. A mathematical tool; uniqueness of the triangulation is not required but the proof relies on its existence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Vietoris--Rips Shadow for Euclidean Graph Reconstruction." pith.science (2026). https://pith.science/paper/FWH2RNO6

@misc{pith2026250601603,
  author       = {Pith},
  title        = {Pith review of: Vietoris--Rips Shadow for Euclidean Graph Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWH2RNO6}},
  note         = {Machine review of arXiv:2506.01603}
}

abstract

The shadow of an abstract simplicial complex $K$ with vertices in $\mathbb{R}^N$ is a subset of $\mathbb{R}^N$ defined as the union of the convex hulls of simplices of $K$. The Vietoris--Rips complex of a metric space $(S,d)$ at scale $\beta$ is an abstract simplicial complex whose each $k$-simplex corresponds to $(k+1)$ points of $S$ within diameter $\beta$. In case $S\subset\mathbb R^2$ and $d(a,b)=\|a-b\|$ the standard Euclidean metric, the natural shadow projection of the Vietoris--Rips complex is already proved by Chambers et al. to induce isomorphisms on $\pi_0$ and $\pi_1$. We extend the result beyond the standard Euclidean distance on $S\subset\mathbb R^N$ to a family of path-based metrics, $d^\varepsilon_{S}$. From the pairwise Euclidean distances of points in $S$, we introduce a family (parametrized by $\varepsilon$) of path-based Vietoris--Rips complexes $R^\varepsilon_\beta(S)$ for a scale $\beta>0$. If $S\subset\mathbb{R}^2$ is Hausdorff-close to a planar Euclidean graph $G$, we provide quantitative bounds on scales $\beta,\varepsilon$ for the shadow projection map of the Vietoris--Rips complex of $(S,d^\varepsilon_S)$ at scale $\beta$ to induce $\pi_1$-isomorphism. This paper first studies the homotopy-type recovery of $G\subset\mathbb R^N$ using the abstract Vietoris--Rips complex of a Hausdorff-close sample $S$ under the $d^\varepsilon_S$ metric. Then, our result on the $\pi_1$-isomorphism induced by the shadow projection lends itself to providing also a geometrically close embedding for the reconstruction. Based on the length of the shortest loop and large-scale distortion of the embedding of $G$, we quantify the choice of a suitable sample density $\varepsilon$ and a scale $\beta$ at which the shadow of $R^\varepsilon_\beta(S)$ is homotopy-equivalent and Hausdorff-close to $G$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

On Homotopy Types of Euclidean Rips Complexes.Discrete & Computational Geometry, 58(3):526–542, October 2017

Michał Adamaszek, Florian Frick, and Adrien Vakili. On Homotopy Types of Euclidean Rips Complexes.Discrete & Computational Geometry, 58(3):526–542, October 2017. Publisher: Springer New York LLC

work page 2017
[2]

Stability and computation of medial axes: a state-of-the-art report

Dominique Attali, Jean-Daniel Boissonnat, and Herbert Edelsbrunner. Stability and computation of medial axes: a state-of-the-art report. InMathematical foundations of scientific visualization, computer graphics, and massive data exploration, Math. Vis., pages 109–125. Springer, Berlin, 2009. EUCLIDEAN GRAPH RECONSTRUCTION 25

work page 2009
[3]

Vietoris-Rips complexes also provide topologically correct re- constructions of sampled shapes

Dominique Attali, Andr ´e Lieutier, and David Salinas. Vietoris-Rips complexes also provide topologically correct re- constructions of sampled shapes. InProceedings of the twenty-seventh annual symposium on Computational geometry, pages 491–500, 2011

work page 2011
[4]

Chambers, Vin de Silva, Jeff Erickson, and Robert Ghrist

Erin W. Chambers, Vin de Silva, Jeff Erickson, and Robert Ghrist. Vietoris–Rips complexes of planar point sets. Discrete & Computational Geometry, 44(1):75–90, Jul 2010

work page 2010
[5]

A sampling theory for compact sets in Euclidean space

Fr ´ed´eric Chazal, David Cohen-Steiner, and Andr´e Lieutier. A sampling theory for compact sets in Euclidean space. InProceedings of the twenty-second annual symposium on Computational geometry, pages 319–326, 2006

work page 2006
[6]

Mathematical theory of medial axis transform.Pacific J

Hyeong In Choi, Sung Woo Choi, and Hwan Pyo Moon. Mathematical theory of medial axis transform.Pacific J. Math., 181(1):57–88, 1997

work page 1997
[7]

Helly , Radon, and Carath´eodory type theorems

J ¨urgen Eckhoff. Helly , Radon, and Carath´eodory type theorems. InHandbook of Convex Geometry, pages 389–448. Elsevier, 1993

work page 1993
[8]

On the reconstruction of geodesic subspaces ofR n.International Journal of Computational Geometry & Applications, 32(01n02):91–117, 2022

Brittany Terese Fasy , Rafal Komendarczyk, Sushovan Majhi, and Carola Wenk. On the reconstruction of geodesic subspaces ofR n.International Journal of Computational Geometry & Applications, 32(01n02):91–117, 2022

work page 2022

Show all 16 references

[9]

On the Vietoris-Rips complexes and a cohomology theory for metric spaces.Annals of Mathematics Studies, 138:175–188, 1995

Jean-Claude Hausmann et al. On the Vietoris-Rips complexes and a cohomology theory for metric spaces.Annals of Mathematics Studies, 138:175–188, 1995

1995
[10]

Homotopy reconstruction via the ˇCech complex and the Vietoris-Rips complex

Jisu Kim, Jaehyeok Shin, Fr ´ed´eric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy reconstruction via the ˇCech complex and the Vietoris-Rips complex. In36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum f¨ur Inform...

2020
[11]

Topological stability and Latschev-type reconstruction theo- rems for spaces of curvature bounded above.arXiv:2406.04259 [math.AT], 2024

Rafal Komendarczyk, Sushovan Majhi, and Will Tran. Topological stability and Latschev-type reconstruction theo- rems for spaces of curvature bounded above.arXiv:2406.04259 [math.AT], 2024

2024
[12]

Jung’s theorem for Alexandrov spaces of curvature bounded above.Annals of Global Analysis and Geometry, 15:263–275, 1997

Urs Lang and Viktor Schroeder. Jung’s theorem for Alexandrov spaces of curvature bounded above.Annals of Global Analysis and Geometry, 15:263–275, 1997

1997
[13]

Vietoris–Rips complexes of metric spaces near a metric graph.Journal of Applied and Computa- tional Topology, pages 1–30, 2023

Sushovan Majhi. Vietoris–Rips complexes of metric spaces near a metric graph.Journal of Applied and Computa- tional Topology, pages 1–30, 2023

2023
[14]

Demystifying Latschev’s theorem: Manifold reconstruction from noisy data.Discrete & Computa- tional Geometry, May 2024

Sushovan Majhi. Demystifying Latschev’s theorem: Manifold reconstruction from noisy data.Discrete & Computa- tional Geometry, May 2024

2024
[15]

Moise.The Jordan curve theorem, pages 31–41

Edwin E. Moise.The Jordan curve theorem, pages 31–41. Springer New York, New York, NY, 1977

1977
[16]

CRC press, 2018

James R Munkres.Elements of algebraic topology. CRC press, 2018

2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

On Homotopy Types of Euclidean Rips Complexes.Discrete & Computational Geometry, 58(3):526–542, October 2017

Michał Adamaszek, Florian Frick, and Adrien Vakili. On Homotopy Types of Euclidean Rips Complexes.Discrete & Computational Geometry, 58(3):526–542, October 2017. Publisher: Springer New York LLC

work page 2017

[2] [2]

Stability and computation of medial axes: a state-of-the-art report

Dominique Attali, Jean-Daniel Boissonnat, and Herbert Edelsbrunner. Stability and computation of medial axes: a state-of-the-art report. InMathematical foundations of scientific visualization, computer graphics, and massive data exploration, Math. Vis., pages 109–125. Springer, Berlin, 2009. EUCLIDEAN GRAPH RECONSTRUCTION 25

work page 2009

[3] [3]

Vietoris-Rips complexes also provide topologically correct re- constructions of sampled shapes

Dominique Attali, Andr ´e Lieutier, and David Salinas. Vietoris-Rips complexes also provide topologically correct re- constructions of sampled shapes. InProceedings of the twenty-seventh annual symposium on Computational geometry, pages 491–500, 2011

work page 2011

[4] [4]

Chambers, Vin de Silva, Jeff Erickson, and Robert Ghrist

Erin W. Chambers, Vin de Silva, Jeff Erickson, and Robert Ghrist. Vietoris–Rips complexes of planar point sets. Discrete & Computational Geometry, 44(1):75–90, Jul 2010

work page 2010

[5] [5]

A sampling theory for compact sets in Euclidean space

Fr ´ed´eric Chazal, David Cohen-Steiner, and Andr´e Lieutier. A sampling theory for compact sets in Euclidean space. InProceedings of the twenty-second annual symposium on Computational geometry, pages 319–326, 2006

work page 2006

[6] [6]

Mathematical theory of medial axis transform.Pacific J

Hyeong In Choi, Sung Woo Choi, and Hwan Pyo Moon. Mathematical theory of medial axis transform.Pacific J. Math., 181(1):57–88, 1997

work page 1997

[7] [7]

Helly , Radon, and Carath´eodory type theorems

J ¨urgen Eckhoff. Helly , Radon, and Carath´eodory type theorems. InHandbook of Convex Geometry, pages 389–448. Elsevier, 1993

work page 1993

[8] [8]

On the reconstruction of geodesic subspaces ofR n.International Journal of Computational Geometry & Applications, 32(01n02):91–117, 2022

Brittany Terese Fasy , Rafal Komendarczyk, Sushovan Majhi, and Carola Wenk. On the reconstruction of geodesic subspaces ofR n.International Journal of Computational Geometry & Applications, 32(01n02):91–117, 2022

work page 2022