---
_id: '5799'
abstract:
- lang: eng
  text: We construct a polyhedral surface called a graceful surface, which provides
    best possible approximation to a given sphere regarding certain criteria. In digital
    geometry terms, the graceful surface is uniquely characterized by its minimality
    while guaranteeing the connectivity of certain discrete (polyhedral) curves defined
    on it. The notion of “gracefulness” was first proposed in Brimkov and Barneva
    (1999) and shown to be useful for triangular mesh discretization through graceful
    planes and graceful lines. In this paper we extend the considerations to a nonlinear
    object such as a sphere. In particular, we investigate the properties of a discrete
    geodesic path between two voxels and show that discrete 3D circles, circular arcs,
    and Mobius triangles are all constructible on a graceful sphere, with guaranteed
    minimum thickness and the desired connectivity in the discrete topological space.
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Partha
  full_name: Bhowmick, Partha
  last_name: Bhowmick
- first_name: Valentin E.
  full_name: Brimkov, Valentin E.
  last_name: Brimkov
citation:
  ama: Biswas R, Bhowmick P, Brimkov VE. On the polyhedra of graceful spheres and
    circular geodesics. <i>Discrete Applied Mathematics</i>. 2017;216:362-375. doi:<a
    href="https://doi.org/10.1016/j.dam.2015.11.017">10.1016/j.dam.2015.11.017</a>
  apa: Biswas, R., Bhowmick, P., &#38; Brimkov, V. E. (2017). On the polyhedra of
    graceful spheres and circular geodesics. <i>Discrete Applied Mathematics</i>.
    Elsevier. <a href="https://doi.org/10.1016/j.dam.2015.11.017">https://doi.org/10.1016/j.dam.2015.11.017</a>
  chicago: Biswas, Ranita, Partha Bhowmick, and Valentin E. Brimkov. “On the Polyhedra
    of Graceful Spheres and Circular Geodesics.” <i>Discrete Applied Mathematics</i>.
    Elsevier, 2017. <a href="https://doi.org/10.1016/j.dam.2015.11.017">https://doi.org/10.1016/j.dam.2015.11.017</a>.
  ieee: R. Biswas, P. Bhowmick, and V. E. Brimkov, “On the polyhedra of graceful spheres
    and circular geodesics,” <i>Discrete Applied Mathematics</i>, vol. 216. Elsevier,
    pp. 362–375, 2017.
  ista: Biswas R, Bhowmick P, Brimkov VE. 2017. On the polyhedra of graceful spheres
    and circular geodesics. Discrete Applied Mathematics. 216, 362–375.
  mla: Biswas, Ranita, et al. “On the Polyhedra of Graceful Spheres and Circular Geodesics.”
    <i>Discrete Applied Mathematics</i>, vol. 216, Elsevier, 2017, pp. 362–75, doi:<a
    href="https://doi.org/10.1016/j.dam.2015.11.017">10.1016/j.dam.2015.11.017</a>.
  short: R. Biswas, P. Bhowmick, V.E. Brimkov, Discrete Applied Mathematics 216 (2017)
    362–375.
date_created: 2019-01-08T20:41:12Z
date_published: 2017-01-10T00:00:00Z
date_updated: 2021-01-12T08:03:33Z
day: '10'
doi: 10.1016/j.dam.2015.11.017
extern: '1'
intvolume: '       216'
language:
- iso: eng
month: '01'
oa_version: None
page: 362-375
publication: Discrete Applied Mathematics
publication_identifier:
  issn:
  - 0166-218X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: On the polyhedra of graceful spheres and circular geodesics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 216
year: '2017'
...
---
_id: '4013'
abstract:
- lang: eng
  text: The shape of a protein is important for its functions, This includes the location
    and size of identifiable regions in its complement space. We formally define pockets
    as regions in the complement with limited accessibility from the outside. Pockets
    can be efficiently constructed by an algorithm based on alpha complexes. The algorithm
    is implemented and applied to proteins with known three-dimensional conformations.
    1998 Published by Elsevier Science B.V. All rights reserved.
acknowledgement: 'The authors thank Ping Fu and Ernst Miicke for their contributions
  to the alpha shapes software in which the pockets software is embedded. '
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Michael
  full_name: Facello, Michael
  last_name: Facello
- first_name: Jie
  full_name: Liang, Jie
  last_name: Liang
citation:
  ama: Edelsbrunner H, Facello M, Liang J. On the definition and the construction
    of pockets in macromolecules. <i>Discrete Applied Mathematics</i>. 1998;88(1-3):83-102.
    doi:<a href="https://doi.org/10.1016/S0166-218X(98)00067-5">10.1016/S0166-218X(98)00067-5</a>
  apa: Edelsbrunner, H., Facello, M., &#38; Liang, J. (1998). On the definition and
    the construction of pockets in macromolecules. <i>Discrete Applied Mathematics</i>.
    Elsevier. <a href="https://doi.org/10.1016/S0166-218X(98)00067-5">https://doi.org/10.1016/S0166-218X(98)00067-5</a>
  chicago: Edelsbrunner, Herbert, Michael Facello, and Jie Liang. “On the Definition
    and the Construction of Pockets in Macromolecules.” <i>Discrete Applied Mathematics</i>.
    Elsevier, 1998. <a href="https://doi.org/10.1016/S0166-218X(98)00067-5">https://doi.org/10.1016/S0166-218X(98)00067-5</a>.
  ieee: H. Edelsbrunner, M. Facello, and J. Liang, “On the definition and the construction
    of pockets in macromolecules,” <i>Discrete Applied Mathematics</i>, vol. 88, no.
    1–3. Elsevier, pp. 83–102, 1998.
  ista: Edelsbrunner H, Facello M, Liang J. 1998. On the definition and the construction
    of pockets in macromolecules. Discrete Applied Mathematics. 88(1–3), 83–102.
  mla: Edelsbrunner, Herbert, et al. “On the Definition and the Construction of Pockets
    in Macromolecules.” <i>Discrete Applied Mathematics</i>, vol. 88, no. 1–3, Elsevier,
    1998, pp. 83–102, doi:<a href="https://doi.org/10.1016/S0166-218X(98)00067-5">10.1016/S0166-218X(98)00067-5</a>.
  short: H. Edelsbrunner, M. Facello, J. Liang, Discrete Applied Mathematics 88 (1998)
    83–102.
date_created: 2018-12-11T12:06:26Z
date_published: 1998-11-09T00:00:00Z
date_updated: 2022-08-25T15:06:30Z
day: '09'
doi: 10.1016/S0166-218X(98)00067-5
extern: '1'
external_id:
  pmid:
  - '9390238'
intvolume: '        88'
issue: 1-3
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://www.sciencedirect.com/science/article/pii/S0166218X98000675?via%3Dihub
month: '11'
oa: 1
oa_version: Published Version
page: 83 - 102
pmid: 1
publication: Discrete Applied Mathematics
publication_identifier:
  issn:
  - 0166-218X
publication_status: published
publisher: Elsevier
publist_id: '2114'
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the definition and the construction of pockets in macromolecules
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 88
year: '1998'
...
---
_id: '4025'
abstract:
- lang: eng
  text: Questions of chemical reactivity can often be cast as questions of molecular
    geometry. Common geometric models for proteins and other molecules are the space-filling
    diagram, the solvent accessible surface and the molecular surface. In this paper
    we present a new approach to triangulating the surface of a molecule under the
    three models, which is fast, robust, and results in topologically correct triangulations.
    Our computations are based on a simplicial complex dual to the molecule models.
    All proposed algorithms are parallelizable.
acknowledgement: 'The research of both authors is partially supported by the Office
  of Naval Research. Herbert Edelsbrunner is also supported through the Alan T. Waterman
  award, grant CCR-9118874. '
article_processing_charge: No
article_type: original
author:
- first_name: Nataraj
  full_name: Akkiraju, Nataraj
  last_name: Akkiraju
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
citation:
  ama: Akkiraju N, Edelsbrunner H. Triangulating the surface of a molecule. <i>Discrete
    Applied Mathematics</i>. 1996;71(1-3):5-22. doi:<a href="https://doi.org/10.1016/S0166-218X(96)00054-6">10.1016/S0166-218X(96)00054-6</a>
  apa: Akkiraju, N., &#38; Edelsbrunner, H. (1996). Triangulating the surface of a
    molecule. <i>Discrete Applied Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/S0166-218X(96)00054-6">https://doi.org/10.1016/S0166-218X(96)00054-6</a>
  chicago: Akkiraju, Nataraj, and Herbert Edelsbrunner. “Triangulating the Surface
    of a Molecule.” <i>Discrete Applied Mathematics</i>. Elsevier, 1996. <a href="https://doi.org/10.1016/S0166-218X(96)00054-6">https://doi.org/10.1016/S0166-218X(96)00054-6</a>.
  ieee: N. Akkiraju and H. Edelsbrunner, “Triangulating the surface of a molecule,”
    <i>Discrete Applied Mathematics</i>, vol. 71, no. 1–3. Elsevier, pp. 5–22, 1996.
  ista: Akkiraju N, Edelsbrunner H. 1996. Triangulating the surface of a molecule.
    Discrete Applied Mathematics. 71(1–3), 5–22.
  mla: Akkiraju, Nataraj, and Herbert Edelsbrunner. “Triangulating the Surface of
    a Molecule.” <i>Discrete Applied Mathematics</i>, vol. 71, no. 1–3, Elsevier,
    1996, pp. 5–22, doi:<a href="https://doi.org/10.1016/S0166-218X(96)00054-6">10.1016/S0166-218X(96)00054-6</a>.
  short: N. Akkiraju, H. Edelsbrunner, Discrete Applied Mathematics 71 (1996) 5–22.
date_created: 2018-12-11T12:06:30Z
date_published: 1996-12-05T00:00:00Z
date_updated: 2022-08-09T14:06:12Z
day: '05'
doi: 10.1016/S0166-218X(96)00054-6
extern: '1'
intvolume: '        71'
issue: 1-3
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://www.sciencedirect.com/science/article/pii/S0166218X96000546?via%3Dihub
month: '12'
oa: 1
oa_version: Published Version
page: 5 - 22
publication: Discrete Applied Mathematics
publication_identifier:
  issn:
  - 0166-218X
publication_status: published
publisher: Elsevier
publist_id: '2102'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Triangulating the surface of a molecule
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 71
year: '1996'
...