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Asymptotic analysis of discrete normals and curvatures of polylines

Published: 12 May 2005 Publication History

Abstract

Accurate estimations of geometric properties of a smooth curve from its discrete approximation are important for many computer graphics and computer vision applications. To assess and improve the quality of such an approximation, we assume that the curve is known in general form. Then we can represent the curve by a Taylor series expansion and compare its geometric properties with the corresponding discrete approximations. In turn we can either prove convergence of these approximations towards the true properties as the edge lengths tend to zero, or we can get hints on how to eliminate the error. In this paper, we propose and study discrete schemes for estimating tangent and normal vectors as well as for estimating curvature and torsion of a smooth 3D curve approximated by a polyline. Thereby we make some interesting findings about connections between (smooth) classical curves and certain estimation schemes for polylines.

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Cited By

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  • (2021)Practical Computation of the Cut Locus on Discrete SurfacesComputer Graphics Forum10.1111/cgf.1437240:5(261-273)Online publication date: 23-Aug-2021
  • (2021)Discrete curvature and torsion from cross-ratiosAnnali di Matematica Pura ed Applicata (1923 -)10.1007/s10231-021-01065-x200:5(1935-1960)Online publication date: 21-Jan-2021
  • (2015)Hydro-Responsive Curling of the Resurrection Plant Selaginella lepidophyllaScientific Reports10.1038/srep080645:1Online publication date: 27-Jan-2015
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Published In

cover image ACM Conferences
SCCG '05: Proceedings of the 21st Spring Conference on Computer Graphics
May 2005
227 pages
ISBN:1595932046
DOI:10.1145/1090122
  • Conference Chair:
  • Bert Jüttler
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 12 May 2005

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Author Tags

  1. curvature
  2. discrete approximation
  3. error analysis
  4. frenet frame
  5. normal vector
  6. polyline
  7. tangent vector
  8. torsion

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SCCG05
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SCCG05: Spring Conference in Computer Graphics
May 12 - 14, 2005
Budmerice, Slovakia

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Overall Acceptance Rate 67 of 115 submissions, 58%

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Cited By

View all
  • (2021)Practical Computation of the Cut Locus on Discrete SurfacesComputer Graphics Forum10.1111/cgf.1437240:5(261-273)Online publication date: 23-Aug-2021
  • (2021)Discrete curvature and torsion from cross-ratiosAnnali di Matematica Pura ed Applicata (1923 -)10.1007/s10231-021-01065-x200:5(1935-1960)Online publication date: 21-Jan-2021
  • (2015)Hydro-Responsive Curling of the Resurrection Plant Selaginella lepidophyllaScientific Reports10.1038/srep080645:1Online publication date: 27-Jan-2015
  • (2012)ConceptureProceedings of the International Symposium on Sketch-Based Interfaces and Modeling10.5555/2331067.2331073(29-37)Online publication date: 4-Jun-2012
  • (2010)Piecewise 3D Euler spiralsProceedings of the 14th ACM Symposium on Solid and Physical Modeling10.1145/1839778.1839810(201-206)Online publication date: 1-Sep-2010
  • (2008)Shape InterrogationShape Analysis and Structuring10.1007/978-3-540-33265-7_1(1-51)Online publication date: 2008
  • (2007)Numerical comparison of some Hessian recovery techniquesInternational Journal for Numerical Methods in Engineering10.1002/nme.203672:8(987-1007)Online publication date: 28-Mar-2007
  • (2006)An accurate vertex normal computation schemeProceedings of the 24th international conference on Advances in Computer Graphics10.1007/11784203_38(442-451)Online publication date: 26-Jun-2006

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