DOI QR코드

DOI QR Code

VISUAL CURVATURE FOR SPACE CURVES

JEON, MYUNGJIN

  • Received : 2015.11.10
  • Accepted : 2015.12.01
  • Published : 2015.12.25

Abstract

For a smooth plane curve, the curvature can be characterized by the rate of change of the angle between the tangent vector and a fixed vector. In this article we prove that the curvature of a space curve can also be given by the rate of change of the locally defined angle between the tangent vector at a point and the nearby point. By using height functions, we introduce turning angle of a space curve and characterize the curvature by the rate of change of the turning angle. The main advantage of the turning angle is that it can be used to characterize the curvature of discrete curves. For this purpose, we introduce a discrete turning angle and a discrete curvature called visual curvature for space curves. We can show that the visual curvature is an approximation of curvature for smooth curves.

Keywords

curvature of space curve;discrete curvature;turning angle;discrete turning angle

References

  1. Elena V. Anoshkina, Alexander G. Belyaev, Hans-Peter Seidel, Asymptotic Analysis of Three-Point Approximations of Vertex Normals and Curvatures, Vision, Modeling, and Visualization 2002, 211-216, Erlangen, Germany, November (2002)
  2. Alexander G. Belyaev, Plane and space curves. curvature, curvature-based fea-tures, Max-Planck-Institut fur Informatik (2004)
  3. V. Borrelli, F. Cazals, J-M. Morvan, On the angular defect of triangulations and the pointwise approximation of curvatures, Computer Aided Geometric Design 20, 319-341 (2003) https://doi.org/10.1016/S0167-8396(03)00077-3
  4. Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall (1976)
  5. D. Coeurjolly, M. Serge and T. Laure, Discrete curvature based on osculating circles estimation, Lecture Notes in Computer Science 2059, 303-312, Springer (2001)
  6. M. Jeon, Approximate tangent vector and geometric cubic Hermite interpolation, Journal of Applied Mathematics and Computing, Vol. 20, No. 1-2, 575-584 (2006)
  7. Torsten Langer, Alexander G. Belyaev, Hans-Peter Seidel, Analysis and design of discrete normals and curvatures, Forschungsbericht Research Report, Max-Planck-Institut Fur Informatik (2005)
  8. Thomas Lewiner, J. D. Gomes Jr., H. Lopes AND M. Craizer, Curvature and Torsion Estimators based on Parametric Curve Fitting, to appear in Computers & Graphics (2005)
  9. Hairong Liu, Longin Jan Latecki, Wenyu Liu, A Unified Curvature Definition for Regular, Polygonal, and Digital Planar Curves, International Journal Computer Vision 80, 104-124(2008) https://doi.org/10.1007/s11263-008-0131-y
  10. Jean-Louis Maltret, Marc Daniel, Discrete curvatures and applications: a survey, Report de recherche 004.2002, Laboratoire des Sciences de l'Information dt des Systemes (2002)
  11. http://thesaurus.maths.org/