THE HEAT EQUATION AND THE FRENET FORMULAS FOR DIGITAL CURVES

Sheng-Gwo Chen, Mei-Hsiu Chi, Jyh-Yang Wu

2007

Abstract

Abstract：In this note we shall discuss the heat equation and the eigenvalue problem for digital curves in the 3D Euclidean space. First, we shall introduce the derivative of a function along a digital curve by the weighted combination method. Then, we can define the Laplace of a function on a digital curve. The Frenet formulas for digital curves will also be discussed. Numerical simulations show our method will provide good estimations for the curvature and torsion.

References

  1. Bjorck, A. Golub, G. H., 1977. Eigenproblems for matrices associated with periodic boundary conditions. SIAM Review Vol. 19, No. 1
  2. Chen, S.-G.., Wu, J.-Y., 2004. Estimating normal vectors and curvatures by centroid weights. Computer Aided Geometric Design, 21, pp. 447-458.
  3. Chen, S.-G.., Wu, J.-Y., 2005. A geometric interpretation of weighted normal vectors and application. Proceeding of the IEEE Computer Society Conference on Computer Graphics, Imaging and Visualization, New Trends, pp.422-425.
  4. do Carmo, M., 1976. Differential Geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, NJ.
  5. Rosenfeld, A. Klette, R. 2002. Digital geometry. Information Sciences 148,p 123-127
  6. Wu, J.-Y., Chen, S.-G. and Chi, M.-H., 2005. A simple effective method for curvatures estimation on triangular meshes, Technical Report WU02, NCCU, Department of Mathematics.
  7. 30 50 70 6: Boundary points with uniform partitions.
  8. a 30 50 70 90 7: Interior points with non-uniform partitions.
  9. nonuniform and closed curve helix(nonuniform and boundary points) 30 50 70 5: 8: Boundary points with non-uniform partitions.
Download


Paper Citation


in Harvard Style

Chen S., Chi M. and Wu J. (2007). THE HEAT EQUATION AND THE FRENET FORMULAS FOR DIGITAL CURVES . In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, ISBN 978-972-8865-71-9, pages 97-102. DOI: 10.5220/0002072900970102


in Bibtex Style

@conference{grapp07,
author={Sheng-Gwo Chen and Mei-Hsiu Chi and Jyh-Yang Wu},
title={THE HEAT EQUATION AND THE FRENET FORMULAS FOR DIGITAL CURVES},
booktitle={Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},
year={2007},
pages={97-102},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002072900970102},
isbn={978-972-8865-71-9},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,
TI - THE HEAT EQUATION AND THE FRENET FORMULAS FOR DIGITAL CURVES
SN - 978-972-8865-71-9
AU - Chen S.
AU - Chi M.
AU - Wu J.
PY - 2007
SP - 97
EP - 102
DO - 10.5220/0002072900970102