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Polygonal Curve Approximation Using Grid Points with Application to a Triangular Mesh Generation with Small Number of Different Edge Lengths

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Algorithmic Aspects in Information and Management (AAIM 2006)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 4041))

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Abstract

For a given x-monotone polygonal curve each of whose edge lengths is between \(\underline{l}\) and \(2\underline{l}\), we consider the problem of approximating it by another x-monotone polygonal curve using points of a square grid so that there exists a small number of different edge lengths and every edge length is between \(\underline{l}\) and \(\beta \underline{l}\), where β is a given parameter satisfying 1≤β≤2. Our first algorithm computes an approximate polygonal curve using fixed square grid points in O((n/α 4)log(n/α)) time. Based on this, our second algorithm finds an approximate polygonal curve as well as an optimal grid placement simultaneously in O((n 3/α 12)log2(n/α)) time, where α is a parameter that controls the closeness of approximation. Based on the approximate polygonal curve, we shall give an algorithm for finding a uniform triangular mesh for an x-monotone polygon with a constant number of different edge lengths.

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References

  1. Agarwal, P.K., Aronov, B., Sharir, M., Suri, S.: Selecting distances in the plane. Algorithmica 9(5), 495–514 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agarwal, P.K., Har-Peled, S., Mustafa, N.H., Wang, Y.: Near-linear time approximation algorithms for curve simplification. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 29–41. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Agarwal, P.K., Sharir, M., Toledo, S.: Applications of parametric searching in geometric optimization

    Google Scholar 

  4. Agarwal, P.K., Varadarajan, K.R.: Efficient algorithms for approximating polygonal chains. Discrete & Comp., Geom. 23(2), 273–291 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aronov, B., Asano, T., Katoh, N., Mehlhorn, K., Tokuyama, T.: Polyline fitting of planar points under min-sum criteria. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 77–88. Springer, Heidelberg (2004); Int. J. of Comput. Geom. & Appl. (to appear)

    Chapter  Google Scholar 

  6. Chan, W.S., Chin, F.: Approximation of polygonal curves with minimum number of line segments or minimum error. Int. J. Comput. Geometry Appl. 6(1), 59–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, D.Z., Daescu, O.: Space-efficient algorithms for approximating polygonal curves in two-dimensional space. Int. J. Comput. Geometry Appl. 13(2), 95–111 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Edelsbrunner, H., Guibas, L.J., Pach, J., Pollack, R., Seidel, R., Sharir, M.: Arrangements of curves in the plane - topology, combinatorics and algorithms. Theor. Comput. Sci. 92(2), 319–336 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goodrich, M.T.: Efficient piecewise-linear function approximation using the uniform metric. Discrete & Comput. Geom. 14(4), 445–462 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gudmundsson, J., Narasimhan, G., Smid, M.H.M.: Distance-preserving approximations of polygonal paths. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 217–228. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  11. Guibas, L.J., Hershberger, J., Mitchell, J.S.B., Snoeyink, J.: Approximating polygons and subdivisions with minimum link paths. Int. J. Comput. Geometory Appl. 3(4), 383–415 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Imai, H., Iri, M.: An optimal algorithm for approximating a piecewise linear function. Info. Proc. Letters 9(3), 159–162 (1986)

    MATH  MathSciNet  Google Scholar 

  13. Imai, H., Iri, M.: Polygonal approximations of a curve - formulatons and algorithms. In: Toussaint, G.T. (ed.) Comp. Morphology, pp. 71–86. North-Holland, Amsterdam (1988)

    Google Scholar 

  14. Katoh, N., Ohsaki, M., Xu, Y.: A uniform triangle mesh generation of curved surfaces. In: Akiyama, J., Kano, M., Tan, X. (eds.) JCDCG 2004. LNCS, vol. 3742. Springer, Heidelberg (2005)

    Google Scholar 

  15. Lee, D.T., Robert, I., Drysdale, L.: Generalization of voronoi diagrams in the plane. SIAM J. Comput. 10(1), 73–87 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Melkman, A., O’Rourke, J.: On polygonal chain approximation. In: Toussaint, G.T. (ed.) Comp. Morphology, pp. 87–95. North-Holland, Amsterdam (1988)

    Google Scholar 

  18. Tanigawa, S., Katoh, N.: Finding a triangular mesh with a constant number of different edge lengths. In: Proc. of the 17th Canad. Conf. on Comput. Geom. (2005), http://cccg.cs.uwindsor.ca/papers/41.pdf

  19. Xu, Y.F., Dai, W., Katoh, N., Ohsaki, M.: Triangulating a convex polygon with small number of non-standard bars. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 481–489. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Architecture and Architectural Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo, Kyoto, 615-8540, Japan

    Shin-ichi Tanigawa & Naoki Katoh

Authors
  1. Shin-ichi Tanigawa
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  2. Naoki Katoh
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Clear Water Bay, Hong Kong University of Science and Technology, Hong Kong, China

    Siu-Wing Cheng

  2. Department of Computer Science, City University of Hong Kong,  

    Chung Keung Poon

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© 2006 Springer-Verlag Berlin Heidelberg

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Tanigawa, Si., Katoh, N. (2006). Polygonal Curve Approximation Using Grid Points with Application to a Triangular Mesh Generation with Small Number of Different Edge Lengths. In: Cheng, SW., Poon, C.K. (eds) Algorithmic Aspects in Information and Management. AAIM 2006. Lecture Notes in Computer Science, vol 4041. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11775096_16

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  • DOI: https://doi.org/10.1007/11775096_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35157-3

  • Online ISBN: 978-3-540-35158-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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