Skip to main content

Advertisement

Springer Nature Link
Log in

Modeling on triangulations with geodesic curves

  • Special Issue Article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

This paper discusses the problem of modeling on triangulated surfaces with geodesic curves. In the first part of the paper we define a new class of curves, called geodesic Bézier curves, that are suitable for modeling on manifold triangulations. As a natural generalization of Bézier curves, the new curves are as smooth as possible. In the second part we discuss the construction of C 0 and C 1 piecewise Bézier splines. We also describe how to perform editing operations, such as trimming, using these curves. Special care is taken to achieve interactive rates for modeling tasks. The third part is devoted to the definition and study of convex sets on triangulated surfaces. We derive the convex hull property of geodesic Bézier curves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Aleksandrov, A.D.: A.D. Aleksandrov Selected Works. Intrinsic Geometry of Convex Surfaces. Chapman & Hall/CRC, London/Boca Raton (2006). S.S. Kutateladze (ed.)

    Google Scholar 

  2. Aleksandrov, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces. Translation of Mathematical Monographs, vol. 15. Am. Math. Soc., Providence (1967)

    MATH  Google Scholar 

  3. Biermann, H., Martin, I.M., Zorin, D., Bernardini, F.: Sharp features on multiresolution subdivision surfaces. Graph. Models 64(2), 61–77 (2002). http://dx.doi.org/10.1006/gmod.2002.0570

    Article  MATH  Google Scholar 

  4. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976)

    MATH  Google Scholar 

  5. Chen, J., Han, Y.: Shortest paths on a polyhedron. In: Proceedings of 6th Annu. ACM Sympos. Comput. Geom., pp. 360–369 (1990)

  6. Coelho, L.C.G., Gattass, M., de Figueiredo, L.H.: Intersecting and trimming parametric meshes on finite-element shells. Int. J. Numer. Methods Eng. 47(4), 777–800 (2000)

    Article  MATH  Google Scholar 

  7. Cohen, E., Riesenfeld, R.F., Elber, G.: Geometric Modeling with Splines: An Introduction. A.K. Peters, Ltd., Natick (2001)

    MATH  Google Scholar 

  8. Crouch, P., Kun, G., Leite, F.S.: The de Casteljau Algorithm on lie groups and spheres. J. Dyn. Control Syst. 5(3), 397–429 (1999)

    Article  MATH  Google Scholar 

  9. de Faget de Casteljau, P.: Outillage Méthodes Calcul, SA André Citroën. Paris, P2108, Internes Dokument (1959)

  10. Farin, G.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann, San Mateo (2002)

    Google Scholar 

  11. Hofer, M., Pottmann, H.: Energy-minimizing splines in manifolds. ACM Trans. Graph. 23(3), 284–293 (2004). http://doi.acm.org/10.1145/1015706.1015716

    Article  Google Scholar 

  12. Kapoor, S.: Efficient computation of geodesic shortest paths. In: Proceedings of 31st Annu. ACM Sympos. Theory Comput., pp. 770–779 (1999)

  13. Kasap, E., Yapici, M., Talay Akyildiz, F.: A numerical study for computation of geodesic curves. Appl. Math. Comput. 171(2), 1206–1213 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kimmel, R., Sapiro, G.: Shortening three-dimensional curves via two-dimensional flows. Comput. Math. Appl. 29(3), 49–62 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. USA 95(15), 8431–8435 (1998). http://citeseer.nj.nec.com/kimmel98computing.html

    Article  MATH  MathSciNet  Google Scholar 

  16. Klassen, E., Srivastava, A., Mio, W., Joshi, S.: Analysis of planar shapes using geodesic paths on shape manifolds. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–384 (2004)

    Article  Google Scholar 

  17. Krishnan, S., Manocha, D.: An efficient surface intersection algorithm based on lower dimensional formulation. ACM Trans. Comput. Graph. 16(1), 74–106 (1997)

    Article  Google Scholar 

  18. Lages, E.: Curso de análise, 2. Projeto Euclides, 5 (1999)

  19. Litke, N., Levin, A., Schröder, P.: Trimming for subdivision surfaces. Comput. Aided Geom. Des. 18(5), 463–481 (2001)

    Article  MATH  Google Scholar 

  20. Martínez, D., Velho, L., Carvalho, P.C.: Computing geodesics on triangular meshes. Comput. Graph. 29(5), 667–675 (2005)

    Article  Google Scholar 

  21. Martínez, D., Carvalho, P.C., Velho, L.: Geodesic Bézier curves: a tool for modeling on triangulations. In: Proceedings of SIBGRAPI 2007–XX Brazilian Symposium on Computer Graphics and Image Processing, pp. 71–78. IEEE Comput. Soc., Los Alamitos (2007)

    Google Scholar 

  22. Mémoli, F., Sapiro, G.: Distance functions and geodesics on submanifolds of ℝd and point clouds. SIAM J. Appl. Math. 65(4), 1227–1260 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The discrete geodesic problem. SIAM J. Comput. 16, 647–668 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  24. Park, F.C., Ravani, B.: Bezier curves on Riemannian manifolds and Lie groups with kinematic applications. J. Mech. Des. 117, 36–40 (1995)

    Article  Google Scholar 

  25. Pottmann, H., Hofer, M.: A variational approach to spline curves on surfaces. Comput. Aided Geom. Des. 22(7), 693–709 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Polthier, K., Schmies, M.: Straightest geodesics on polyhedral surfaces. In: Hege, H.-C., Polthier, K. (eds.) Visualization and Mathematics, pp. 135–150. Springer, Heidelberg (1998)

    Google Scholar 

  27. Rodriguez, R.C., Leite, F.S., Jacubiak, J.: A new geometric algorithm to generate smooth interpolating curves on Riemannian manifolds. LMS J. Comput. Math. 8, 251–266 (2005)

    MathSciNet  Google Scholar 

  28. Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S., Hoppe, H.: Fast exact and approximate geodesics on meshes. Proc. ACM SIGGRAPH 2005, 553–560 (2005)

    Article  Google Scholar 

  29. Wallner, J., Pottmann, H.: Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graph. 25(2), 356–374 (2006)

    Article  MathSciNet  Google Scholar 

  30. Wallner, J., Dyn, N.: Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22(7), 593–622 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wallner, J.: Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24(3), 289–318 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada–IMPA, Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brasil

    Dimas Martínez Morera, Paulo Cezar Carvalho & Luiz Velho

Authors
  1. Dimas Martínez Morera
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Paulo Cezar Carvalho
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Luiz Velho
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Dimas Martínez Morera.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Morera, D.M., Carvalho, P.C. & Velho, L. Modeling on triangulations with geodesic curves. TVC 24, 1025–1037 (2008). https://doi.org/10.1007/s00371-008-0298-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-008-0298-9

Keywords