Straightest Paths on Meshes by Cutting Planes

  • Sungyeol Lee
  • Joonhee Han
  • Haeyoung Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


Geodesic paths and distances on meshes are used for many applications such as parameterization, remeshing, mesh segmentation, and simulations of natural phenomena. Noble works to compute shortest geodesic paths have been published. In this paper, we present a new approach to compute the straightest path from a source to one or more vertices on a manifold mesh with a boundary. A cutting plane with a source and a destination vertex is first defined. Then the straightest path between these two vertices is created by intersecting the cutting plane with faces on the mesh. We demonstrate that our straightest path algorithm contributes to reducing distortion in a shape-preserving linear parameterization by generating a measured boundary.


Straight Path Boundary Vertex Obtuse Angle Measured Boundary Interior Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Chen, J., Han, Y.: Shortest Paths on a Polyhedron; Part I: Computing Shortest Paths. Int. J. Comp. Geom. & Appl. 6(2) (1996)Google Scholar
  2. 2.
    Desbrun, M., Meyer, M., Alliez, P.: Intrinsic Parameterizations of Surface Meshes. In: Eurographics 2002 Conference Proceeding (2002)Google Scholar
  3. 3.
    Floater, M.: Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design (1997)Google Scholar
  4. 4.
    Floater, M.: Mean Value Coordinates. Comput. Aided Geom. Des. (2003)Google Scholar
  5. 5.
    Kaneva, B., O’Rourke, J.: An implementation of Chen and Han’s sortest paths algorithm. In: Proc. of the 12th Canadian Conf. on Computational Geometry (2000)Google Scholar
  6. 6.
    Kanai, T., Suzuki, H.: Approximate Shortest Path on a Polyhedral Surface Based on Selective Refinement of the Discrete Graph and Its Applications. Proc. Geometric Modeling and Processing 2000, HongKong (2000)Google Scholar
  7. 7.
    Kimmel, R., Sethian, J.A.: Computing Geodesic Paths on Manifolds. Proc. Natl. Acad. Sci. USA 95 (1998)Google Scholar
  8. 8.
    Lee, Y., Kim, H., Lee, S.: Mesh Parameterization with a Virtual Boundary. Computer and Graphics 26 (2002)Google Scholar
  9. 9.
    Lee, H., Kim, L., Meyer, M., Desbrun, M.: Meshes on Fire. In: Computer Animation and Simulation 2001, Eurographics (2001)Google Scholar
  10. 10.
    Lee, H., Tong, Y., Desbrun, M.: Geodesics-Based One-to-One Parameterization of 3D Triangle Meshes. IEEE Multimedia 12(1) (January/March 2005)Google Scholar
  11. 11.
    Mitchell, J.S.B.: Geometric Shortest Paths and network optimization. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry. Elsevier Science, Amsterdam (2000)Google Scholar
  12. 12.
    Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The Discrete Geodesic Problem. SIAM J. of Computing 16(4) (1987)Google Scholar
  13. 13.
    Peyré, G., Cohen, L.: Geodesic Re-meshing and Parameterization Using Front Propagation. In: Proceedings of VLSM 2003 (2003)Google Scholar
  14. 14.
    Polthier, K., Schmies, M.: Straightest Geodesics on Polyhedral Surfaces. Mathematical Visualization (1998)Google Scholar
  15. 15.
    Polthier, K., Schmies, M.: Geodesic Flow on Polyhedral Surfaces. In: Proceedings of Eurographics-IEEE Symposium on Scientific Visualization 1999 (1999)Google Scholar
  16. 16.
    Riken, T., Suzuki, H.: Approximate Shortest Path on a Polyhedral Surface Based on Selective Refinement of the Discrete Graph and Its Applications. In: Geometric Modeling and Processing 2000, Hongkong (2000)Google Scholar
  17. 17.
    Sander, P.V., Snyder, J., Gortler, S.J., Hoppe, H.: Texture Mapping Progressive Meshes. In: Proceedings of SIGGRAPH 2001 (2001)Google Scholar
  18. 18.
    Sifri, O., Sheffer, A., Gotsman, C.: Geodesic-based Surface Remeshing. In: Proceedings of 12th Intnl. Meshing Roundtable (2003)Google Scholar
  19. 19.
    Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S., Hoppe, H.: Fast Exact and Approximate Geodesics on Meshes. In: ACM SIGGRAPH 2005 Conference Proceedings (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sungyeol Lee
    • 1
  • Joonhee Han
    • 1
  • Haeyoung Lee
    • 1
  1. 1.Dept. of Computer EngineeringHongik UniversitySeoulKorea

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