This Article 
 Bibliographic References 
 Add to: 
'Meshsweeper': Dynamic Point-to-Polygonal-Mesh Distance and Applications
January-March 2001 (vol. 7 no. 1)
pp. 47-61

Abstract—We introduce a new algorithm for computing the distance from a point to an arbitrary polygonal mesh. Our algorithm uses a multiresolution hierarchy of bounding volumes generated by geometric simplification. Our algorithm is dynamic, exploiting coherence between subsequent queries using a priority process and achieving constant time queries in some cases. It can be applied to meshes that transform rigidly or deform nonrigidly. We illustrate our algorithm with a simulation of particle dynamics and collisions with a deformable mesh, the computation of distance maps and offset surfaces, the computation of an approximation to the expensive Hausdorff distance between two shapes, and the detection of self-intersections. We also report comparison results between our algorithm and an alternative algorithm using an octree, upon which our method permits an order-of-magnitude speed-up.

[1] L. Markosian, J.M. Cohen, T. Crulli, and J. Hughes, “Skin: A Constructive Approach to Modeling Free-Form Shapes,” SIGGRAPH '99 Proc., pp. 393-400, Aug. 1999.
[2] D. Baraff, "Curved Surfaces and Coherence for Non-Penetrating Rigid Body Simulation," Computer Graphics, vol. 24, no. 4, pp. 19-28, Aug. 1990.
[3] E.G. Gilbert, D.W. Johnson, and S.S. Keerthi, A Fast Procedure for Computing the Distance between Objects in Three-Dimensional Space J. Robotics and Automation, vol. 4, no. 2, pp. 193-203, 1988.
[4] M.C. Lin, "Efficient Collision Detection for Animation and Robotics," PhD thesis, Dept. of Electrical Eng. and Computer Science, Univ. of California, Berkeley, Dec. 1993.
[5] F. Aurenhammer, "Voronoi Diagrams: A Survey of a Fundamental Geometric Data Structure," ACM Computing Surveys, vol. 23, no. 3, 1991, pp. 345-405.
[6] K. HoffIII et al., "Fast Computation of Generalized Voronoi Diagrams using Graphics Hardware," Proc. Siggraph 99, ACM Press, 1999, pp. 277-286.
[7] H. Samet, The Design and Analysis of Spatial Data Structures. Addison-Wesley, 1990.
[8] H. Samet, Applications of Spatial Data Structures. Addison-Wesley, 1990.
[9] O. Guenther, “Efficient Structures for Geometric Data Management,” PhD thesis, Univ. of California at Berkeley, 1987.
[10] H. Hoppe, “Progressive Meshes,” Proc. SIGGRAPH '96, pp. 99-108, 1996.
[11] L. Kobbelt, J. Vorsatz, and H.-P. Seidel, “Multiresolution Hierarchies on Unstructured Triangle Meshes,” Computational Geometry: Theory and Applications, vol. 14, pp. 5-24, Dec. 1999.
[12] S. Quinlan, Efficient Distance Computation Between Non-Convex Objects Proc. IEEE Int'l Conf. Robotics and Automation, pp. 3325-3330, 1994.
[13] E. Larsen, S. Gottschalk, M.C. Lin, and D. Manocha, “Fast Proximity Queries with Swept Sphere Volumes,” Technical Report TR99-018, Univ. of North Carolina Chapel Hill, 1999.
[14] D. Johnson and E. Cohen, “Bound Coherence for Minimum Distance Computations,” Proc. Conf. Robotics and Automation, pp. 1843-1848, 1999.
[15] P.J. Besl and N.D. McKay, "A Method for Registration of 3D Shapes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, pp. 239-256, Feb. 1992.
[16] D. Simon, “Fast and Accurate Shape-Based Registration,” PhD thesis, Technical Report CMU-RI-TR-96-45, Robotics Institute, Carnegie Mellon Univ., Dec. 1996.
[17] S. Gottschalk, M. Lin, and D. Manocha, "Obb-Tree: A Hierarchical Structure for Rapid Interference Detection," Proc. ACM Siggraph '96, pp. 171-180, 1996.
[18] D. Johnson and E. Cohen, “A Framework for Efficient Minimum Distance Computation,” Proc. Conf. Robotics and Automation, pp. 3678-3683, 1998.
[19] J.T. Klosowski, M. Held, J.S.B. Mitchell, H. Sowizral, and K. Zikan, Efficient Collision Detection Using Bounding Volume Hierarchies of k-dops IEEE Trans. Visualization and Computer Graphics, vol. 4, no. 1, pp. 21-36, Jan.-Mar. 1998.
[20] J. Xia and A. Varshney, "Dynamic View-Dependent Simplification for Polygonal Models," Proc. IEEE Visualization 96, ACM Press, New York, 1996, pp. 327-334.
[21] H. Hoppe, “View-Dependent Refinement of Progressive Meshes,” Proc. SIGGRAPH '97, pp. 189-198, 1997.
[22] L.D. Floriani, P. Magillo, and E. Puppo, “Building and Traversing a Surface at Variable Resolution,” Proc. IEEE Visualization '97, pp. 103-110, Nov. 1997.
[23] A. Maheshwahri, P. Morin, and J.-R. Sack, “Progressive TINs: Algorithms and Applications,” Proc. Fifth Int'l Workshop Advances in Geographic Information Systems, pp. 24-29, 1997.
[24] A. Guéziec, G. Taubin, F. Lazarus, and W. Horn, “A Framework for Streaming Geometry in VRML,” IEEE Computer Graphics and Applications, vol. 19, no. 2, pp. 68-78, Mar.-Apr. 1999.
[25] L. De Floriani and P. Magillo, “Horizon Computation on a Hierarchical Triangulated Terrain Model,” The Visual Computer, vol. 11, pp. 134-149, 1995.
[26] U. Ramer, “An Iterative Procedure for the Polygonal Approximation of Plane Curves,” Computer Graphics and Image Processing, vol. 1, pp. 244-256, 1972.
[27] D.H. Douglas and T.K. Peucker, “Algorithms for the Reduction of the Number of Points Required to Represent a Digitized Line or Its Caricature,” The Canadian Cartographer, vol. 10, no. 2, pp. 112-122, Dec. 1973.
[28] J. Cohen, M. Olano, and D. Manocha, “Appearance-Preserving Simplification,” Proc. Siggraph, pp. 115-122, July 1998.
[29] C. Bajaj and D. Schikore, “Error-Bounded Reduction of Triangle Meshes with Multivariate Data,” Proc. Visual Data Exploration and Analysis III, pp. 34-45, Mar. 1996.
[30] A. Guéziec, Locally Toleranced Surface Simplification IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 2, pp. 168-189, Apr.-June 1999.
[31] M. Garland and P.S. Heckbert, "Surface Simplification Using Quadric Error Metrics," Proc. Siggraph 97, ACM Press, New York, 1997, pp. 209-216.
[32] H. Hoppe, New Quadric Metric for Simplifying Meshes with Appearance Attributes Proc. IEEE Visualization '99, pp. 59-66, 1999.
[33] R. Ronfard and J. Rossignac, “Full-Range Approximation of Triangulated Polyhedra,” Computer Graphics Forum, vol. 15, no. 3, pp. C67-C76, 1996.
[34] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[35] D. Eberly, “Magic Software,” http:/, 2000.
[36] J. Cohen, M. Lin, D. Manocha, and M. Ponamgi, "I-Collide: An Interactive and Exact Collision Detection System for Large-Scale Environments," Proc. ACM Interactive 3D Graphics Conf., pp. 189-196, 1995.
[37] P.M. Hubbard, "Interactive Collision Detection," Proc. IEEE Symp. Research Frontiers in Virtual Reality, Oct. 1993.
[38] N. Max, “Visualizing Hilbert Curves,” Proc. Visualization '98, pp. 447-450, Oct. 1998.
[39] R. Dafner, D. Cohen-Or, and Y. Matias, “Context-Based Space Filling Curves,” Computer Graphics Forum, vol. 19, no. 3, pp. C209-C217, 2000.
[40] A. Kaufman, “Efficient Algorithms for 3-D Scan Conversion of Parametric Curves, Surfaces, and Volumes,” Computer Graphics, vol. 21, pp. 171-179, 1987.
[41] P.E. Danielsson, “Euclidean Distance Mapping,” Computer Graphics and Image Processing, vol. 14, pp. 227-248, 1980.
[42] D. Baraff, A. Witkin, and M. Kass, “Physically Based Modelling,” Course Notes 36, ACM SIGGRAPH '99, Aug. 1999.
[43] M.M. Chow, “Optimized Geometry Compression for Real-Time Rendering,” Proc. IEEE Visualization '97, pp. 346-354, Nov. 1997.
[44] G. Taubin et al., “Geometry Coding and VRML,” Proc. IEEE, vol. 86, no. 6, pp. 1228-1243, June 1998.
[45] T. Moeller, “A Fast Triangle-Triangle Intersection Test,” J. Graphics Tools, vol. 2, no. 2, pp. 25-30, 1997.
[46] P. Volino and N.M. Thalmann, “Efficient Self-Collision Detection on Smoothly Discretized Surface Animations Using Geometrical Shape Regularity,” Computer Graphics Forum, vol. 13, no. 3, pp. C155-C166, 1994.

Index Terms:
Triangular mesh, closest point, multiresolution hierarchy, priority process, dynamic queries.
André Guéziec, "'Meshsweeper': Dynamic Point-to-Polygonal-Mesh Distance and Applications," IEEE Transactions on Visualization and Computer Graphics, vol. 7, no. 1, pp. 47-61, Jan.-March 2001, doi:10.1109/2945.910820
Usage of this product signifies your acceptance of the Terms of Use.