Theoretically-based algorithms for robustly tracking intersection curves of deforming surfaces

This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for continuously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. The set of intersection curves of two deforming surfaces over all time is formulated as an implicit 2-manifold I in an augmented (by time domain) parametric space R^5. Hyperplanes corresponding to some fixed time instants may touchI at some isolated transition points, which delineate transition events, i.e. the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of five constraints in five variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper-tangent bounding cones. The actual transition events are computed by contouring the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the Euclidean space in which the surfaces are embedded.

[1]  Helmut Pottmann,et al.  Optimal slicing of free-form surfaces , 2002, Comput. Aided Geom. Des..

[2]  Michael E. Hohmeyer,et al.  A surface intersection algorithm based on loop detection , 1991, SMA '91.

[3]  Michael I. Jordan,et al.  Surface/surface intersection , 1987, Comput. Aided Geom. Des..

[4]  Nicholas M. Patrikalakis,et al.  Topological and differential-equation methods for surface intersections , 1992, Comput. Aided Des..

[5]  R. L. Magedson,et al.  Solutions of tangential surface and curve intersections , 1989 .

[6]  Robert E. Barnhill,et al.  A marching method for parametric surface/surface intersection , 1990, Comput. Aided Geom. Des..

[7]  Gershon Elber,et al.  Global curvature analysis and segmentation of volumetric data sets using trivariate B-spline functions , 2004, Geometric Modeling and Processing, 2004. Proceedings.

[8]  Leslie C. Wilson.  Infinitely determined map germs , 1981 .

[9]  Takashi Maekawa,et al.  An overview of offset curves and surfaces , 1999, Comput. Aided Des..

[10]  J. Damon.  Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case , 2004, Compositio Mathematica.

[11]  Johannes Wallner,et al.  Self-Intersections of Offset Curves and Surfaces , 2001, Int. J. Shape Model..

[12]  Binh Pham,et al.  Offset curves and surfaces: a brief survey , 1992, Comput. Aided Des..

[13]  Chandrajit L. Bajaj,et al.  Tracing surface intersections , 1988, Comput. Aided Geom. Des..

[14]  James N. Damon,et al.  Determining the Geometry of Boundaries of Objects from Medial Data , 2005, International Journal of Computer Vision.

[15]  Nicholas M. Patrikalakis,et al.  Computation of singularities and intersections of offsets of planar curves , 1993, Comput. Aided Geom. Des..

[16]  Nicholas M. Patrikalakis,et al.  Computation of the solutions of nonlinear polynomial systems , 1993, Comput. Aided Geom. Des..

[17]  T. Sederberg,et al.  Improved test for closed loops in surface intersections , 1989 .

[18]  Nicholas M. Patrikalakis,et al.  Surface to Surface Intersections , 2004 .

[19]  Alfred M. Bruckstein,et al.  Shape offsets via level sets , 1993, Comput. Aided Des..

[20]  Xiuzi Ye,et al.  Differential geometry of intersection curves of two surfaces , 1999, Comput. Aided Geom. Des..

[21]  Xiuzi Ye,et al.  Robust interval algorithm for surface intersections , 1997, Comput. Aided Des..

[22]  Gershon Elber,et al.  Geometric constraint solver using multivariate rational spline functions , 2001, SMA '01.

[23]  IRkGuoliang Xu.  Curvature Computations of 2-manifolds in , 2003 .

[24]  B. G. Prakash,et al.  Computing offsets of trimmed NURBS surfaces , 2003, Comput. Aided Des..

[25]  Michael E. Hohmeyer.  A surface intersection algorithm based on loop detection , 1991, Int. J. Comput. Geom. Appl..

[26]  Gershon Elber,et al.  Error bounded variable distance offset operator for free form curves and surfaces , 1991, Int. J. Comput. Geom. Appl..

[27]  唐敏,et al.  Intersections of two offset parametric surfaces based on topology analysis , 2004 .

[28]  Bernd Hamann.  Visualization and modeling contours of trivariate functions , 1991 .

[29]  Gershon Elber,et al.  Comparing Offset Curve Approximation Methods , 1997, IEEE Computer Graphics and Applications.

[30]  J. Damon.  Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case , 2004, Compositio Mathematica.

[31]  Rida T. Farouki,et al.  Analytic properties of plane offset curves , 1990, Comput. Aided Geom. Des..

[32]  Karim Abdel-Malek,et al.  Determining intersection curves between surfaces of two solids , 1996, Comput. Aided Des..

[33]  Alexis Gourdon,et al.  Computing the Differential Characteristics of Isointensity Surfaces , 1995, Comput. Vis. Image Underst..

[34]  Thomas W. Sederberg,et al.  Loop detection in surface patch intersections , 1988, Comput. Aided Geom. Des..

[35]  B. O'neill.  Elementary Differential Geometry , 1966 .

[36]  Jaroslaw Rossignac,et al.  Proceedings of the first ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, Austin, Texas, USA, June 05-07, 1991 , 1991, Symposium on Solid Modeling and Applications.

[37]  J. Mather,et al.  Stability of C∞ mappings, I, the division theorem , 1968 .

[38]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[39]  Dong-Soo Kim,et al.  Surface slicing algorithm based on topology transition , 2001, Comput. Aided Des..

[40]  Jan J. Koenderink,et al.  Solid shape , 1990 .

[41]  Ron Goldman,et al.  Curvature formulas for implicit curves and surfaces , 2005, Comput. Aided Geom. Des..