On the determination of starting points for parametric surface intersections

Two numerical algorithms for computing starting points on the curve of intersection between two parametric surfaces are presented. The problem of determining intersection curves between two surfaces is analytically formulated by parametrizing inequality constraints into equality constraints and augmenting the constraint function. The first method uses an iterative optimization formulation and an iterative conjugate gradient algorithm to minimize a function comprising the vector of coordinates and a weighted constraint term. The second method uses the Moore-Penrose pseudo inverse of the constraint function to determine a starting point. Numerical examples are presented to validate both methods. Both methods require an initial point on one of the surfaces. Numerical examples illustrating the validity of the presented methods are discussed. The local versus the global views of the intersection problem in terms of iterative and recursive subdivision methods are addressed. Difficulties in determining more than one point are also illustrated using examples. The two algorithms are compared by studying their computational complexity. The Moore-Penrose inverse method has showed superior efficiency in the computational complexity, number of iterations needed, and time for conversion to a starting point. It is also shown that the Moore-Penrose inverse converges to a starting point in cases where the iterative optimization method does not.

[1]  Gregor Müllenheim.  Convergence of a surface/surface intersection algorithm , 1990, Comput. Aided Geom. Des..

[2]  B. J. Davies,et al.  Surface Engineering Geometry for Computer Aided Design and Manufacture , 1987 .

[3]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design , 1989 .

[4]  Gregor Müllenheim.  On determining start points for a surface/surface intersection algorithm , 1991, Comput. Aided Geom. Des..

[5]  F. A. Adkins,et al.  Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces , 1996 .

[6]  Itzhak Wilf,et al.  Quadric-surface intersection curves: shape and structure , 1993, Comput. Aided Des..

[7]  Christoph M. Hoffmann,et al.  Geometric and Solid Modeling: An Introduction , 1989 .

[8]  J. A. Gregory.  The Mathematics of Surfaces. , 1987 .

[9]  B. Noble.  Applied Linear Algebra , 1969 .

[10]  K. Cheng.  Using plane vector fields to obtain all the intersection curves of two general surfaces , 1989 .

[11]  Wayne Tiller,et al.  Applications of power series in computational geometry , 1986 .

[12]  Chaman L. Sabharwal,et al.  Implementation of a divide-and-conquer method for intersection of parametric surfaces , 1985, Comput. Aided Geom. Des..

[13]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

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

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

[16]  Vibeke Skytt,et al.  Recursive subdivision and iteration in intersections and related problems , 1989 .

[17]  Wolfgang W. Bein,et al.  Surface intersection using parallelism , 1994, Comput. Aided Geom. Des..

[18]  Roger Fletcher,et al.  A Rapidly Convergent Descent Method for Minimization , 1963, Comput. J..

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

[20]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[21]  Tor Dokken.  Finding intersections of B-spline represented geometries using recursive subdivision techniques , 1985, Comput. Aided Geom. Des..

[22]  Joe D. Warren,et al.  On computing the intersection of a pair of algebraic surfaces , 1989, Comput. Aided Geom. Des..

[23]  Robert Schaback,et al.  A parallel multistage method for surface/surface intersection , 1993, Comput. Aided Geom. Des..

[24]  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.

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

[26]  Michael Gleicher,et al.  An interval refinement technique for surface intersection , 1992 .

[27]  Christoph M. Hoffmann,et al.  A dimensionality paradigm for surface interrogations , 1990, Comput. Aided Geom. Des..

[28]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .

[29]  Dinesh Manocha,et al.  A new approach for surface intersection , 1991, SMA '91.

[30]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[31]  Thomas W. Sederberg.  Planar piecewise algebraic curves , 1984, Comput. Aided Geom. Des..

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

[33]  Christoph M. Hoffmann,et al.  On local implicit approximation and its applications , 1989, TOGS.

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

[35]  Robert E. Barnhill,et al.  Geometry Processing for Design and Manufacturing , 1992 .

[36]  Robert P. Markot,et al.  Procedural method for evaluating the intersection curves of two parametric surfaces , 1991, Comput. Aided Des..

[37]  Rida T. Farouki,et al.  The characterization of parametric surface sections , 1986, Comput. Vis. Graph. Image Process..

[38]  Shigeru Aomura,et al.  Intersection of Arbitrary Surfaces. , 1991 .

[39]  A. Requicha,et al.  Piecewise-circular curves for geometric modeling , 1987 .

[40]  K. Georg.  On Tracing an Implicitly Defined Curve by Quasi-Newton Steps and Calculating Bifurcation by Local Perturbations , 1981 .