Intersection of parametric surfaces in the Bernstein—Be´zier representation
Computer-Aided Design
The characterization of parametric surface sections
Computer Vision, Graphics, and Image Processing
Piecewise-circular curves for geometric modeling
IBM Journal of Research and Development
On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
Loop detection in surface patch intersections
Computer Aided Geometric Design
Computer Aided Geometric Design
Automatic parameterization of rational curves and surfaces III: algebraic plane curves
Computer Aided Geometric Design
Geometric method of intersecting natural quadrics represented in trimmed surface form
Computer-Aided Design
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
A marching method for parametric surface/surface intersection
Computer Aided Geometric Design
Recursive subdivision and iteration in intersections and related problems
Mathematical methods in computer aided geometric design
Using plane vector fields to obtain all the intersection curves of two general surfaces
Theory and practice of geometric modeling
A dimensionality paradigm for surface interrogations
Computer Aided Geometric Design
Linear programming and convex hulls made easy
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Geometric Hermite approximation of surface patch intersection curves
Computer Aided Geometric Design
Interval analysis for computer graphics
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Surface-to-Surface Intersections
IEEE Computer Graphics and Applications - Special issue on computer-aided geometric design
Algebraic and numeric techniques in modeling and robotics
Algebraic and numeric techniques in modeling and robotics
Computing selected solutions of polynomial equations
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Algebraic pruning: a fast technique for curve and surface intersection
Computer Aided Geometric Design
Exploiting topological and geometric properties for selective subdivision
SCG '85 Proceedings of the first annual symposium on Computational geometry
IEEE Computer Graphics and Applications
Solving Systems of Polynomial Equations
IEEE Computer Graphics and Applications
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Topologically reliable display of algebraic curves
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
Implicit and parametric curves and surfaces for computer aided geometric design
Implicit and parametric curves and surfaces for computer aided geometric design
Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic
SMA '97 Proceedings of the fourth ACM symposium on Solid modeling and applications
Fast collision detection using QuOSPO trees
I3D '99 Proceedings of the 1999 symposium on Interactive 3D graphics
Normal bounds for subdivision-surface interference detection
Proceedings of the conference on Visualization '01
Ray-Tracing Triangular Trimmed Free-Form Surfaces
IEEE Transactions on Visualization and Computer Graphics
Using Linked Volumes to Model Object Collisions, Deformation, Cutting, Carving, and Joining
IEEE Transactions on Visualization and Computer Graphics
Fast and Accurate Collision Detection for Virtual Environments
Dagstuhl '97, Scientific Visualization
Computing intersections of rational patches
Machine Graphics & Vision International Journal
Shape modeling with point-sampled geometry
ACM SIGGRAPH 2003 Papers
ACM SIGGRAPH 2003 Papers
Selfintersections of a bézier bicubic surface
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A complete and non-overlapping tracing algorithm for closed loops
Computer Aided Geometric Design
On the geometry of parametrized bicubic surfaces
Journal of Symbolic Computation
Trajectory generation for open-contoured structures in robotic fibre placement
Robotics and Computer-Integrated Manufacturing
Intersection and self-intersection of surfaces by means of Bezoutian matrices
Computer Aided Geometric Design
ACM SIGGRAPH 2008 papers
Performing efficient NURBS modeling operations on the GPU
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Improved Binary Space Partition merging
Computer-Aided Design
Accelerating geometric queries using the GPU
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
A complete and non-overlapping tracing algorithm for closed loops
Computer Aided Geometric Design
Computing minimum distance between two implicit algebraic surfaces
Computer-Aided Design
Some applications of scalar and vector fields to geometric processing of surfaces
Computers and Graphics
Technical Section: Method of trimming PDE surfaces
Computers and Graphics
A new differential approach for parametric-implicit surface intersection
ICCS'03 Proceedings of the 1st international conference on Computational science: PartI
Critical point analysis using domain lifting for fast geometry queries
Computer-Aided Design
Computer Aided Geometric Design
Fast intersections for subdivision surfaces
ICCSA'06 Proceedings of the 6th international conference on Computational Science and Its Applications - Volume Part I
A marching method for computing intersection curves of two subdivision solids
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Approximating algebraic space curves by circular arcs
Proceedings of the 7th international conference on Curves and Surfaces
Weaker conditions for the convergence of Newton's method
Journal of Complexity
Trimming for subdivision surfaces
Computer Aided Geometric Design
Secant-type methods and nondiscrete induction
Numerical Algorithms
Journal of Symbolic Computation
Robotics and Computer-Integrated Manufacturing
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We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algbraic formulation. In particular, we propose and matrix computations. We present algorithms to compute a start point on each component of the intersection curve (both open and closed components), detect the presence of singularities, and find all the curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.