Comparison of three curve intersection algorithms
Computer-Aided Design
Improperly parametrized rational curves
Computer Aided Geometric Design
An introduction to splines for use in computer graphics & geometric modeling
An introduction to splines for use in computer graphics & geometric modeling
On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Algorithm for algebraic curve intersection
Computer-Aided Design
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Solutions of tangential surface and curve intersections
Computer-Aided Design
Regular curves and proper parametrizations
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Hidden curve removal for free form surfaces
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
Fat arcs: a bounding region with cubic convergence
Computer Aided Geometric Design
Numerically stable implicitization of cubic curves
ACM Transactions on Graphics (TOG)
LAPACK's user's guide
Detecting cusps and inflection points in curves
Computer Aided Geometric Design
An efficient algorithm for infallible polynomial complex root isolation
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
On computing condition numbers for the nonsymmetric eigenproblem
ACM Transactions on Mathematical Software (TOMS)
Algebraic and numeric techniques in modeling and robotics
Algebraic and numeric techniques in modeling and robotics
Algorithms for Intersecting Parametric and Algebraic Curves
Algorithms for Intersecting Parametric and Algebraic Curves
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
Implicit and parametric curves and surfaces for computer aided geometric design
Implicit and parametric curves and surfaces for computer aided geometric design
Interactive display of large-scale NURBS models
I3D '95 Proceedings of the 1995 symposium on Interactive 3D graphics
Numeric-symbolic algorithms for evaluating one-dimensional algebraic sets
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Fast and Accurate Collision Detection for Virtual Environments
Dagstuhl '97, Scientific Visualization
Computing the topology of a real algebraic plane curve whose equation is not directly available
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Global minimization of rational functions and the nearest GCDs
Journal of Global Optimization
Computing nearest Gcd with certification
Proceedings of the 2009 conference on Symbolic numeric computation
A subdivision method for computing nearest gcd with certification
Theoretical Computer Science
GPU-based parallel solver via the Kantorovich theorem for the nonlinear Bernstein polynomial systems
Computers & Mathematics with Applications
Computing curve intersection by homotopy methods
Journal of Computational and Applied Mathematics
Bernstein Bezoutians and application to intersection problems
Computer Aided Geometric Design
Computer Aided Geometric Design
A geometric strategy for computing intersections of two spatial parametric curves
The Visual Computer: International Journal of Computer Graphics
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The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics and geometric and solid modeling. Previous algorithms are based on techniques from elimination theory or subdivision and iteration. The former is, however, restricted to low-degree curves. This is mainly due to issues of efficiency and numerical stability. In this article we use elimination theory and express the resultant of the equations of intersection as matrix determinant. The matrix itself rather than its symbolic determinant, a polynomial, is used as the representation. The problem of intersection is reduced to that of computing the eigenvalues and eigenvectors of a numeric matrix. The main advantage of this approach lies in its efficency and robustness. Moreover, the numerical accuracy of these operations is well understood. For almost all cases we are able to compute accurate answers in 64-bit IEEE floating-point arithmetic.