Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Interpolating polynomials from their values
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
Simple multivariate polynomial multiplication
Journal of Symbolic Computation
On the complexity of sparse elimination
Journal of Complexity
Implicitization of parametric curves and surfaces by using multidimensional Newton formulae
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
On computing the dual of a plane algebraic curve
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Studying cyclides with Laguerre geometry
Computer Aided Geometric Design
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Improved Sparse Multivariate Polynomial Interpolation Algorithms
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
Numerical Implicitization of Parametric Hypersurfaces with Linear Algebra
AISC '00 Revised Papers from the International Conference on Artificial Intelligence and Symbolic Computation
Lagrange interpolation on subgrids of tensor product grids
Mathematics of Computation
Comparison of Distance Measures for Planar Curves
Algorithmica
Computing the Fréchet distance between simple polygons
Computational Geometry: Theory and Applications
Computing the minimum distance between two Bézier curves
Journal of Computational and Applied Mathematics
Approximate parametrization of plane algebraic curves by linear systems of curves
Computer Aided Geometric Design
Implicit polynomial support optimized for sparseness
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartIII
An implicitization challenge for binary factor analysis
Journal of Symbolic Computation
Matrix-based implicit representations of rational algebraic curves and applications
Computer Aided Geometric Design
Approximate implicitization of triangular Bézier surfaces
Proceedings of the 26th Spring Conference on Computer Graphics
An output-sensitive algorithm for computing projections of resultant polytopes
Proceedings of the twenty-eighth annual symposium on Computational geometry
Implicitization of curves and surfaces using predicted support
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Rational Hausdorff divisors: A new approach to the approximate parametrization of curves
Journal of Computational and Applied Mathematics
Hi-index | 5.23 |
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation given a superset of its terms. For predicting these terms, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods in Maple, and some in Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples. In building a square or rectangular interpolation matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complex numbers offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive.