Algebraic properties of plane offset curves
Computer Aided Geometric Design
Algorithms for intersecting parametric and algebraic curves I: simple intersections
ACM Transactions on Graphics (TOG)
Algorithms for intersecting parametric and algebraic curves II: multiple intersections
Graphical Models and Image Processing
Polynomial roots from companion matrix eigenvalues
Mathematics of Computation
An improved upper complexity bound for the topology computation of a real algebraic plane curve
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Implicitization of parametric curves and surfaces by using multidimensional Newton formulae
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Tracing index of rational curve parametrizations
Computer Aided Geometric Design
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Various New Expressions for Subresultants and Their Applications
Applicable Algebra in Engineering, Communication and Computing
Solving Polynomials with Small Leading Coefficients
SIAM Journal on Matrix Analysis and Applications
Geometric applications of the Bezout matrix in the Lagrange basis
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
IEEE Transactions on Computers
New algorithms for matrices, polynomials and matrix polynomials
New algorithms for matrices, polynomials and matrix polynomials
Shifting planes to follow a surface of revolution
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
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The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so-called ''polynomial algebra by values'' approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.