A new approach for surface intersection
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants
Computer Aided Geometric Design
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Applied numerical linear algebra
Applied numerical linear algebra
On the validity of implicitization by moving quadrics for rational surfaces with no base points
Journal of Symbolic Computation
Mathematical Methods for Curves and Surfaces
Curve/surface intersection problem by means of matrix representations
Proceedings of the 2009 conference on Symbolic numeric computation
Implicitizing rational hypersurfaces using approximation complexes
Journal of Symbolic Computation
Matrix-based implicit representations of rational algebraic curves and applications
Computer Aided Geometric Design
On SVD for estimating generalized eigenvalues of singular matrixpencil in noise
IEEE Transactions on Signal Processing
The surface/surface intersection problem by means of matrix based representations
Computer Aided Geometric Design
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We introduce and study a new implicit representation of rational Bezier curves and surfaces in the 3-dimensional space. Given such a curve or surface, this representation consists of a matrix whose entries depend on the space variables and whose rank drops exactly on this curve or surface. Our approach can be seen as an extension of the moving lines implicitization method introduced by Sederberg, from non-singular matrices to the more general context of singular matrices. In the first part of this paper, we describe the construction of these new implicit matrix representations and their main geometric properties, in particular their ability to solve efficiently the inversion problem. The second part of this paper aims to show that these implicitization matrices adapt geometric problems, such as intersection problems, to the powerful tools of numerical linear algebra, in particular to one of the most important: the singular value decomposition. So, from the singular values of a given implicit matrix representation, we introduce a real evaluation function. We show that the variation of this function is qualitatively comparable to the Euclidean distance function. As an interesting consequence, we obtain a new determinantal formula for implicitizing a rational space curve or surface over the field of real numbers. Then, we show that implicit matrix representations can be used with numerical computations, in particular there is no need for symbolic computations to use them. We give some rigorous results explaining the numerical stability that we have observed in our experiments. We end the paper with a short illustration on ray tracing of parameterized surfaces.