Computer-Aided Design
Automatic parametrization of rational curves and surfaces II: cubics and cubicoids
Computer-Aided Design
Parametrization of cubic algebraic surfaces
Proceedings on Mathematics of surfaces II
The displacement method for implicit blending surfaces in solid models
ACM Transactions on Graphics (TOG) - Special issue on computer-aided design
Curves and surfaces for computer aided geometric design
Curves and surfaces for computer aided geometric design
Blending of implicit surfaces with functional splines
Computer-Aided Design
Symbolic parametrization of curves
Journal of Symbolic Computation
On the choice of pencils in the parametrization of curves
Journal of Symbolic Computation
Cyclides in Surface and Solid Modeling
IEEE Computer Graphics and Applications - Special issue on computer-aided geometric design
An algebraic approach to curves and surfaces on the sphere and on other quadrics
Selected papers of the international symposium on Free-form curves and free-form surfaces
Rational parametrizations of algebraic curves using a canonical divisor
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Computing rational parametrizations of canal surfaces
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Rational parametrization of surfaces
Journal of Symbolic Computation
Applications of Laguerre geometry in CAGD
Computer Aided Geometric Design
Blending two cones with Dupin cyclides
Computer Aided Geometric Design
Minkowski pythagorean hodographs
Computer Aided Geometric Design
Studying cyclides with Laguerre geometry
Computer Aided Geometric Design
Proper parametrization of real tubular surfaces
Journal of Symbolic Computation
Implicit Gn-blending of vertices
Computer Aided Geometric Design
The parametrization of canal surfaces and the decomposition of polynomials into a sum of two squares
Journal of Symbolic Computation - Special issue on computer algebra and mechanized reasoning: selected St. Andrews' ISSAC/Calculemus 2000 contributions
Gn-continous connections between normal ringed surfaces
Computer Aided Geometric Design
Rational Parametrization of Canal Surface by 4 Dimensional Minkowski Pythagorean Hodograph Curves
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
Clifford algebra, Lorentzian geometry, and rational parametrization of canal surfaces
Computer Aided Geometric Design
Minimal rational parametrizations of canal surfaces
Computing - Special issue on Geometric Modeling (Dagstuhl 2005)
On the computation of the topology of a non-reduced implicit space curve
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
The implicit equation of a canal surface
Journal of Symbolic Computation
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Computation of the topology of real algebraic space curves
Journal of Symbolic Computation
Local parametrization of cubic surfaces
Journal of Symbolic Computation
On rational Minkowski Pythagorean hodograph curves
Computer Aided Geometric Design
Graphical Models
Blends of canal surfaces from polyhedral medial transform representations
Computer-Aided Design
Parameterizing rational offset canal surfaces via rational contour curves
Computer-Aided Design
Journal of Computational and Applied Mathematics
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A canal surface is the envelope of a one-parameter family of spheres centered at the spine curve m(t) and with the radii described by the function r(t). It was proved in Peternell and Pottmann (1997) [9] that any canal surface to a rational spine curve and a rational radius function possesses a rational parameterization. Then a symbolic method for generating rational parameterizations of canal surfaces was developed in Landsmann et al. (2001) [21]. Indeed, this method leads to the problem of decomposing a polynomial into a sum of two squares over reals, which is solved numerically in general. Hence, approximate techniques generating a parameterization within a certain region of interest are also worth studying. In this paper, we present a method for the computation of approximate rational parameterizations of canal surfaces. A main feature of our approach is a combination of symbolic and numerical techniques yielding approximate topology-based parameterizations of contour curves which are then applied to compute an approximate parameterization of the given canal surface. The algorithm is mainly suitable for implicit blend surfaces of the canal-surface-type.