IBM Journal of Research and Development
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 curves and surfaces with rational offsets
Computer Aided Geometric Design
Computing rational parametrizations of canal surfaces
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Parametric generalized offsets to hypersurfaces
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Applications of Laguerre geometry in CAGD
Computer Aided Geometric Design
A Laguerre geometric approach to rational offsets
Computer Aided Geometric Design
Blending two cones with Dupin cyclides
Computer Aided Geometric Design
Minkowski pythagorean hodographs
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
Polynomial and Rational Pythagorean-Hodograph Curves Reconciled
Proceedings of the 6th IMA Conference on the Mathematics of Surfaces
Rational Parametrization of Canal Surface by 4 Dimensional Minkowski Pythagorean Hodograph Curves
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
C1 Hermite interpolation using MPH quartic
Computer Aided Geometric Design
Clifford algebra, Lorentzian geometry, and rational parametrization of canal surfaces
Computer Aided Geometric Design
Analytic and algebraic properties of canal surfaces
Journal of Computational and Applied Mathematics - Special issue: The international symposium on computing and information (ISCI2004)
Rational hypersurfaces with rational convolutions
Computer Aided Geometric Design
Convolution surfaces of quadratic triangular Bézier surfaces
Computer Aided Geometric Design
The implicit equation of a canal surface
Journal of Symbolic Computation
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Rational surfaces with linear normals and their convolutions with rational surfaces
Computer Aided Geometric Design
G1 Hermite interpolation by Minkowski Pythagorean hodograph cubics
Computer Aided Geometric Design
On convolutions of algebraic curves
Journal of Symbolic Computation
On rational Minkowski Pythagorean hodograph curves
Computer Aided Geometric Design
Computer Aided Geometric Design
Graphical Models
A unified Pythagorean hodograph approach to the medial axis transform and offset approximation
Journal of Computational and Applied Mathematics
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A canal surface is the envelope of a 1-parameter set of spheres centered at the spine curve m(t) and with the radii described by the function r(t). Any canal surface given by rational m(t) and r(t) possesses a rational parameterization. However, an arbitrary rational canal surface does not have to fulfill the PN (Pythagorean normals) condition. Most (exact or approximate) parameterization methods are based on a construction of a rational unit normal vector field guaranteeing rational offsets. In this paper, we will study a condition which guarantees that a given canal surface has rational contour curves, which are later used for a straightforward computation of rational parameterizations of canal surfaces providing rational offsets. Using the contour curves in the parameterization algorithm brings another extra feature; the parameter lines do not unnecessarily wind around the canal surface. Our approach follows a construction of rational spatial MPH curves from the associated planar PH curves introduced in Kosinka and Lavicka (2010) [28] and gives it to the relation with the contour curves of canal surfaces given by their medial axis transforms. We also present simple methods for computing approximate PN parameterizations of given canal surfaces and rational offset blends between two canal surfaces.