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
Offset-rational parametric plane curves
Computer Aided Geometric Design
Pipe surfaces with rational spine curve are rational
Computer Aided Geometric Design
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
Studying cyclides with Laguerre geometry
Computer Aided Geometric Design
Rational surfaces with linear normals and their convolutions with rational surfaces
Computer Aided Geometric Design
Minimal rational parametrizations of canal surfaces
Computing - Special issue on Geometric Modeling (Dagstuhl 2005)
On quadratic two-parameter families of spheres and their envelopes
Computer Aided Geometric Design
On generalized ln-surfaces in 4-space
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
MOS surfaces: medial surface transforms with rational domain boundaries
Proceedings of the 12th IMA international conference on Mathematics of surfaces XII
On rational Minkowski Pythagorean hodograph curves
Computer Aided Geometric Design
A unified Pythagorean hodograph approach to the medial axis transform and offset approximation
Journal of Computational and Applied Mathematics
Blends of canal surfaces from polyhedral medial transform representations
Computer-Aided Design
Research on 3D medial axis transform via the saddle point programming method
Computer-Aided Design
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The present paper investigates two-parameter families of spheres in R^3 and their corresponding two-dimensional surfaces @F in R^4. Considering a rational surface @F in R^4, the envelope surface @J of the corresponding family of spheres in R^3 is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface @J and its offset surfaces admit rational parameterizations if and only if @F is a rational sub-variety of a rational isotropic hyper-surface in R^4. The close relation between the envelope surfaces @J and rational offset surfaces in R^3 is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces @F in R^4 whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.