Automatic parameterization of rational curves and surfaces 1: conics and conicoids
Computer-Aided Design
Automatic parametrization of rational curves and surfaces II: cubics and cubicoids
Computer-Aided Design
The complexity of robot motion planning
The complexity of robot motion planning
Parametrization of cubic algebraic surfaces
Proceedings on Mathematics of surfaces II
Computer graphics: principles and practice (2nd ed.)
Computer graphics: principles and practice (2nd ed.)
Geometric computations with algebraic varieties of bounded degree
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Introduction—towards a new framework for vision
Geometric invariance in computer vision
Invariant theory and enumerative combinatorics of young tableaux
Geometric invariance in computer vision
Invariant signatures for planar shape recognition under partial occlusion
CVGIP: Image Understanding
Shastra: Multimedia Collaborative Design Environment
IEEE MultiMedia
ACM Transactions on Graphics (TOG)
Geometric Modeling with Algebraic Surfaces
Proceedings of the 3rd IMA Conference on the Mathematics of Surfaces
Triangulation and display of rational parametric surfaces
VIS '94 Proceedings of the conference on Visualization '94
Proper parametrization of surfaces with a rational pencil
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Universal parametrization and interpolation on cubic surfaces
Computer Aided Geometric Design
Implicitization and parametrization of quadratic surfaces with one simple base point
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
A remark on parameterizing nonsingular cubic surfaces
Computer Aided Geometric Design
Local parametrization of cubic surfaces
Journal of Symbolic Computation
Approximate rational parameterization of implicitly defined surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Characterization of rational ruled surfaces
Journal of Symbolic Computation
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Real cubic algebraic surfaces may be described by either implicit or parametric equations. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of 27 lines on them. We provide a method of extracting all these lines by constructing and solving a polynomial of degree 27. Simple roots of this polynomial correspond to real lines on the surface, and real skew lines are used to form rational parametrizations for three of these families. Complex conjugate skew lines are used to parametrize surfaces from the fourth family. The parametrizations for these four families involve quotients of polynomials of degree no higher than four. Each of these parametrizations covers the whole surface except for a few points, lines, or conic sections. The parametrization for the fifth family, as noted previously in the literature, requires a square root. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with “base” points where the parametrizations are ill-defined.