Implicitization, parameterization and singularity computation of Steiner surfaces using moving surfaces

Authors:
Xuhui Wang;Falai Chen
Affiliations:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China and School of Mathematics, Hefei University of Technology, Hefei, Anhui, 230009, China;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China
Venue:
Journal of Symbolic Computation
Year:
2012

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Abstract

A Steiner surface is a quadratically parameterizable surface without base points. To make Steiner surfaces more applicable in Computer Aided Geometric Design and Geometric Modeling, this paper discusses implicitization, parameterization and singularity computation of Steiner surfaces using the moving surface technique. For implicitization, we prove that there exist two linearly independent moving planes with total degree one in the parametric variables. From this fact, the implicit equation of a Steiner surface can be expressed as a 3x3 determinant. The inversion formula and singularities for the Steiner surface can also be easily computed from the moving planes. For parameterization, we first compute the singularities of a Steiner surface in implicit form. Based on the singularities, we can find some special moving planes, from which a quadratic parameterization of the Steiner surface can be retrieved.