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

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 movin...

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Bibliographic Details
Published in:Journal of symbolic computation 2012-06, Vol.47 (6), p.733-750
Main Authors: Wang, Xuhui, Chen, Falai
Format: Article
Language:English
Subjects:
Implicitization
Inversion formula
Moving surface
Parameterization
Singularity
Steiner surface
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Summary: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 3×3 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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2011.12.029