First steps towards radical parametrization of algebraic surfaces

Abstract

This is the author’s\ud version of a work that was accepted for publication in\ud Computer Aided Geometric Design. Changes resulting from the publishing\ud process, such as peer review, editing, corrections,\ud structural formatting, and other quality control mechanisms may not be\ud reflected in this document.\ud Changes may have been made to this work since it was submitted for\ud publication.\ud A definitive version was subsequently published in Computer Aided\ud Geometric Design Volume 30, Issue 4, pp. 374-388 (2013) .\ud DOI: 10.1016/j.cagd.2012.12.004We introduce the notion of radical parametrization of a surface, and we provide algorithms\ud to compute such type of parametrizations for families of surfaces, like: Fermat surfaces,\ud surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible\ud surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces\ud containing a pencil of low-genus curves. In addition, we prove that radical parametrizations\ud are preserved under certain type of geometric constructions that include offset and\ud conchoids.The authors are supported by by the Spanish Ministerio de Ciencia e Innovaci´on under\ud the Project MTM2008-04699-C03-01, and by the Ministerio de Econom´ıa y Competitividad\ud under the project MTM2011-25816-C02-01; and the Austrian Science Fund (FWF)\ud P22766-N18, respectively. The first author is a member of the of the Research Group\ud asynacs (Ref. ccee2011/r34