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A Survey of the Representations of Rational Ruled Surfaces

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Abstract

The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Plücker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them.

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Chun-Ming Yuan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Chun-Ming Yuan & Li-Yong Shen

  3. Grupo ASYNACS, Dpto. de Física y Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Madrid, Spain

    Sonia Pérez-Díaz

Authors
  1. Chun-Ming Yuan
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  2. Sonia Pérez-Díaz
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  3. Li-Yong Shen
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Corresponding author

Correspondence to Li-Yong Shen.

Additional information

This paper was supported by Beijing Natural Science Foundation under Grant No. Z190004, the National Natural Science Foundation of China under Grant No. 61872332, and the University of Chinese Academy of Sciences and by FEDER/Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación/MTM2017-88796-P (Symbolic Computation: New challenges in Algebra and Geometry together with its applications). The second author belongs to the Research Group ASYNACS (Ref. CCEE2011/R34).

This paper was recommended for publication by Editor FENG Ruyong.

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Yuan, CM., Pérez-Díaz, S. & Shen, LY. A Survey of the Representations of Rational Ruled Surfaces. J Syst Sci Complex 34, 2357–2377 (2021). https://doi.org/10.1007/s11424-020-0018-8

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  • DOI: https://doi.org/10.1007/s11424-020-0018-8

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