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Birational Quadratic Planar Maps with Generalized Complex Rational Representations

Author

Listed:
  • Xuhui Wang

    (Department of Mathematics, Hohai University, Nanjing 211100, China)

  • Yuhao Han

    (Department of Mathematics, Hohai University, Nanjing 211100, China)

  • Qian Ni

    (School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, China)

  • Rui Li

    (Department of Mathematics, Hohai University, Nanjing 211100, China)

  • Ron Goldman

    (Department of Computer Science, Rice University, Houston, TX 77251, USA)

Abstract

Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.

Suggested Citation

  • Xuhui Wang & Yuhao Han & Qian Ni & Rui Li & Ron Goldman, 2023. "Birational Quadratic Planar Maps with Generalized Complex Rational Representations," Mathematics, MDPI, vol. 11(16), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3609-:d:1221422
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