Birational Quadratic Planar Maps with Generalized Complex Rational Representations

Xuhui Wang (), Yuhao Han, Qian Ni, Rui Li and Ron Goldman
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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

Mathematics, 2023, vol. 11, issue 16, 1-13

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.

Keywords: birational map; ? -basis; syzygy; inverse equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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