Birational Quadratic Planar Maps with Generalized Complex Rational Representations

Autor: Xuhui Wang, Yuhao Han, Qian Ni, Rui Li, Ron Goldman
Jazyk: angličtina
Rok vydání: 2023
Předmět:
Zdroj: Mathematics, Vol 11, Iss 16, p 3609 (2023)
Druh dokumentu: article
ISSN: 2227-7390
33883483
DOI: 10.3390/math11163609
Popis: 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.
Databáze: Directory of Open Access Journals
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