Browsing by Author "Wang, Xuhui"

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    Birational Quadratic Planar Maps with Generalized Complex Rational Representations
    (MDPI, 2023) Wang, Xuhui; Han, Yuhao; Ni, Qian; Li, Rui; Goldman, Ron
    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.
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    Rational curves over generalized complex numbers
    (Elsevier, 2019) Du, Juan; Goldman, Ron; Wang, Xuhui
    Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.