Planar Cubic Pythagorean-Hodograph Hyperbolic Curves
Received:March 02, 2021  Revised:September 04, 2021
Key Word: Pythagorean-hodograph curve   algebraic hyperbolic B\'{e}zier curve   $G^{1}$ Hermite interpolation  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11801225), University Science Research Project of Jiangsu Province (Grant No.18KJB110005) and the Research Foundation for Advanced Talents of Jiangsu University (Grant No.14JDG034).
Author NameAffiliation
Yongxia HAO School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
Lianxing LIAO School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
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      The purpose of this paper is to develop a method to construct the Pythagorean-hodograph hyperbolic (PH-H) curves based on the good geometric properties of PH curves and algebraic hyperbolic curves. The definition of Pythagorean-hodograph hyperbolic curves is given and their properties are examined. By using hyperbolic basis functions and algebraic B\'{e}zier basis functions respectively, two different necessary and sufficient conditions for a planar cubic algebraic hyperbolic B\'{e}zier curve to be a PH curve are obtained. Moreover, cubic PH-H curves are applied in the problem of $G^{1}$ Hermite interpolation with determined closed form solutions. Several examples serve to illustrate our method.
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