A note on Pythagorean hodograph quartic spiral  

作　　者：ZHENG Zhi-hao WANG Guo-zhao 

机构地区：[1]School of Mathematical Sciences, Zhejiang University

出　　处：《Applied Mathematics(A Journal of Chinese Universities)》2018年第2期234-252,共19页高校应用数学学报（英文版）（B辑）

基　　金：Supported by the National Natural Science Foundation of China(61272300)

摘　　要：By using the geometric constraints on the control polygon of a Pythagorean hodo- graph （PH） quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.By using the geometric constraints on the control polygon of a Pythagorean hodo- graph （PH） quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

关 键 词：Pythagorean hodograph quartic curve SPIRAL CURVATURE transition curve 

分 类 号：O186.1[理学—数学]