存在圆的渐开线通过两个三重点的条件

THE CONDITION OF TWO TRIPLE POINTS WHICH CAN BE PASSED BY AN INVOLUTE OF A CIRCLE

设p(x,y)是平面上一点,如果给定两个实数m和k与p对应,则p((x,y,),m,k)称为三重点.在平面上给定两个三重点p_i((x_i,y_i),m_i,k_i),(i=0,1;x_0<x_1)满足容许条件,即劣弧??和劣弧??位于弦p_0p_1的同一侧.本文给出了存在圆的渐开线过p_0和p_1并在p_0和p_1处的斜率分别为m_0和m_1,曲率分别为k_0和k_1的充分必要条件;同时还给出了一种计算通过两个三重点的C^2-类曲线的方法.

A point P(x,y) in plane with two real numbers m and k is called a triple point, denoted by P((x,y),m,k). A C^2-curve Г passes a triple point P((x,y),m,k) if Г passes P(x,y), and the slope and curvature of Г at P(x,y) are m and k respectively. In this paper, we give a sufficient and necessary condition of two triple points which can be passed by an involute of a circle, and a new method to calculate a C^2-curve passing two given triple points and with the curvature nearly monotone.