圆锥曲线沿直线滚动所形成曲线的研究

运动平面∑在一个静止平面∑0上运动，可理解为绕其瞬时转动中心作无穷小的旋转，据此可得出抛物线、椭圆和双曲线沿直线滚动时，其上任一点的轨迹曲线的法线在运动的任何时刻都通过此刻相应的切点(瞬时转动中心)的结论．并利用圆锥曲线的性质，推导出其焦点轨迹曲线方程，并得出结论：圆锥曲线沿直线滚动时，其抛物线的焦点轨迹曲线为一条悬链线，而椭圆和双曲线的焦点轨迹，其形状与缩短和伸长摆线相似．

The Research Of The Curve Formed By The Conic Section Rolled On Line

The movement of the kinematic plane on a immobile plane means that the plane infinitesimally rotate round the instantaneous velocity pole. According to the situation introduced above, we can reach the conclusion that when parabola, ellipse and hyperbola roll on a line, the normal of the trochoid of any point on them get across to the point of tangency (instantaneous velocity pole) at any time in the movement. Using the character of the conic section, we can educe the equation of trochoid of the focus:So we can make a conclusion: when the conic section roll on a line, the trochoid of the parabola's focus is a catenary, but the shape of the trochoid of ellipse and hyperbola's focus are similar to the shape of shorten or elongate cycloid.