抛物线与双曲线

开普勒把行星的轨道描述为椭圆，牛顿后来修改了这些椭圆，因为他指出这些轨道是特殊的圆锥截面，如抛物线和双曲线。抛物线和双曲线有许多相似之处，但也有区别，因为有不同的方程来解决涉及这些圆锥截面的几何问题。为了更好地理解抛物线和双曲线之间的区别，我们需要了解这些圆锥曲线。

Image Courtesy: http://cseligman.com

A section is a surface or the outline of that surface formed by cutting a solid figure with a plane. If the solid figure happens to be a cone, the resulting curve is called a conic section. The kind and shape of the conic section is determined by the angle of intersection of the plane and the axis of the cone. When the cone is cut at right angles to the axis, we get a circular shape. When cut at less than a right angle but more than the angle made by the side of the cone results in an ellipse. When cut parallel to the side of the cone, the curve obtained is a parabola and when cut nearly parallel to the axis that to the side, we get a curve known as hyperbola. As you can see from the figures, circles and ellipses are closed curves whereas parabolas and hyperbolas are open curves. In the case of a parabola, the two arms eventually become parallel to each other whereas in the case of a hyperbola it is not so.

因为圆和抛物线是通过在特定角度切割圆锥而形成的，所以所有的圆在形状上是相同的，所有抛物线的形状都是相同的。在双曲线和椭圆的情况下，平面和轴之间的角度范围很广，这就是为什么它们的形状会有很大的变化。四种圆锥曲线的方程如下。

圆-x2+y2=1

椭圆-x2/a2+y2/b2=1

抛物线-y2=4ax