某个偏锥面跟平面的交线

圆锥曲线在射影几何里最原始的定义是:

圆锥曲线（英语：conic section），又称圆锥截痕、圆锥截面、二次平面曲线，是数学、几何学中通过平切圆锥（严格为一个正圆锥面和一个平面完整相切）得到的曲线，包括圆，椭圆，抛物线，双曲线及一些退化类型。 圆锥曲线在约公元前200年时就已被命名和研究了，其发现者为古希腊的数学家阿波罗尼奥斯，那时阿波罗尼阿斯对它们的性质已做了系统性的研究。

圆锥曲线应用最广泛的定义为（椭圆，抛物线，双曲线的统一定义）：动点到一定点（焦点）的距离与其到一定直线（准线）的距离之比为常数（离心率e）的点的集合是圆锥曲线。对于0 < e < 1得到椭圆，对于e = 1得到抛物线，对于e > 1得到双曲线。 在笛卡尔坐标系内，二元二次方程的图像可以表示圆锥曲线，并且所有圆锥曲线都以这种方式引出。方程有如下形式

Ax2+Bxy+Cy2+Dx+Ey+F=0 ; 有着参数 A ,，B,和 C ,不得皆等于0,。

如果B2−4AC0 ,,方程表示圆； 如果 B2−4AC=0 ,，方程表示抛物线； 如果 B2−4AC>0 ,，方程表示双曲线； 如果还有 A=−C ,，方程表示直角双曲线。

Wolfram mathworld里面是:

The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9).

The ellipse and hyperbola are known as central conics.

Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible.

A conic section may more formally be defined as the locus of a point P that moves in the plane of a fixed point F called the focus and a fixed line d called the conic section directrix (with F not on d ) such that the ratio of the distance of P from F to its distance from d is a constant e called the eccentricity. If e=0, the conic is a circle, if 0