A conic section is defined as the curve of the intersection of a plane with a right circular cone of two nappes. There are three types of curves that occur in this way: the parabola, the ellipse, and the hyperbola. The resulting curves depend upon the inclination of the axis of the cone to the cutting plane. The Greek mathematician Apollonius studied conic sections geometrically using this concept. In this section, we shall give an analytic definition of a conic section, and as special cases of this definition, we shall obtain the three types of curves.
When considering conic sections geometrically, a cone is thought of as having two nappes extending in both directions. A portion of a right circular cone of two nappes is shown in the figure below.
A generator (or element) of the cone is a line lying in the cone, and all generators of a cone contain the point V, called the vertex of the cone.
In figure a below, we have a cone and a cutting plane which is parallel to one and only one generator of the cone. This conic is a parabola. If the cutting plane is parallel to two generators, this intersects nappes of the cone, and a hyperbola is obtained. An ellipse is obtained if the cutting plane is parallel to no generator, in which case the cutting plane intersects each generator, as shown in figure c.
A special case of the ellipse is a circle, which is obtained if the cutting plane, which intersects each generator, is also perpendicular to the axis of the cone. Degenerate cases of the conic sections include a point, a straight line, and two intersecting straight lines. A point is obtained if the cutting plane contains the vertex of the cone but does not contain a generator. This is a degenerate ellipse. If the cutting plane contains the vertex of the cone and only one generator, then a straight line is obtained, and this is a degenerate parabola. A degenerate hyperbola is obtained when the cutting plane contains the vertex of the cone and two generators, giving us two intersecting straight lines.