Conic Sections: The Double Cone

The quadratic curves are circles, ellipses, parabolas, and hyperbolas. They are called conic sections because each one is the intersection of a double cone and an inclined plane.

If the plane is perpendicular to the cone's axis, the intersection is a circle. If it is inclined at an angle greater than zero but less than the half-angle of the cone, it is an (eccentric) ellipse. If the plane's inclination is equal to this half-angle, the intersection is a parabola. If it exceeds the half-angle, it is a hyperbola.

When the plane passes through the apex of the cone, the intersection is a point, one line, or a double line.