Intersecting a Rotating Cone with a Plane

If the center of the cone is in the plane, the intersection is a point, a straight line, or a pair of straight lines, depending on the angle of the axis of the cone.

If the center of the cone is not in the plane, the intersection is a conic section. Let be the angle of the cone, that is, the angle between the axis and one of the generating lines of the cone. You get a circle if the angle is or , an ellipse if the angle is between and (or between and ), a parabola if the angle is , and a hyperbola if the angle is within of .