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.[more]
When the plane passes through the apex of the cone, the intersection is a point, one line, or a double line.[less]
Mathematical note: If the half-angle of the cone is then the eccentricity of the intersection is , where is the angle of inclination of the plane. Its projection in the plane is also a conic section, with focus at the origin and eccentricity . Note that the eccentricity of the intersection and of its projection are equal for precisely two values of , namely 0 and , corresponding to eccentricities of 0 and 1, and thus to the circle and parabola, respectively.
Snapshot 1: circle
Snapshot 2: ellipse
Snapshot 3: parabola
Snapshot 4: hyperbola