Three-dimensional space is translated to two-dimensional as the intersections of a cone and planes By slicing through a cone, a sharp knife produces cross-sections of: circles, if the cut is made exactly horizontal (perpendicular to the altitude) ellipses, if the cut is made less than horizontal hyperbolas, if the cut is made exactly vertical (parallel to the altitude) parabolas, if the cut is made less than vertical (parallel to the cone’s side). Image: conic.gif wyzant.com :