It's intuitive to know how arbitrary circles, and ellipses, hyperbola of varying eccentricities can be obtained through a plane and double cone intersection. However, I am not able to figure how one might obtain straight lines that intersect at arbitrary angles (or even parallel lines for that matter) as conics?

I currently understand that only a pair of straight lines can be obtained as a conic that intersects at an angle which is same as the vertex angles of the double cone, but I am not able to find a visual model that can show otherwise.


1 Answer 1


Given two lines in a common plane in Euclidean space, there are only two possible cases. Either they are parallel and have no intersection or else they intersect at exactly one point. In the first case, extending to projective space, the parallel lines do intersect at a single point at infinity. Consider the locus of all points equidistant from the two lines. In the first case the locus is a line parallel to both and midway between them. In the second case they are on two perpendicular lines passing through the common intersection point. Pick any one of them. In both cases use the line of equidistant points as a rotation axis to rotate the two lines into a surface of revolution. In the first case it is an infinite circular cylinder.

In the second case it is a circular double cone. In the first case, for any plane that intersects the cylinder, the points of intersection form a conic section -- either a circle, ellipse, or two parallel lines. For the second case, every plane intersects the double cone in at least one point. The points of intersection form a conic section -- either a single point, a circle, an ellipse, a parabola, a hyperbola, a single line, or a pair of intersecting lines.

In the case of intersecting lines, the plane passes through the vertex of the double cone and the angle of their intersection is less than or equal to the angle of the two generating lines. The angles are only equal if the rotation axis lies in the intersecting plane. As for visualization, try a web search for conic section images.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .