Expert: Paul Klarreich - 6/25/2010

Question
According to Kepler, "when the angles between the plane and the cone are slowly increased one will get ellipse, parabola and hyperbola. But after ellipse, one can get parabola when the plane is parallel to one side of the cone." But I can get an unbounded curve(I'm not sure that it is parabola. If it is not, then what it is?)when my plane cuts the base of the cone at all angles(not parallel) less than the angle when the plane touches the other cone which is at the end of the first cone. Please explain me... I'm waiting eagerly for your reply.

Answer
Questioner:    shameem
Private: No
Subject:  
Question: According to Kepler, "when the angles between the plane and the cone are slowly increased one will get ellipse, parabola and hyperbola. But after ellipse, one can get parabola when the plane is parallel to one side of the cone." But I can get an unbounded curve(I'm not sure that it is parabola. If it is not, then what it is?)when my plane cuts the base (NO SUCH THING - THE CONE IS UNBOUNDED AND HAS TWO PIECES THAT TOUCH AT THEIR COMMON VERTEX) of the cone at all angles(not parallel) less than the angle when the plane touches the other cone (PIECE) which is at the end of the first cone. Please explain me... I'm waiting eagerly for your reply


That is how you get the hyperbola.  If the plane rotates past parallel to an 'edge', then it must cut the other piece (I think it is called a nape.) and the intersection has two parts.