Asymptotes of hyperbolic sections of a given cone

[imurme8]

A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only those hyperbolas whose asymptotes make a sufficiently small angle occur as plane sections."

It seems to me that all hyperbolic sections of the same cone must have asymptotes that make exactly the same angle with each other (the angle formed by two antipodal generators of the cone). Is this incorrect? The wording in the book suggests their angles fall a range of values.

 

[tiny-tim]

hi imurme8!

It seems to me that all hyperbolic sections of the same cone must have asymptotes that make exactly the same angle with each other (the angle formed by two antipodal generators of the cone).

no, the asymptotes will be parallel to the intersection of the cone with the parallel plane through the vertex …

tilt the plane away from the "vertical", and you reduce the angle

 

[imurme8]

Thank you, I see it now. :)