# Getting the minimum radius of curvature of a conic section

I am fairly new to this forum and since I am not directly from a mathmetics background I recently ran into a problem I cannot solve.

What I am trying to do is to intersect a cone at a specific angle and want to receive the minimum radius of curvature. I know how to do it for a cylinder and I also know that I will get either an ellipse, a parabola or a hyperbole for a conic section, but I cannot find a source for either the euqtion of the intersection nor the minimum radius of curvature/curvature itself.

I found some theoretical proofs for the different intersections, but unfortunately the math behind it was a bit too high for me.

Is there maybe a short and clear answer to this question (for a stupid engineer as myself)? Or can someone refer me to an other sourve where I could the infromation from? I would be more than happy to hear from you! Thanks in advance!

• It’s “hyperbola,” not “hyperbole,” (which is a type of figure of speech). – amd May 6 '19 at 1:08

First of all, the minimum radius of curvature for a conic section occurs at a vertex, and it has the same length as the semi-latus rectum of the conic, that is $$b^2/a$$ for an ellipse or hyperbola, where $$a$$ and $$b$$ are as usual the semiaxes.
The values of $$a$$ and $$b$$ depend not only on the inclination of the intersecting plane, but also on the distance of that plane from the vertex of the cone. I'll express them as a function of the distances $$m$$ and $$n$$ of the vertices (of the ellipse or hyperbola) from the vertex of the cone. If $$u$$ is the semi-aperture angle of the cone, we have: $$4a^2=m^2+n^2\mp2mn\cos2u,\quad b^2=mn\sin^2u,$$ where sign $$-$$ must be taken for an ellipse and sign $$+$$ for a hyperbola. You can find the proof (for an ellipse, but that for a hyperbola is analogous) in this answer.
It follows that the minimum radius of curvature is $$r_{MIN}={2mn\sin^2u\over\sqrt{m^2+n^2\mp2mn\cos2u}}$$ For a parabola, you just need to take the limit of the above result for $$n\to\infty$$: $$r_{MIN}=2m\sin^2u.$$
• $m$ and $n$ are the distances from the vertex $V$ of the cone to the two vertices $A$ and $B$ of the ellipse or hyperbola (endpoints of major axis). Points $A$ and $B$ do lie on the cone surface, so distances $m=AV$ and $n=BV$ are indeed "on the cone itself" (if I understand what you mean). $m$ and $n$ can then be easily related to other parameters, such as the distance of the plane from $V$ and the angle between the plane and the axis of the cone. I don't know modern books treating such matters in detail. – Intelligenti pauca May 7 '19 at 10:31