Getting the minimum radius of curvature of a conic sectionI 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? Or can someone refer me to an other sourve where I could the infromation from?

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Kason Gonzales · Accepted Answer

Step 1First 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&#039;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:  4      a    2    =      m    2    +      n    2    ∓  2  m  n  cos  ⁡  2  u  ,        b    2    =  m  n      sin    2    ⁡  u  , where sign − must be taken for an ellipse and sign + for a hyperbola.Step 2It follows that the minimum radius of curvature is       r          M      I      N        =                    2        m        n                  sin          2                ⁡        u                              m          2                +                  n          2                ∓        2        m        n        cos        ⁡        2        u            .For a parabola, you just need to take the limit of the above result for   n  →  ∞:      r          M      I      N        =  2  m      sin    2    ⁡  u  .

Getting the minimum radius of curvature of a conic section

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