Intersection of cone and cylinder layout formula for sheet metal application

Question

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the parts. So I need the generic mathematical formula for the intersecting curve for the cylinder, where the formula describes the curved edge when the cylinder is laid flat and then rolled up so that it exactly meets the truncated cone. The cylinder could intersect at any angle.

Answer 1

I agree that this is a nontrivial problem.

That said, here's a start on a very easy preliminary question that may be of some use.

If you slice a cylinder with a plane and unroll the cylinder you get a sine curve. Here's a photo from an early edition of Hugo Steinhaus' Mathematical Snapshots:

Finding the equation of that sine curve given the orientation of the plane and the diameter of the cylinder is probably straightforward.

The cross section of the cylinder is an ellipse. If you're not limited to cones on a circular base you could build one on that elliptical base. (It's convenient that the cone can be rolled from a flat sheet too.)

In full generality you're looking at the intersection of two developable quadric surfaces. Finding an equation for the unrolled curve strikes me as a relatively difficult problem in algebraic or differential geometry. I doubt the existence of a nice closed form solution (an expert in one of those areas might know).

Your manufacturing question may have a satisfactory answer if you can imagine asking for more than the above naive example but less than full generality.

Perhaps you could program cad software or sage or mathematica to give you a picture you could then use to drive the machine that cuts your sheet metal.

Answer 2

Place the cone with the vertex at the origin, with axis along $Oz$, and with opening angle $\beta$ (which can be computed from the diameters and height of the truncated cone). The equation is then

$x^2 + y^2 = z^2 \tan^2\beta$.

Now take a cylinder of radius $r$, also with axis $Oz$. It can be parametrized as $x = r\cos{u}, y=r\sin{u}, z=v$. A curve on the cylinder corresponds to a curve in the parameter space $(u,v)$; something like $v = f(u)$ would determine the curve as $(r\cos{u}, r\sin{u}, f(u))$.

Now rotate the cylinder an angle $\alpha$ about the $x$-axis and shift it along the $z$-axis by $h$, depending on the position of the axis of the cylinder and angle with the axis of the cone. The new parametrization looks something like $ x = r\cos{u}, y = r\cos{\alpha}\sin{u} - v \sin{\alpha}, z = r\sin{\alpha}\sin{u} + v\cos{\alpha} + h. $

The intersection between the cylinder and the cone gives a quadratic equation in $v$. Not pretty, but solvable - with an explicit formula for a solution in terms of $u$ and parameters $\alpha$, $\beta$, $r$, $h$.