I am interested in a system of two intersecting cylinders of different sizes, where the smaller cylinder (green in the photo below) is at an arbitrary angle relative to the larger cylinder. In this system, the cylinders are centered on each other (ie. their centerlines will always intersect at one point).
Unwrapping the green cylinder results in the following: Unwrapped Cylinder
If one were to cut the green cylinder using just a plane instead of another cylinder, the resulting unfolded cylinder curve is a sinusoid (see "Unwrapping Curves from Cylinders and Cones" by Tom Apostol), but in the case of a cylinder cutting another cylinder, it is clear from the picture above this is no longer true.
I am wondering if a general equation can be derived which can describe the two-dimensional unwrapped curve of the green cylinder in terms of an angle variable, and the two different radii of the cylinders.
This an elaboration of @Blue comment above.
Attach a coordinate system $Oxyz$ to the base of the green cylinder with the origin at a point on its axis. Let the radius of the green cylinder be $r$.
Now, let the bigger gray cylinder have a radius $R \gt r$, and an unit vector axis $a$ that passes through the point $P_0 = (0, 0 , z_0)$.
The parametric equation of the smaller green cylinder is
$ P = (x,y,z) = (r \cos(\phi), r \sin(\phi), z ) $
And the algebraic equation of the bigger gray cylinder is
$ (P - P_0)^T (I - {a a}^T ) (P - P_0) = R^2 $
where the point $P$ has been substituted into the equation.
Now, for each $\phi$, the last equation is a quadratic equation in $z$, and will have two solutions for $z$, using the quadratic formula. Sorting these two solutions (smaller and bigger) for each $\phi$ results in two curves which are the near and far intersections of the green cylinder with the bigger cylinder.