Finding the Intersection of two conical meshes

I have developed a basic gui for use in designing a sun sensor comprised of solar cells on 5 faces of a cuboidal satellite. Prior to signal processing, we find parts of the vector from the current on each face, which follows a sinusoidal pattern based on the angle the sun makes with said face.

In simplistic terms, each face can be considered to give us a hollow cone of potential areas for the sun. Normally, we would use alternative methods for finding the intersection of the three cones from the three active faces. However, since one side lacks solar cells, I want to find the parabola created by the intersection of the two remaining cones when a third input is unavailable.

So the general matlab problem is thus, if I have two hollow cones (represented as meshes) originating in about the same location, but at right angles to one another, how can I plot their intersection?

I have also tried to simply model the parabola based upon three known points that I can find via other methods, however I cannot figure out how to find or plot such a curve in 3D, polyfit only operates in 2D except for surfaces which require far more known points than are possible to find.

I have attached an image for clarity in what purpose I wish fulfilled.

Though it is not properly commented yet (I am the only member of the team on this project so I am waiting until I finish to comment) here is the current code as well.