Boolean of spheres, how to develop a triangulated new surface

I working on an experiement involving creating a new surface from the boolean of several spheres interesected. 

I have attached an image to show my intentions.

I have managed to set up the boolean operation part in grasshopper, that is not an issue.

What I have been struggling with is how to approach a subdivision, and triangularization of the surface, so that all the intersection curves have the same u and v, so that the triangular panels would fit at the seams.

It is important to keep the interesection curves, as in the figure in the right side of the attached image, since it is what I find interesting in boolean operation of spheres.

So my main question, does anybody have an good idea on how to approach this dilemma?