get list of vertices and related triangles, preserving circle order (how?), finding bound vertices
find C_0 for all vertices, add thresholds to the Big List o' Vertices
for all of C_0 below other threshold, find longest trail (related triangles blitz), multiply, start at vertex, span out - if a vertex's branches span outside the sphere, keep it, otherwise, dump it. circle preservation important. add also to the aforementioned Big List
dijkstra on everything in the Big List, save the intermediate points - any chance of crossing?
link all the bounds (well, not link, but add to the big list)
MST on the rest of them, breaking adjacencies and making vertices (possible?)
use tree nodes (and internal threshold points?) as points for the next algorithm
I *wrote* it, and I'm not sure what it's telling me to do half the time.