Have only skimmed over the thread, for the triangulation part look at the delaunay alrorithm:
There are definitely implementations in CGAL and PCL
http://www.cgal.org/
http://www.pointclouds.org/
For the scattered data approximation part, you might have a look into RBF interpolation, that works well even for very sparse / clustered data:
Time to get some breakfast...