Mapping surface to non-continuous vertices

I’m working on a program that has to read in a bunch of data from a file and displays the output. When I render it using GL_POINTS it works like it is supposed to. Unfortunately the vertecies are not continuous, so when I try to render with GL_POLYGONS or GL_LINE_STRIP I get polygons or lines that cut through the object every which way. I can’t change the way the data sets are made because they are generated by another program I didnt write.

I’ve tried everything I can think of, including glMap functions, and nothing works like I want it too. In my mind I can visualize mapping the surface like dropping a blanket over the outside of the points, but opengl doesnt seem to want to cooperate.

Does anyone have any suggestions on how to make this work efficiently? (the program uses very large data sets, so performance is an issue)

Crosspost answered in the advanced forum.