Mesh triangulations

I am interested in triangulating meshes, but I am encountering issues that I couldn't solve by any research yet.

When I use Rhino's triangulation command or Weaverbird, the result I get is something like this:

Basically the polygonal faces are split into triangles. But what I want to achieve is something like this:

This actually resembles a geodesic division of a sphere. How could I possibly create such triangulations on free form surfaces?