Thanks, Warren! That gives me a way to incrementally add new faces! Suppose we have a face which is a square. Embed a new square the middle of the given square, where the new square is half as big as the given square, and aligned with the given square. Link each vertex of the new square with the corresponding vertex of the given square. Now "push out" the new square from the plane of the given square. Once this little square is "pushed out", I can now deform it to better approximate the underlying surface. ASCII art: -------------- |\ /| | \--------/ | | | | | | | | | | | | | | | | | | /--------\ | |/ \| -------------- At 09:22 AM 1/6/2013, Warren Smith wrote:
"Henry Baker" <hbaker1@pipeline.com> Q. Is it possible to "quadrangulate" a 3D surface with _only_ (planar) quadrilaterals "hinged" together ? (A quadrilateral has to have 4 non-trivial sides, so no triangles as limiting cases.)
--First answer: A triangle can be divided into 3 quads using 4 extra vertices at edge midpoints and some central point. And any convex polyhedron can be regarded as having triangle faces only (happens automatically by perturbing all vertices by epsilon randomly).
--Additional remark: Let F=#quads, E=#edges, V=#vertices. F-E+V=2 (Euler) and E=2*F for a quadrangle-faced polyhedron with same topology as a sphere. So V = 2+F.