Yes, Marc's projection mechanism would work as an approximation, but would produce nearly 100% non-planar quadrilaterals. To some extent, I'm trying to understand the mathematics of planar quadrilaterals, so Warren's Euler formula provided me with significant insight about how one might go about proving the non-existence of some types of quadrangularization. Marc's octahedral skeleton is the dual of the cubical skeleton. I can take such a cubical skeleton and "push out" all of the faces, and then push out all of the resulting faces, in a recursive manner to produce a pretty decent approximation to the sphere. The original cubic skeleton has 6 faces. Since "pushing out" one face results in 5 new faces, one level of recursion gives us 6*5=30 total faces. Recursively pushing out n times results in 6*5^n total faces. Note that my "pushing out" algorithm approximates the sphere _from the inside_, in the sense that the 8 vertices of the cube lie on the surface of the sphere, and each new "pushed out" face will also have all of its vertices on the surface of the sphere. BTW, my recursive mechanism provides a (new???) set of coordinates for every point on the surface of the sphere. My recursive mechanism also provides for a (new???) way of "projecting" (not really a mathematical projection in the usual sense) the surface of a sphere onto a set of reasonably compact planar quadrilaterals. Given all of the globe-mapping algorithms that have been suggested over the previous centuries, I'd be surprised if this one hasn't already been suggested at some point. Note that the little planar bits are (almost certainly) _not_ precisely perpendicular to a radius vector from the center of the sphere. This is a problem for computer graphics, because these little faces won't reflect correctly. We might have to optimize the choices involved in "pushing out", so as to make the planar quadrilaterals as compact as possible. Here the definition of "compact" might be to minimize the enclosing radius of the quadrilateral planar pieces at a given depth k of the recursion. At 12:58 PM 1/6/2013, Marc LeBrun wrote:
="Henry Baker" <hbaker1@pipeline.com> Good, but we have a convex surface modelled by a non-convex surface. Can we do better?
Well, how do we define "better"? On the one hand we seem to be talking about approximating surfaces ever more closely, but on the other we seem to be groping to converge on discovering some unstated intuitive definition.
Duality. I note that a triangulated surface in which every vertex has exactly 4 edges coming out (I believe) is dual to a quadrangulated surface where each vertex meets exactly 3 planar quadrilaterals.
Then you just get the dual of the problem: "all" one has to do is find an algorithm for triangulating with a net that's all degree-4.
I think if the surface is genus 0 this implies an octahedral "skeleton".
Anyway, why wouldn't this work:
Make a cubic cage with square mesh sides, put a light inside, insert in the target shape, then take the quads defined by the shadows on the surface?
If not close enough, or produces non-planar quads, use (locally) finer mesh.
(If the shape is too lumpy or floppy or the like first smoothly deform it to be rounder, project as above, then deform back...)