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Whilst working on this problem, I've come across the work of Konrad Polthier on generalizing the notion of curature to discrete surfaces; much of his work is covered in his thesis, "Polyhedral surfaces of constant mean curvature". However, I have very little background knowledge of curvature and am struggling to follow much of what is being said.

I want to work with polyhedral surfaces more general than simplicial complexes (i.e. having possibly non-triangular faces), so I can no longer move the vertices around independently. Polthier mentions in passing that mean curvature can be defined on the edges of a surface rather than its vertices; this presumably corresponds to the fact that the realization space of a polyhedral graph has one degree of freedom for each edge, a conclusion which I reached as follows:

Intuitively, we can move any face of a polyhedron by defining a linear normal vector field on the plane in which it lies; this is equivalent to choosing three points in the face, and displacing them each by some amount in the normal direction. If the displacement is small and the polyhedron is "simple" (three faces meeting at each vertex), the new polyhedron will be combinatorially the same and the degrees of freedom are 3F = E + 6. However, having an extra face meeting at a vertex corresponds to removing an edge by contracting it to a point, and reduces the degrees of freedom by 1 if we want the new polyhedron to remain combinatorially the same; so the degrees of freedom are E + 6 in general. Finally, six of these degrees of freedom correspond to scaling, rotation and translation, i.e. similarity-preserving transformations. So the "shape" only really depends on E values.

Polthier defines mean curvature on edges in terms of their length and dihedral angle, but it's not clear to me whether this curvature can be varied independently on each edge in a canonical way (or even at all!) in the case of general polyhedral surfaces. The fact that there's one degree of freedom for each edge even in the general case makes me feel that there should be some natural correspondence, but I can't see what it is...

I tried explicitly working out the change in area which would result from moving a single face of a polyhedron in the aforementioned way, with the hope of deriving mean curvature as the gradient, but I failed because I'm shockingly bad at doing geometry with vectors.

Ultimately, I'm hoping to connect the "missing" degrees of freedom in non-simple polyhedra with the conjecture that they should never be global optimizers of surface area for a fixed number of faces; and, hopefully, show that the global optimizer (which is proven to exist) will never have faces with a large number of edges, because these would have large internal angles and small dihedral angles on their edges.

Sorry if any of this is vaguely worded. I'd greatly appreciate any help!

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2 Answers

I am a little puzzled by the Polthier reference, since at least one definition of PL mean curvature goes back to at least T. Regge's classic paper

MR0127372 (23 #B418) Regge, T. General relativity without coordinates. (Italian summary) Nuovo Cimento (10) 19 1961 558–571,

and now that I think of it, to Allendoerfer + Weil's classic

MR0007627 (4,169e) Allendoerfer, Carl B.; Weil, André The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Amer. Math. Soc. 53, (1943). 101–129. 52.0X

If you are looking for how mean curvature changes after deformations, you can look at

MR1668323 (2000a:53133) Almgren, Frederic J., Jr.; Rivin, Igor(4-WARW-MI) The mean curvature integral is invariant under bending. (English summary) The Epstein birthday schrift, 1–21 (electronic), Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998. 53C65 (49Q20 53A07)

or the papers that cite it.

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Sorry, I should have made it clear: I'm not trying to keep the faces of constant shape. I only want to preserve the combinatorial properties of the skeleton (since otherwise, extra complications will be introduced). Polthier was simply the name which came up the most in my searches, so I decided to look though his publications. –  Robin Saunders Jan 31 '11 at 0:37
The Almgren/Rivin reference (see also Rivin/Schlenker and Cooper/Rivin), see also the many papers by Bobenko et al talk about the general bending process. There is no assumption on faces being of "constant shape". –  Igor Rivin Jan 31 '11 at 0:51
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There is a recent paper by John Sullivan entitled "Curvatures of Smooth and Discrete Surfaces" in the collection Discrete Differential Geometry, but available also in an earlier arXiv version. Here is the Abstract:

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.

It may be that this paper will be of some help in your investigation.

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The Berlin School (Polthier and Bobenko have already been mentioned, you have just mentioned Sullivan, others are Pinkall, Springborn, many others) seems to be carrying the torch in this business. –  Igor Rivin Jan 31 '11 at 7:27
And the book I cite is a collection edited by Bobenko, Schröder, Sullivan, and Ziegler: definitely the Berlin School. –  Joseph O'Rourke Jan 31 '11 at 13:53
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