Timeline for Realizations of polyhedra and discrete mean curvature (followup question)

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Oct 24, 2011 at 23:34	comment	added	Robin Saunders		In general it is not possible to vary all vertices freely, hence my interest in variations of edges. In order for area-minimizing polyhedra to have constant mean curvature on edges, the mean curvature on the edges of such polyhedra must reduce to the form L*tan(θ/2). As Sullivan acknowledges, there are multiple ways to define mean curvature in the discrete case, depending on what identities one wishes to hold. All of them will be roughly linear in θ for small θ, so that all of the identities hold in the smooth limit where θ → 0 – but for nonzero θ, they are still distinct.
Oct 24, 2011 at 23:33	comment	added	Robin Saunders		Thanks. As it happens, I've been reading through the work of Sullivan, Polthier and others from (mostly) TU-Berlin, but for my purposes they have certain shortcomings. While these authors define mean curvature on an edge, in the context of variational problems they always use the curvature at a vertex (sometimes defined as the sum over the surrounding edges) and assume vertices can be varied freely – that is, the surface is triangulated. The exception is when “CMC” surfaces are defined as dual to (triangulated) spherical minimal surfaces; but in this case, they are no longer area-minimizing.
Oct 24, 2011 at 19:51	history	edited	Joseph O'Rourke	CC BY-SA 3.0	grammar
Oct 24, 2011 at 19:27	history	answered	Joseph O'Rourke	CC BY-SA 3.0