15,239 research outputs found
A design tool for globally developable discrete architectural surfaces using Ricci flow
This paper presents an approach for the design of discrete architectural surfaces that are globally developable; that is, having zero Gaussian curvature at every interior node. This kind of architectural surface is particularly suitable for fast fabrication at a low cost, since their curved geometry can be developed into a plane. This highly non-linear design problem is broken down into two sub-problems: (1) find the member lengths of a triangular mesh that lead to zero Gaussian curvature, by employing the discrete surface Ricci flow developed in the field of discrete differential geometry; (2) realize the final geometry by solving an optimization problem, subject to the constraints on member lengths as well as the given boundary. It is demonstrated by the numerical examples that both of these two sub-problems can be solved with small computational costs and sufficient accuracy. In addition, the Ricci flow algorithm has an attractive feature-the final design is conformal to the initial one. Conformality could result in higher structural performance, because the shape of each panel is kept as close as possible to its initial design, suppressing possible distortion of the panels. This paper further presents an improved circle packing scheme implemented in the discrete surface Ricci flow to achieve better conformality, while keeping its simplicity in algorithm implementation as in the existing Thurston's scheme
The Simplicial Ricci Tensor
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of
gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the
moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the
Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton
to define a non-linear, diffusive Ricci flow (RF) that was fundamental to
Perelman's proof of the Poincare conjecture. Analytic applications of RF can be
found in many fields including general relativity and mathematics. Numerically
it has been applied broadly to communication networks, medical physics,
computer design and more. In this paper, we use Regge calculus (RC) to provide
the first geometric discretization of the Ric. This result is fundamental for
higher-dimensional generalizations of discrete RF. We construct this tensor on
both the simplicial lattice and its dual and prove their equivalence. We show
that the Ric is an edge-based weighted average of deficit divided by an
edge-based weighted average of dual area -- an expression similar to the
vertex-based weighted average of the scalar curvature reported recently. We use
this Ric in a third and independent geometric derivation of the RC Einstein
tensor in arbitrary dimension.Comment: 19 pages, 2 figure
Combinatorial Calabi flows on surfaces
For triangulated surfaces, we introduce the combinatorial Calabi flow which
is an analogue of smooth Calabi flow. We prove that the solution of
combinatorial Calabi flow exists for all time. Moreover, the solution converges
if and only if Thurston's circle packing exists. As a consequence,
combinatorial Calabi flow provides a new algorithm to find circle packings with
prescribed curvatures. The proofs rely on careful analysis of combinatorial
Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure
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