Abstract. Given a smooth surface patch we construct an approximating piecewise linear structure. More precisely, we produce a mesh for which virtually all vertices have valency three. We present two methods for the construction of meshes whose facets are tangent to the original surface. These two methods can deal with elliptic and hyperbolic surfaces, respectively. In order to describe and to derive the construction, which is based on a projective duality, we use the so-called support function representation of the surface and of the mesh, where the latter one has a piecewise linear support function.
This paper deals with the problem: What sort of boundary conditions are to be prescribed on the edge of a shell that is to be in a statically determinate membrane state? Triangulated single layer trusses have been suggested as a model for membrane shells by (Calladine 1983) and have been investigated by (Szabo and Tarnai 1992). However, these efforts have not solved the problem completely. The paper presents a triangular single layer truss system – the stringer system – which differs from the above mentioned models in two ways: the systematic topology makes it relatively simple to control whether or not the system is statically and geometrically/kinematically determinate, the geometry corresponds to the curvature of the considered surface and ensures that the system can be applied as a valid static and geometric model for the membrane shell. In most cases, the stringer system regarded as a static model of membrane shells leads to the same results as the theory of partial differential equations, as investigated by (Tarnai 1980a, 1981, 1983). However, some new boundary conditions for shells with positive and zero Gaussian curvature have been found and the types of supports and the connection between statical and geometrical determinacy have been further generalised. The paper is based on parts of the Ph.D. Thesis by (Almegaard 2003).
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