By Schaaf W.L.

ISBN-10: 0873531205

ISBN-13: 9780873531207

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3 Support Functions of Surfaces The function h : N(Ω) → R : h = H ◦ N−1 (7) which is obtained by composing the inverse of the map N defined in (3) with the function H defined in (6), is called the support function of the given surface patch. Support functions are a classical tool in the field of convex geometry [2,5,6]. Recently, curves and surfaces with polynomial support functions have been studied by three of the authors [15]. If a support function h ∈ C 1 (D, R) is given, where D ⊆ S2 , then the associated surface patch can be constructed by computing the envelope of the planes {x : h(n) = n · x}, n ∈ D.

Since a redistribution of the stresses is not possible in glass, fracture will occur – hence the brittle behavior. As described earlier, bending stresses are minimized in shell structures, which are appropriately shaped and supported, and the load is transferred primarily via in-plane stresses (membrane stresses). This allows for a better utilization of the capacity of the structural material, since stresses are distributed evenly over the thickness of the structure instead of concentrated at the surfaces.

The support function is somehow “in–between” both objects, and it allows to describe both of them in a simple way. The use of the support functions automatically leads to a restriction of the class of dual meshes; these meshes have to be star–shaped with respect to the origin. A possible application of the meshes described by piecewise linear support function comes from architecture, where free–form surfaces are often approximated by planar facets [4]. Motivated by applications in architectural design, constructions of quadrangular meshes with planar facets have recently been described in [10].

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A bibliography of recreational mathematics (5 vols.) by Schaaf W.L.


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