4,269 research outputs found
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Smooth parametric surfaces and n-sided patches
The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed
Dynamic wormholes, anti-trapped surfaces, and energy conditions
Adapting and extending a suggestion due to Page, we define a wormhole throat
to be a marginally anti-trapped surface, that is, a closed two-dimensional
spatial hypersurface such that one of the two future-directed null geodesic
congruences orthogonal to it is just beginning to diverge. Typically a dynamic
wormhole will possess two such throats, corresponding to the two orthogonal
null geodesic congruences, and these two throats will not coincide, (though
they do coalesce into a single throat in the static limit). The divergence
property of the null geodesics at the marginally anti-trapped surface
generalizes the ``flare-out'' condition for an arbitrary wormhole. We derive
theorems regarding violations of the null energy condition (NEC) at and near
these throats and find that, even for wormholes with arbitrary time-dependence,
the violation of the NEC is a generic property of wormhole throats. We also
discuss wormhole throats in the presence of fully antisymmetric torsion and
find that the energy condition violations cannot be dumped into the torsion
degrees of freedom. Finally by means of a concrete example we demonstrate that
even temporary suspension of energy-condition violations is incompatible with
the flare-out property of dynamic throats.Comment: 32 pages in plain LaTex, no figures. Additional text and references
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A pentagonal surface patch for computer-aided geometric design
A vector valued interpolation scheme for a pentagon is described which is compatible with surface patches which have a rectangular domain of definition. Such a scheme could be useful in computer- aided geometric design problems, where a pentagonal patch occurs within a rectangular patch framework
A common geometric data-base approach for computer-aided manufacturing of wind-tunnel models and theoretical aerodynamic analysis
A more automated process to produce wind tunnel models using existing facilities is discussed. A process was sought to more rapidly determine the aerodynamic characteristics of advanced aircraft configurations. Such aerodynamic characteristics are determined from theoretical analyses and wind tunnel tests of the configurations. Computers are used to perform the theoretical analyses, and a computer aided manufacturing system is used to fabricate the wind tunnel models. In the past a separate set of input data describing the aircraft geometry had to be generated for each process. This process establishes a common data base by enabling the computer aided manufacturing system to use, via a software interface, the geometric input data generated for the theoretical analysis. Thus, only one set of geometric data needs to be generated. Tests reveal that the process can reduce by several weeks the time needed to produce a wind tunnel model component. In addition, this process increases the similarity of the wind tunnel model to the mathematical model used by the theoretical aerodynamic analysis programs. Specifically, the wind tunnel model can be machined to within 0.008 in. of the original mathematical model. However, the software interface is highly complex and cumbersome to operate, making it unsuitable for routine use. The procurement of an independent computer aided design/computer aided manufacturing system with the capability to support both the theoretical analysis and the manufacturing tasks was recommended
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A Combinatorial Parametric Engineering Model for Solid Freeform Fabrication
Fabricated parts are often represented as compact connected smooth 3-manifolds with
boundary, where the boundaries consist of compact smooth 2-manifolds. This class of mathematical
structures includes topological spaces with enclosed voids and tunnels. Useful information about these
structures are coded into level functions (Morse functions) which map points in the 3-manifold onto their
height above a fixed plane. By definition, Morse functions are smooth functions, all of whose critical
points are nondegenerate. This information is presented by the Reeb graph construction that develops a
topologically informative skeleton of the manifold whose nodes are the critical points of the Morse function
and whose edges are associated with the connected components between critical slices. This approach
accurately captures the SFF process: using a solid geometric model of the part, defining surface
boundaries; selecting a part orientation; forming planar slices, decomposing the solid into a sequence of
thin cross-sectional polyhedral layers; and then fabricating the part by producing the polyhedra by additive
manufacturing. This note will define a qualitative and combinatorial parametric engineering model of the
SFF part design process. The objects under study will be abstract simplicial complexes K with boundary
âK. Systems of labeled 2-surfaces in K, called slices, will be associated with the cross-sectional polyhedral
layers. The labeled slices are mapped into a family of digraph automata, which, unlike cellular automata,
are defined not on regular lattices with simple connectivities (cells usually have either 4 or 8 cell
neighborhoods) but on unrestricted digraphs whose connectivities are irregular and more complicated.Mechanical Engineerin
Smectic blue phases: layered systems with high intrinsic curvature
We report on a construction for smectic blue phases, which have quasi-long
range smectic translational order as well as three dimensional crystalline
order. Our proposed structures fill space by adding layers on top of a minimal
surface, introducing either curvature or edge defects as necessary. We find
that for the right range of material parameters, the favorable saddle-splay
energy of these structures can stabilize them against uniform layered
structures. We also consider the nature of curvature frustration between mean
curvature and saddle-splay.Comment: 15 pages, 11 figure
Multisided generalisations of Gregory patches
We propose two generalisations of Gregory patches to faces of any valency by using generalised barycentric coordinates in combination with two kinds of multisided BĂ©zier patches. Our first construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G1 continuity between adjoining BĂ©zier patches is generalised so that the novel Gregory S-patches of any valency can be smoothly joined to one another. Our second construction makes a minor adjustment to the generalised BĂ©zier patch structure to allow for cross-boundary derivatives to be defined independently per side. We show that the corresponding blending functions have the inherent ability to blend ribbon data much like the rational blending functions of Gregory patches. Both constructions take as input a polygonal mesh with vertex normals and provide G1 surfaces interpolating the input vertices and normals. Due to the full locality of the methods, they are well suited for geometric modelling as well as computer graphics applications relying on hardware tessellation
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