5214346
Efficient and stable numerical algorithms on equilibrium equations for geometric modeling
In this paper the applications of equilibrium equation to geometric modeling is exploited and efficient numerical algorithms are proposed for solving the equilibrium equation. First we show that from diverse geometric modeling applications the equilibrium system can be extracted as the central framework. Second, by exploiting in-depth the special structures inherent in the geometric applications, we present simplified analytic solutions to the resulting geometric equilibrium equations via system decomposition. Finally, given the observation that the geometric equilibrium systems are extremely sensitive to both perturbations in input data and round off errors, efficient, stable and accurate numerical algorithms are proposed.
1143041
Quasi-Developable Mesh Surface Interpolation via Mesh Deformation
We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and over-determined linear system which is subsequently solved by a robust least-squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental data has shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.
24231103
Biorthogonal wavelet construction for hybrid quad/triangle meshes
Ever since its introduction by Stam and Loop, the quad/triangle subdivision scheme, which is a generalization of the well-known Catmull–Clark subdivision and Loop subdivision, has attracted a great deal of interest due to its flexibility of allowing both quads and triangles in the same model. In this paper, we present a novel biorthogonal wavelet—constructed through the lifting scheme—that accommodates the quad/triangle subdivision. The introduced wavelet smoothly unifies the Catmull–Clark subdivision wavelet (for quadrilateral meshes) and the Loop subdivision wavelet (for triangular meshes) in a single framework. It can be used to flexibly and efficiently process any complicated semi-regular hybrid meshes containing both quadrilateral and triangular regions. Because the analysis and synthesis algorithms of the wavelet are composed of only local lifting operations allowing fully in-place calculations, they can be performed in linear time. The experiments demonstrate sufficient stability and fine fitting quality of the presented wavelet, which are similar to those of the Catmull–Clark subdivision wavelet and the Loop subdivision wavelet. The wavelet analysis can be used in various applications, such as shape approximation, progressive transmission, data compression and multiresolution edit of complex models.
1920147
On the intersection of a set of direction cones
The problem of passing a convex polyhedron through a window is described as: Given a convex polyhedron P = (pr, p2,. . . , p,) and a convex polygon W = (WI, w,, *. ., w,) on a plane h not intersecting P, find all the directions for translating (single translation) P through W. Toussaint [l] proposed an O(nm log nm) algorithm to solve this problem. He first showed that P can pass W with a single translation in direction 8 if, and only if, each vertex of P can be passed through W with a single translation in direction 8. Based on this observation, he posed the algorithm as follows: Consider vertex pi of P. All directions for translating pi from its initial position through window W are defined by all the vectors emanating at pi and intersecting H in W. Therefore the cone determined by the half-lines from pi through wj, j = 1,2,. . . , m specifies all such directions for pi. This cone is named as direction cone of pi and is denoted as CONE(pi, W). Construct a 3D euclidean direction space D and translate all the cones CONE(pi, W), i = 1,2,. . . , n in D such that the pi all overlap with the origin of D. Then the intersection of all the cones in D gives another cone which is the set of directions for simultaneous translation of all the pi, and hence of P. Each cone can be computed in O(m) time and thus all the cones can be found in 0( nm) time. All the cones can be translated to D in O(nm) time. To compute the intersection of the cones, Toussaint views each cone as the intersection of M half spaces determined by the planes coplanar with the M faces of each cone. The interior half space contains the cone. Therefore the solution cone is the intersection of all the interior half spaces determined by all the cones in D. Since the
195710929
A Supply Chain Network Design Model for Deteriorating Items
We consider a supply chain network design problem for deteriorating items, involving a single supplier, some potential distribution centers and multiple retailers. This problem is to determine how many distribution centers to set up, where to locate them, how to serve the retailers using these distribution centers, and to determine the optimal inventory policies for the whole supply chain network. The goal is to minimize the total deterioration, inventory, transportation and facilities location cost. We formulate this problem as a nonlinear mix-integer programming model. This model is a NP-Hard problem, so we present a Lagrangian relaxation solution algorithm and a Genetic algorithm to solve our model respectively. Finally, numerical examples are given to illustrate the application of these two algorithms, and our computational results show that Lagrangian relaxation algorithm is more efficient for our problem.