Trajectory fitting in function space with application to analytic modeling of surfaces

A theory for representing a parameter-dependent function as a function trajectory is described. Additionally, a theory for determining a piecewise analytic fit to the trajectory is described. An example is given that illustrates the application of the theory to generating a smooth surface through a discrete set of input cross-section shapes. A simple procedure for smoothing in the parameter direction is discussed, and a computed example is given. Application of the theory to aerodynamic surface modeling is demonstrated by applying it to a blended wing-fuselage surface

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Stretching short biopolymers by fields and forces

We study the mechanical properties of semiflexible polymers when the contour length of the polymer is comparable to its persistence length. We compute the exact average end-to-end distance and shape of the polymer for different boundary conditions, and show that boundary effects can lead to significant deviations from the well-known long-polymer results. We also consider the case of stretching a uniformly charged biopolymer by an electric field, for which we compute the average extension and the average shape, which is shown to be trumpetlike. Our results also apply to long biopolymers when thermal fluctuations have been smoothed out by a large applied field or force.Comment: 10 pages, 7 figure

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

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