Radial Basis Function (RBF)-based Parametric Models for Closed and Open Curves within the Method of Regularized Stokeslets

The method of regularized Stokeslets (MRS) is a numerical approach using regularized fundamental solutions to compute the flow due to an object in a viscous fluid where inertial effects can be neglected. The elastic object is represented as a Lagrangian structure, exerting point forces on the fluid. The forces on the structure are often determined by a bending or tension model, previously calculated using finite difference approximations. In this paper, we study Spherical Basis Function (SBF), Radial Basis Function (RBF) and Lagrange-Chebyshev parametric models to represent and calculate forces on elastic structures that can be represented by an open curve, motivated by the study of cilia and flagella. The evaluation error for static open curves for the different interpolants, as well as errors for calculating normals and second derivatives using different types of clustered parametric nodes, are given for the case of an open planar curve. We determine that SBF and RBF interpolants built on clustered nodes are competitive with Lagrange-Chebyshev interpolants for modeling twice-differentiable open planar curves. We propose using SBF and RBF parametric models within the MRS for evaluating and updating the elastic structure. Results for open and closed elastic structures immersed in a 2D fluid are presented, showing the efficacy of the RBF-Stokeslets method.Comment: 23 pages, 11 figures, 1 tabl

Area-Preserving Geometric Hermite Interpolation

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic B\'ezier polynomials. We show there exists a family of such curves which are $5^{th}$ order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly

A Note on Robust Biarc Computation

A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical configurations. The proposed algorithm uses an algebraic construction which is reduced to the solution of a single $2$ by $2$ linear system. Singular angles configurations are treated smoothly by using the pseudoinverse matrix when solving the linear system. The proposed algorithm is compared with the Matlab's routine \texttt{rscvn} that solves geometrically the same problem. Numerical experiments show that Matlab's routine sometimes fails near singular configurations and does not select the correct solution for large angles, whereas the proposed algorithm always returns the correct solution. The proposed solution smoothly depends on the geometrical parameters so that it can be easily included in more complex algorithms like splines of biarcs or least squares data fitting.Comment: 10 pages, 4 figure

Curve Reconstruction in Riemannian Manifolds: Ordering Motion Frames

In this article we extend the computational geometric curve reconstruction approach to curves in Riemannian manifolds. We prove that the minimal spanning tree, given a sufficiently dense sample, correctly reconstructs the smooth arcs and further closed and simple curves in Riemannian manifolds. The proof is based on the behaviour of the curve segment inside the tubular neighbourhood of the curve. To take care of the local topological changes of the manifold, the tubular neighbourhood is constructed in consideration with the injectivity radius of the underlying Riemannian manifold. We also present examples of successfully reconstructed curves and show an applications of curve reconstruction to ordering motion frames.Comment: 20 Figures, 27 Equations, 12 page

Geometry of rational helices and its applications

The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this to related subjects. One of these applications is Geometric C1 Hermite interpolation (i.e. interpolation of end points with associated unit tangents) by rational helices. Furthermore, we investigate the existence of rational rotation minimizing frames (RRMFs) on rational helices. A rational approximation procedure to rotation minimizing frames (RMFs) is suggested. Subsequently, we deploy the approximate frame for modeling a rational sweep surface. The resulting algorithms are illustrated by several examples.Comment: 20 pages, 8 figure

Bounds for spiral and piecewise spiral splines

This note is the updated outline of the article "Interpolational properties of planar spiral curves", Fund. and Applied Math., 2001, Vol.7, N.2, 441-463, published in Russian. The main result establishes boundary regions for spiral and piecewise spiral splines, matching given data. The width of such region can serve as the measure of fairness of the point set, subjected to interpolation. Application to tolerance control of curvilinear profiles is discussed.Comment: 15 pages, 10 figure

Modelling Character Motions on Infinite-Dimensional Manifolds

In this article, we will formulate a mathematical framework that allows us to treat character animations as points on infinite dimensional Hilbert manifolds. Constructing geodesic paths between animations on those manifolds allows us to derive a distance function to measure similarities of different motions. This approach is derived from the field of geometric shape analysis, where such formalisms have been used to facilitate object recognition tasks. Analogously to the idea of shape spaces, we construct motion spaces consisting of equivalence classes of animations under reparametrizations. Especially cyclic motions can be represented elegantly in this framework. We demonstrate the suitability of this approach in multiple applications in the field of computer animation. First, we show how visual artifacts in cyclic animations can be removed by applying a computationally efficient manifold projection method. We next highlight how geodesic paths can be used to calculate interpolations between various animations in a computationally stable way. Finally, we show how the same mathematical framework can be used to perform cluster analysis on large motion capture databases, which can be used for or as part of motion retrieval problems

An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1811.1016

Compactly-Supported Smooth Interpolators for Shape Modeling with Varying Resolution

In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.Comment: 26 pages, 12 figure

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given