On the approximation order of a space data-dependent PH quintic Hermite interpolation scheme

Dealing with Pythagorean Hodograph quintic Hermite interpolation in the space, we deepen the analysis of the so-called CC criterion proposed in Farouki et al. (2008) for fixing the two free angular parameters characterizing the set of possible solutions, which remarkably influence the shape of the chosen interpolant. Such criterion is easy to implement, guarantees the reproduction of the standard cubic Hermite interpolant when it is a PH curve and usually allows the selection of interpolants with good shape. Here we first rigorously prove that the PH interpolant it selects doesn?t depend on the unit pure vector chosen for representing its hodograph in quaternion form. Then we evaluate the corresponding interpolation scheme from a theoretical point of view, proving with the help of symbolic computation that it has fourth approximation order. A selection of experiments related to the spline implementation of the method confirms our analysis.     

A Parametric Quartic Spline Interpolant to Position, Tangent and Curvature

We describe a scheme that produces a convexity-preserving parametric quartic spline interpolant to position, tangent and curvature data. The resulting interpolant is curvature continuous and the scheme is local and 6-th order accurate. Moreover, the interpolant depends continuously on the given data.Patent Pending     

Interpolation of fuzzy data by Hermite polynomial

We consider the interpolation of fuzzy data by a differentiable fuzzy-valued function. We do it by setting some conditions on interpolant and first derivative of the interpolant. We give a numerical method for calculating this function.     

The map and blend scattered data interpolant on a sphere

The map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere. This technique consists of two different mappings of the sphere to planar domains, solving two corresponding scattered data interpolation problems on the planar domains, and then the interpolant on the sphere is formed by blending these two planar interpolants. If the user has software for solving the scattered data interpolation problem on a planar domain, only a small programming effort is needed to implement the map and blend interpolant on the sphere. Although any interpolant to scattered data on a planar domain can be used in our general technique, we use the multiquadric radial basis method.     

A unified, integral construction for coordinates over closed curves

We propose a simple generalization of Shephard's interpolation to piecewise smooth, convex closed curves that yields a family of boundary interpolants with linear precision. Two instances of this family reduce to previously known interpolants: one based on a generalization of Wachspress coordinates to smooth curves and the other an integral version of mean value coordinates for smooth curves. A third instance of this family yields a previously unknown generalization of discrete harmonic coordinates to smooth curves. For closed, piecewise linear curves, we prove that our interpolant reproduces a general family of barycentric coordinates considered by Floater, Hormann and Kós that includes Wachspress coordinates, mean value coordinates and discrete harmonic coordinates.     

Some results on linear rational trigonometric interpolation

We present here formulae for calculating the p^th derivative of a linear rational trigonometric interpolant written in barycentric form. We give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function.     

An iterative method for computing multivariate C piecewise polynomial interpolants

This paper describes a successive overrelaxation (SOR) method for computing a multivariate C¹ piecewise polynomial interpolant. Given a collection of points in R^m together with a triangulation of those points, the scheme described requires only the values of the function to be interpolated at the given points. The result is a C¹ interpolant whose restriction to each of the triangles in the triangulation is a polynomial of degree n.     

New interpolants for runge-kutta algorithms using second derivatives

In this paper a general interpolant for the Explicit Runge-Kutta methods is proposed. These interpolants are based on second derivatives on mesh-points of the integration interval, and first derivatives on interior points of each step. These first derivatives can be produced using lower order interpolants. Here an interpolant with 0(h ⁶) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.     

Generating Symbolic Interpolants for Scattered Data with Normal Vectors

Algorithms to generate a triangular or a quadrilateral interpolant with G1-continuity are given in this paper for arbitrary scattered data with associated normal vectors over a prescribed triangular or quadrilateral decomposition. The interpolants are constructed with a general method to generate surfaces from moving Bezier curves under geometric constraints. With the algorithm, we may obtain interpolants in complete symbolic parametric forms, leading to a fast computation of the interpolant. A dynamic interpolation solid modelling software package DISM is implemented based on the algorithm which can be used to generate and manipulate solid objects in an interactive way.     

On learning vector-valued functions

In this letter, we provide a study of learning in a Hilbert space of vectorvalued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.     

On the interpolation of concentric curvature elements

A convex G² Hermite interpolation problem of concentric curvature elements is considered in this paper. It is first proved that there is no spiral arc solution with turning angle less than or equal to π and then, that any convex solution admits at least two vertices. The curvature and the evolute profiles of such an interpolant are analyzed. In particular, conditions for the existence of a G² convex interpolant with prescribed extremal curvatures are given.     

NURBS to Avoid Boundary Orientation Poses in Serial Manipulators

A procedure to build NURBS motion interpolants to avoid boundary orientation poses for serial manipulator architectures is presented. As an example, the PUMA architecture was used. The procedure, which emerged from B‐spline curves theory, enables a local change of the NURBS motion interpolant. The change may be introduced in any arbitrary neighborhood of the chosen boundary orientation pose. Therefore, when tracking a trajectory, one may change NURBS motion interpolant value at any time instant, leaving its remaining values untouched. Recommendation for further research pertains to exploiting the flexibility of NURBS applied to robot kinematics. © 2003 Wiley Periodicals, Inc.     

Least eccentric ellipses for geometric Hermite interpolation

We present a rational Bézier solution to the geometric Hermite interpolation problem. Given two points and respective unit tangent vectors, we provide an interpolant that can reproduce a circle if possible. When the tangents permit an ellipse, we produce one that deviates least from a circle. We cast the problem as a theorem and provide its proof, and a method for determining the weights of the control points of a rational curve. Our approach targets ellipses, but we also present a cubic interpolant that can find curves with inflection points and space curves when an ellipse cannot satisfy the tangent constraints.     

Gaussian Process Models for Indoor and Outdoor Sensor-Centric Robot Localization

This paper presents an approach to building a map from a sparse set of noisy observations, taken from known locations by a sensor with no obvious geometric model. The basic approach is to fit an interpolant to the training data, representing the expected observation, and to assume additive sensor noise. This paper takes a Bayesian view of the problem, maintaining a posterior over interpolants rather than simply the maximum-likelihood interpolant, giving a measure of uncertainty in the map at any point. This is done using a Gaussian process (GP) framework. The approach is validated experimentally both in an indoor office environment and an outdoor urban environment, using observations from an omnidirectional camera mounted on a mobile robot. A set of training data is collected from each environment and processed offline to produce a GP model. The robot then uses that model to localize while traversing each environment.      

An Interpolant with Tension Defined over Triangles

A C ¹ interpolation scheme is described, which is defined over triangular grids. The interpolant is computed on the basis of curves with tension, which permit local control over the shape of the resulting surface.     

平面三次PH曲线偶C1Hermite插值解的构造和形状分析

一般情况下,三次PH曲线偶的C^1 Hemite插值问题有四个不同的解。在这四个解中,只有一条曲线能很好地满足几何设计的要求。已有的插值算法都是依赖于构造出所有四个解,利用绝对旋转指标或弹性弯曲能量来找出这条“好”的插值曲线。本文提出一种新的方法以区分这些解,即用由三次PH曲线偶和惟一经典三次插值曲线的速端曲线形成的闭环的弯曲数来区分。对于“合理”的Hemite数据,本文还给出了不需计算和比较所有的四个解便可直接构造“好”的三次PH曲线偶的方法。     

一种有理二次插值函数的凸性分析

在给定的插值数据条件下,利用一种带参数的分母为二次的有理二次插值方法,通过调整插值函数中的参数,给出了插值曲线的保凸方法和该方法得以实现的充分必要条件。这种条件是对参数的简单的线性的不等式约束,容易在计算机辅助设计中得到实际应用。     

A bivariate C Clough-Tocher scheme

A Clough-Tocher like interpolation scheme is described for values of position, gradient and hessian at scattered points in two variables. The domain is assumed to have been triangulated. The interpolant has local support, is globally twice differentiable, piecewise polynomial, and reproduces polynomials of degree up to three exactly.     

Geometric interpolants with different degrees of smoothness

Geometric interpolation is a basic task in geometric modelling. In this paper, the geometric interpolant with different degrees of smoothness is introduced. The method provides the lower degree, flexile interpolation curves and construction is simple. Moreover, convexity, regularity and the construction of some special geometric interpolants are also discussed.