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     

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.     

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。     

仿射不变参数化

仿射不变技术的目的是指图形在经历各种仿射变换后,它的某些性质保持不变,即这些性质本身不受变换的影响。将仿射不变技术用于参数化,由此得到的参数在各种仿射变换前后都保持不变。用这种参数做曲线插值时,可为实际应用带来很大的方便。以三次样条函数实现参数曲线,并通过计算实例对新方法与目前常用的参数化方法的插值结果进行了比较,在效率和显示结果上,新的方法是令人满意的。     

C^2保单调或保形的插值多项式样条算法

本文讨论多段多项式的Ｃ＾２连续保形或保单调插值，在每相邻两个型值点之间，构造一段五次或五次以上的多项式，通过在某些段提高多项式次数，使得这个分段多项式插值函数Ｃ＾２连续且保形或保单调。     

The singular cases for γ-spline interpolation

We derive a natural extension of Boehm's free-form γ-spline, the G² interpolating γ-spline. Primarily, the conditions under which singularities in the spline formulation occur are investigated. Also, the effect that these singularities have on the interpolant are studied. Comparisons are made to the behavior of the interpolating ν-spline.     

Cardinal interpolation: A bivariate polynomial example

The general interpolation problem over a linear space is solved by providing explicit formulas for the cardinal basis of the space. As an example of this technique, the cardinal form of a bivariate degree-nine polynomial interpolating to function and derivative values through order four at various points on a triangle is derived. The piecewise polynomial interpolant over an arbitrary triangulated domain in has C² continuity.     

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。     

A trivariate clough—tocher scheme for tetrahedral data

An interpolation scheme is described for values of position, gradient and Hessian at scattered points in three variables. The domain is assumed to have been tesselated into tetrahedra. The interpolant has local support, is globally once differentiable, piecewise polynomial, and reproduces polynomials of degree up to three exactly. The scheme has been implemented in a FORTRAN research code.     

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.     

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.     

Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures

The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the “ordinary” cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how “close” the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the “ordinary” cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.     

Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.     

Dual Bézier curves and convexity preserving interpolation

A convexity preserving interpolation problem is analyzed from a geometrical point of view. A dualization of the usual Bézier techniques allows us to define a subdivision algorithm which generates certain conic sections. This algorithm can be used to define a rational convexity preserving interpolant. We also describe some particular dual Bézier curves which are particularly suitable for the design of convex functions.     

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

Smooth interpolation over hypercubes

Multivariate interpolation problems, which occur often in scientific research, can sometimes be approached using Coons' patches. Coons' patches are smooth, local interpolants to lower dimensional data sets. (For example, they interpolate curves of data when considering 3-dimensional problems.) However, they do not interpolate as desired unless all the mixed partial derivatives, the ‘twists’, are equal. The twists are not equal in many cases of practical importance, such as for wire frame data. Gregory (1983) has developed a compatibly corrected Coons' patch for 3-dimensional surfaces. We generalize this interpolant, ‘Gregory's square’, to the case of functions of n variables. The interpolant we propose is built up inductively from one to n-dimensions, requiring, at each step, only one additional term to be defined. This is the key to the whole process and involves the definition of a general ‘twist operator’.     

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.     

A bivariate C2 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.     

Smooth closed surfaces with discrete triangular interpolants

Discrete interpolants which involve cross boundary derivatives in an attempt to form C¹ surfaces have the following major problem: Requiring C¹ joins between patches makes sense only if the patch domains are adjacent in the domain space. This makes it impossible to form C¹ closed surfaces, or indeed any surface which contains more connections than can be achieved in the domain.

This paper develops a method of forming smooth closed (or otherwise complexly connected) surfaces from a discrete triangular interpolant by relaxing the C¹ property of an interpolant to ‘Visually C¹”.

The only constraint on the scheme is that the data to be interpolated define a unique tangent plane at each vertex where several triangles meet. Then each patch can be calculated independently of its neighbors, using only data defined at its vertices, and the domain for each triangular patch can be chosen without regarding the connectivity of the patch with others. This last feature could be of great interest to a designer of a surface since one can choose the domain of each patch to be an equilateral triangle, and give it no further thought.