Hermite Interpolation Problem Research Articles

Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of T and like a Bernstein basis for imposing smoothness across the edges of T. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of T. Different C2 cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.

Partition of the space of planar quintic Pythagorean-hodograph curves

The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve r(t) may be constructed from a complex quadratic pre–image polynomial w(t) by integration of r′(t)=w2(t), and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of w(t). Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial w(t) passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on r(t) are incurred by a close proximity of w(t) to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.

G1 interpolation of v-asymmetric data with arc-length constraints by Pythagorean-hodograph cubic splines

The interpolation problem of a sequence of points with prescribed segment arc-length by PH cubic spline is addressed. It turns out to be a v-asymmetric G1 Hermite interpolation problem with specified arc length. Using rectifying control polygon and reduction to a canonical form, it is shown that the problem can be expressed in terms of finding the real solutions to a system of nonlinear equations in two variables. A detailed and thorough analysis of the resulting system of nonlinear equations and closed form solutions are provided for any given data. It is confirmed that this construction of G1 Hermite interpolants of specified arc length admits at least two formal solutions, both of which have attractive shape properties and some (if any) must be discarded due to undesired looping behavior. The algorithm developed herein offers simple and efficient closed-form solutions to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. Moreover, the algorithm is illustrated by several numerical examples.

Spherical Bessel functions and critical lengths

The critical length of a space of functions can be described as the supremum of the length of the intervals where Hermite interpolation problems are unisolvent for any choice of nodes. We analyze the critical length for spaces containing products of algebraic polynomials and trigonometric functions. We show the relation of these spaces with spherical Bessel functions and bound above their critical length by the first positive zero of a Bessel function of the first kind.

On the intersection of left and right ideals in the ring of quaternion polynomials

Given a left ideal [Formula: see text] and a right ideal [Formula: see text] in the ring of univariate quaternion polynomials, we present an explicit formula parametrizing the intersection [Formula: see text]. The motivation comes from interpolation theory where any such intersection appears to be the solution set of a two-sided homogeneous Lagrange–Hermite interpolation problem.

G1 Hermite interpolation method for spatial PH curves with PH planar projections

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.

Cd Hermite interpolations with spatial Pythagorean hodograph B-splines

Following the recent results of Albrecht et al. (2017, 2020), the problem of Hermite interpolation by clamped spatial Pythagorean hodograph (PH) B-spline curves is thoroughly investigated in this paper. The constructed interpolants are obtained by using beneficial properties of B-spline basis functions and via solving special quadratic and linear equations in quaternion algebra. All the designed procedures are purely symbolic and the main contribution lies in the unifying approach to the formulated problem. The results are confirmed by several computed examples.

Interpolation of Hermite data by clamped Minkowski Pythagorean hodograph B-spline curves

In this paper, the problem of Hermite interpolation by clamped Minkowski Pythagorean hodograph (MPH) B-spline curves is considered. Using the properties of B-splines, our intention is to use the MPH curves of degrees lower than in algorithms designed before. Special attention is devoted to C1∕C2 Hermite interpolation by MPH B-spline cubics/quintics. The resulting interpolants are obtained by exploiting properties of B-spline basis functions and via solving special quadratic and linear equations in Clifford algebra Cℓ2,1. All the presented algorithms are purely symbolic. The results are confirmed by several applications, in particular we use them to generate an approximate conversion of a given analytic curve to MPH B-spline curve with a high order of approximation, then to an efficient approximation of the medial axis transform of a planar domain leading to NURBS representation of the (trimmed) offsets of the domain boundaries, and to skinning of systems of circles in plane.

Singular cases of planar and spatial C1 Hermite interpolation problems based on quintic Pythagorean-hodograph curves

A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.

On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

Convergence rates of a Hermite generalization of Floater–Hormann interpolants

Cirillo and Hormann (2018) introduce an iterative approach to the Hermite interpolation problem, which, starting from the Lagrange interpolant, successively adds m corrections terms to interpolate the data up to the mth derivative. The method is general enough to be applied to any interpolant in linear form with a sufficiently continuous set of basis functions, but Cirillo and Hormann focus their attention on Floater–Hormann interpolants, a family of barycentric rational interpolants that are based on a particular blend of local polynomial interpolants of degree d. They show that the resulting iterative rational Hermite interpolants converge at the rate of O(h(m+1)(d+1)) as the mesh size h converges to zero for m=1,2, and their numerical results suggest that the same rate holds for m>2. In this paper we prove this convergence rate for any m≥1.

Optimal [formula omitted] Hermite interpolation for 3D curves

We consider a Hermite interpolation problem for a 3D curve where the functional to be minimized is defined as the integral of squared norm of the third parametric derivative, subject to G2 continuity constraints at the end points. The first order necessary optimality condition of the variational problem leads to a parametric transition curve with quintic polynomials. The determination of coefficients is given by a polynomial system with 2 unknowns. Stationary points correspond to positive roots of the resultant which is a degree 9 polynomial.Although the formulated variational problem is non-convex, the proposed approach leads to the global solution, which can be computed in a reliable and fast manner.

Linear computational approach to interpolations with polynomial Minkowski Pythagorean hodograph curves

Methods using Pythagorean hodographs both in Euclidean plane and Minkowski space are often used in geometric modelling when necessary to solve the problem of rationality of offsets of planar domains. A main justification for studying and formulating approximation and interpolation algorithms based on the called Minkowski Pythagorean hodograph (MPH) curves is the fact that they make the trimming procedure of inner offsets considerably simpler. This is why one can find many existing techniques in literature. In this paper a simple computational approach to parametric/geometric Hermite interpolation problem by polynomial MPH curves in R 2 , 1 is presented and an algorithm to construct such interpolants is described. The main idea is to construct first not a tangent but a normal vector space satisfying the prescribed MPH property that matches the given first order conditions, and then to compute a curve possessing this constructed normal vector space and satisfying all the remaining interpolation conditions. Compared to other methods using special formalisms (e.g. Clifford algebra), the presented approach is based only on solving systems of linear equations. The results are confirmed by number of examples.

Spaces of polynomial and nonpolynomial spline-wavelets

This paper, discusses spaces of polynomial and nonpolynomial splines suitable for solving the Hermite interpolation problem (with first-order derivatives) and for constructing a wavelet decomposition. Such splines we call Hermitian type splines of the first level. The basis of these splines is obtained from the approximation relations under the condition connected with the minimum of multiplicity of covering every point of (α, β) (almost everywhere) with the support of the basis splines. Thus these splines belong to the class of minimal splines. Here we consider the processing of flows that include a stream of values of the derivative of an approximated function which is very important for good approximation. Also we construct a splash decomposition of the Hermitian type splines on a non-uniform grid.

Bivariate Hermite interpolation on the exponential curve

We study Hermite interpolation problems on the exponential curve $$y=e^x$$ in $$\mathbb {R}^2$$ . We construct some kind of regular Hermite interpolation schemes and investigate continuity properties of interpolation polynomials with respect to interpolation conditions.

An iterative approach to barycentric rational Hermite interpolation

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at $$n+1$$n+1 nodes and then successively adds m corr...

Fractional Hermite interpolation using RBFs in high dimensions over irregular domains with application

In the interpolation method, in some cases, one often has a number of data points and its derivatives, which are obtained by sampling or experimentation. In this case, the problem of finding an approximating function passing through these points and coinciding with given values of its derivatives at these points is generally known as “Hermite interpolation”. The Hermite interpolation is mostly a method of interpolating data points as a polynomial function that is faced with some challenges in high dimensions and on irregular domains. Radial basis functions take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. So, they can be applied as a suitable tool to high dimensional Hermite interpolation problem on irregular domains. In many applied systems, commonly available derivatives information is presented using fractional order derivatives instead of integer ones. For this purpose, in this paper, we assume that the values of an unknown function and its fractional derivatives at some distinct points are presented. Therefore, we intend to apply a new approach, which we call it as “fractional Hermite interpolation” with radial basis functions in high dimensions. Optimal recovery conditions for the fractional Hermite interpolant are investigated. Then, the existence and uniqueness of the solution in this type of generalized interpolation are proved. In order to increase the accuracy and stability of the method, Hilbert Schmidt's theory has also been used. The main advantages of the used method are its simplicity and efficiency in high dimensions, and over irregular domains. Finally, numerical results in one, two and three dimensions and a real-world problem are presented to support our theoretical analysis.

Linear Combination of Independent Exponential Random Variables

In this paper we prove a recursive identity for the cumulative distribution function of a linear combination of independent exponential random variables. The result is then extended to probability density function, expected value of functions of a linear combination of independent exponential random variables, and other functions. Our goal is on the exact and approximate calculation of the above mentioned functions and expected values. We study this computational problem from different views, namely as a Hermite interpolation problem, and as a matrix function evaluation problem. Examples are presented to illustrate the applicability and performance of the methods.