Hermite Data Research Articles

Gaussian quadrature formulae are strongly asymptotically optimal for a class of infinitely differentiable functions

This paper investigates the optimal quadrature formulae of a class F∞ of infinitely differentiable functions on [−1,1]. We obtain the strong equivalences of the optimal worst case errors of F∞ for standard information and Hermite data. We proved that the Gaussian quadrature formulae are strongly asymptotically optimal.

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.

On [formula omitted] and [formula omitted] Hermite interpolation by spatial Algebraic-Trigonometric Pythagorean Hodograph curves with polynomial parametric speed

In this paper we focus on the class of Algebraic-Trigonometric Pythagorean Hodograph curves (ATPH for short) that is characterized by a purely polynomial parametric speed. Within such a class of ATPH curves, we first construct interpolants to spatial G1 Hermite data equipped with curvature values. With respect to the solutions proposed in [24], the G1 Hermite ATPH interpolants we here propose are characterized by C0- and C1-continuous curvature plots. Secondly, we investigate the existence of ATPH interpolants to spatial G2 Hermite data and show that solutions exist under some restrictions on the Hermite input data.

G2 Hermite Interpolation by Segmented Spirals

A curve with single-signed, monotonically increasing or decreasing curvatures is referred to as a planar spiral. G2 Hermite data are spiral G2 Hermite data for which only interpolation by a spiral is possible. In this study, we design segmented spirals to geometrically interpolate arbitrary C-shaped G2 Hermite data. To separate the data into two or three spiral data sets, we add one or two new points, related tangent vectors and curvatures. We provide different approaches in accordance with the various locations of the external homothetic centers of two end-curvature circles. We then match new data by constructing two or three segmented spirals. We generate at most three piecewise spirals for arbitrary C-shaped data. Furthermore, we illustrate the suggested techniques with several examples.

Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces

In the past few decades polynomial curves with Pythagorean hodograph (PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining, and robotics. This work deals with classes of PH curves built upon exponential-polynomial spaces (EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau--like algorithm and two recently proposed methods: Woźny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong.

Interpolation of 3D data streams with C2 PH quintic splines

The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic splines is proposed. To facilitate real-time computations, an efficient local data stream interpolation algorithm is introduced to successively construct each spline segment as a quintic PH biarc interpolating second- and first-order Hermite data at the initial and final end-point, respectively. A C2 smooth connection between successive spline segments is obtained by taking the locally required second-order derivative information from the previous segment. Consequently, the data stream spline interpolant is globally C2 continuous and can be constructed for arbitrary C1 Hermite data configurations. A simple and effective selection of the free parameters that arise in the local interpolation problem is proposed. The developed theoretical analysis proves its fourth approximation order while a selection of numerical examples confirms the same accuracy for the spline extension of the scheme. In addition, the performances of the method are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.

Confluent Vandermonde with Arnoldi

In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by Brubeck and Trefethen (2022). The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.

Curve-guided 5-axis CNC flank milling of free-form surfaces using custom-shaped tools

A new method for 5-axis flank milling of free-form surfaces is proposed. Existing flank milling path-planning methods typically use on-market milling tools whose shape is cylindrical or conical, and is therefore not well-suited for meeting fine tolerances for manufacturing of benchmark free-form surfaces like turbine blades, gears, or blisks. In contrast, our optimization-based framework incorporates the shape of the tool into the optimization cycle and looks not only for the milling paths, but also for the shape of the tool itself. Given a free-form reference surface and a guiding path that roughly indicates the motion of the milling tool, tangential movability of quadruplets of spheres centered along a straight line is analyzed to indicate possible shapes and their motions. This results in G1 Hermite data in the space of rigid body motions that are interpolated and further optimized, both in terms of the motion and the shape of the milling tool itself. We demonstrate our algorithm on synthetic free-form surfaces and industrial benchmark datasets, showing that the use of custom-shaped tools is capable of meeting fine industrial tolerances and outperforms the use of classical, on-market tools.

A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces W_{infty }^{n}[a,b], nin mathbb{N} in space L_{infty }[a,b] and weighted spaces L_{p,omega }[a,b], 1le p< infty with ω a continuous-integrable weight function in (a,b) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L_{infty } (or L_{p,omega }[a,b], 1le p<infty ) are optimal for W_{infty }^{n}[a,b] in L_{infty }[a,b] (or L_{p,omega }[a,b], 1le p<infty ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames

We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH C1 spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.

Manoeuvring speed estimation of a lane-change system using geometric hermite interpolation

Autonomous vehicles demand a continuous path to provide the passengers with physical ease and relaxation. Smooth experience in autonomous vehicles is very essential and can be accomplished by having good trajectories. The trajectory generated by the interpolation algorithm for Geometric Hermite is provided in this paper. The proposed algorithm calculates a path based on a set of geometric Hermite data that consists of predefined points including prescribed tangent angles and curvature values. The generated path has the potential to determine the maximum speed allowed for safe autonomous driving on roads by providing curvature information. It describes an alternative way to estimate design speed permitted on the road for autonomous lane change systems.

Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the R2,1 Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These are special curves in R2,1 that define rational boundaries for the corresponding domain. A method to use RE and MPH curves for skinning purposes, i.e., for circle-based modeling, has been developed recently. In this paper, we continue this study by proposing a new, more flexible way how these curves can be used for skinning a discrete set of circles. We give a thorough overview of our algorithm, and we show a significant advantage of using RE and MPH curves for skinning purposes: as opposed to traditional skinning methods, unintended intersections can be detected and eliminated efficiently.

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.

Applying Rational Envelope curves for skinning purposes

Special curves in the Minkowski space such as Minkowski Pythagorean hodograph curves play an important role in computer-aided geometric design, and their usages are thoroughly studied in recent years. Bizzarri et al. (2016) introduced the class of Rational Envelope (RE) curves, and an interpolation method for G1 Hermite data was presented, where the resulting RE curve yielded a rational boundary for the represented domain. We now propose a new application area for RE curves: skinning of a discrete set of input circles. We show that if we do not choose the Hermite data correctly for interpolation, then the resulting RE curves are not suitable for skinning. We introduce a novel approach so that the obtained envelope curves touch each circle at previously defined points of contact. Thus, we overcome those problematic scenarios in which the location of touching points would not be appropriate for skinning purposes. A significant advantage of our proposed method lies in the efficiency of trimming offsets of boundaries, which is highly beneficial in computer numerical control machining.

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.

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

Provided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.

A new selection scheme for spatial Pythagorean hodograph quintic Hermite interpolants

For given C1 Hermite data, there exists a two-parameter family of Pythagorean hodograph (PH) quintic curves which interpolate the data (two end-points and end-derivatives) as observed by Farouki et al. (2002b). As “good” candidate curves for a selection problem, we propose a special type of PH quintic interpolating curves called extremal interpolants and prove that the extremal interpolants preserve planarity, i.e., they are planar curves if the data are planar. Since there are only four distinct extremal interpolants, the selection problem, when only considering extremal interpolants as possible candidates, is reduced to picking one curve from finite candidates.Due to the preservation of planarity, extremal interpolants coincide with one of p0,0(t),p0,π(t),pπ,0(t),pπ,π(t) if the data are planar, where pϕ0,ϕ2(t) denotes the parametrization proposed by Šír and Jüttler (2005). However, any of the four extremal interpolants is generically not identical to the interpolants for non-planar data, and empirical results suggest that being compared with the unique cubic interpolant, the best curve is one of extremal interpolants among all extremal interpolants and p0,0(t),p0,π(t),pπ,0(t),pπ,π(t).

$\boldsymbol{G^2[C^1]}$ Hermite interpolation using septic PH curves

In this paper, we examine the problem of $G^2[C^1]$ Hermite interpolation using septic Pythagorean hodograph (PH) curves. PH curves are a special class of polynomial parametric curves, which have a polynomial arc length function and rational offsets, and are thus widely used in computer-aided design. According to different factorizations of their first derivative in complex form, septic PH curves are classified into three classes. The curves in the first class are all regular, and their construction under any $G^2[C^1]$ condition has already been studied. Therefore, in this paper we focus on the remaining two classes. The number of septic PH curves in the second class is even and no more than six. The existence of septic PH curves in the third class is dependent on the initial Hermite data, and users may specify a real parameter to determine the resultant curve. In addition, we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.

Implicit surface reconstruction based on generalized radial basis functions interpolant with distinct constraints

• We propose a generalized radial basis function interpolant with difference constraints. • Various types of constraints are derived to satisfy the interpolation conditions . • An adaptive iterative algorithm is used for approximating gradient and tangent constraints. • The numerical results reveal the improved performance of our method. In this work, we present a new algorithm for reconstruction of implicit surfaces from a set of cloud points with normals (Hermite data), based on the generalized radial basis functions interpolant with various types of constraints. Our key contribution is a novel construction of difference constraints in the interpolant to satisfy certain linear functional data, named as the domain constraints, the difference constraints of the gradient and the difference constraints of the tangent. To avoid ambiguous constraints, we apply an iterative algorithm to determine the adaptive distance in the difference constraints. The extended interpolant can not only provide an exact interpolation to the function values, but also approximate the function's derivatives by constructing a difference operator along the gradient or tangent direction. We interpolate the sampling points using our method by constructing a signed distance field in the geometry domain. Compared to the Hermite–Birkhoff interpolation with RBFs, the interpolant we use has faster solution efficiency. The experimental results of several data sets show the reliability and performance of our method in a variety of practical scenarios.

Existence of Pythagorean-hodograph quintic interpolants to spatial G1 Hermite data with prescribed arc lengths

A unique feature of polynomial Pythagorean–hodograph (PH) curves is the ability to interpolate G1 Hermite data (end points and tangents) with a specified total arc length. Since their construction involves the solution of a set of non–linear equations with coefficients dependent on the specified data, the existence of such interpolants in all instances is non–obvious. A comprehensive analysis of the existence of solutions in the case of spatial PH quintics with end derivatives of equal magnitude is presented, establishing that a two–parameter family of interpolants exists for any prescribed end points, end tangents, and total arc length. The two free parameters may be exploited to optimize a suitable shape measure of the interpolants, such as the elastic bending energy.