Geometric Hermite Interpolation Research Articles

Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions

It has been recently proved that every proper restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points P1≠P2, in the plane, along with corresponding unit directions d1,d2 that satisfy d1⋅(P2−P1)≥0 and d2⋅(P2−P1)≥0, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension λ>0 that connects P1 to P2 with initial direction d1 and terminal direction d2.

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.

Geometric Hermite interpolation in $$\mathbb {R}^{n}$$ by refinements

Geometric Hermite interpolation in $$\mathbb {R}^{n}$$ by refinements

Geometric Hermite interpolation by a family of spatial algebraic–trigonometric PH curves

This paper proposes to define a family of spatial algebraic–trigonometric Pythagorean Hodograph (ATPH) curves by integrals of scaled unit tangent vector fields which are originally defined as sphere curves. The obtained ATPH curves have only polynomial parametric speeds but the curves can be employed to represent several typical non-polynomial curves without rational form. A simple algorithm for geometric Hermite interpolation by the proposed spatial ATPH curves without or with arc length constraint has been given. Given two boundary points and two unit tangents at the points, possibly with a prescribed arc length, a unique interpolating ATPH curve can be obtained by solving a simple linear system.

Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution.

Area-preserving geometric Hermite interpolation

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic Bézier polynomials. We show there exists a family of such curves which are 5th 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 4th 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.

Elastic Splines II: Unicity of Optimal s-Curves and Curvature Continuity

Given points $$P_1,P_2,\ldots ,P_m$$ in the complex plane, we are concerned with the problem of finding an interpolating curve with minimal bending energy (i.e., an optimal interpolating curve). It was shown previously that existence is assured if one requires that the pieces of the interpolating curve be s-curves. In the present article, we also impose the restriction that these s-curves have chord angles not exceeding $$\pi /2$$ in magnitude. With this setup, we have identified a sufficient condition for the curvature continuity of optimal interpolating curves. This sufficient condition relates to the stencil angles $$\{\psi _j\}$$ , where $$\psi _j$$ is defined as the angular change in direction from segment $$[P_{j-1},P_j]$$ to segment $$[P_j,P_{j+1}]$$ . An angle $$\Psi $$ ( $$\approx 37^\circ $$ ) is identified, and we show that if the stencil angles satisfy $$\vert {\psi _j} \vert <\Psi $$ , then optimal interpolating curves are curvature continuous. We also prove that the angle $$\Psi $$ is sharp. As with the previous article (Borbely and Johnson in Constr Approx 40:189–218, 2014), much of our effort is concerned with the geometric Hermite interpolation problem of finding an optimal s-curve $$c_1(\alpha ,\beta )$$ that connects $$0+i0$$ to $$1+i0$$ with prescribed chord angles $$(\alpha ,\beta )$$ . Whereas existence was previously shown, and sometimes uniqueness, the present article begins by establishing uniqueness when $$\vert {\alpha } \vert ,\vert {\beta } \vert \le \pi /2$$ and $$\vert {\alpha -\beta } \vert <\pi $$ . We also prove two fundamental identities involving the initial and terminal signed curvatures of $$c_1(\alpha ,\beta )$$ and partial derivatives, with respect to $$\alpha $$ or $$\beta $$ , of the bending energy of $$c_1(\alpha ,\beta )$$ .

Geometric Hermite interpolation by a family of intrinsically defined planar curves

This paper proposes techniques of interpolation of intrinsically defined planar curves to Hermite data. In particular, a family of planar curves corresponding to which the curvature radius functions are polynomials in terms of the tangent angle are used for the purpose. The Cartesian coordinates, the arc lengths and the offsets of this type of curves can be explicitly obtained provided that the curvature functions are known. For given G1 or G2 boundary data with or without prescribed arc lengths the free parameters within the curvature functions can be obtained just by solving a linear system. By choosing low order polynomials for representing the curvature radius functions, the interpolating curves can be spirals that have monotone curvatures or fair curves with small numbers of curvature extremes. Several examples of shape design or curve approximation using the proposed method are presented.

A constructive framework for minimal energy planar curves

Given points P1,P2,…,Pn in the plane, we are concerned with the problem of finding a fair curve which interpolates the points. We assume that we have a method in hand, called a basic curve method, for solving the geometric Hermite interpolation problem of fitting a regular C∞ curve between two points with prescribed tangent directions at the endpoints. We also assume that we have an energy functional which defines the energy of any basic curve. Using this basic curve method repeatedly, we can then construct G1 curves which interpolate the given points P1,P2,…,Pn. The tangent directions at the interpolation points are variable and the idea is to choose them so that the energy of the resulting curve (i.e., the sum of the energies of its pieces) is minimal. We give sufficient conditions on the basic curve method, the energy functional, and the interpolation points for (a) existence, (b) G2 regularity, and (c) uniqueness of minimal energy interpolating curves. We also identify a one-parameter family of basic curve methods, based on parametric cubics, whose minimal energy interpolating curves are unique and G2 under suitable conditions. One member of this family looks very promising and we suggest its use in place of conventional C2 parametric cubic splines.

Geometric Hermite interpolation by logarithmic arc splines

This paper considers the problem of G1 curve interpolation using a special type of discrete logarithmic spirals. A “logarithmic arc spline” is defined as a set of smoothly connected circular arcs. The arcs of a logarithmic arc spline have equal angles and the curvatures of the arcs form a geometric sequence. Given two points together with two unit tangents at the points, interpolation of logarithmic arc splines with a user specified winding angle is formulated into finding the positive solutions to a vector equation. A practical algorithm is developed for computing the solutions and construction of interpolating logarithmic arc splines. Compared to known methods for logarithmic spiral interpolation, the proposed method has the advantages of unbounded winding angles, simple offsets and NURBS representation.

GEOMETRIC HERMITE INTERPOLATION FOR PLANAR PYTHAGOREAN-HODOGRAPH CUBICS

We solve the geometric Hermite interpolation problem with planar Pythagorean-hodograph cubics. For every Hermite data, we determine the exact number of the geometric Hermite interpolants and represent the interpolants explicitly. We also present a simple criterion for determining whether the interpolants have a loop or not.

PYTHAGOREAN-HODOGRAPH CUBICS AND GEOMETRIC HERMITE INTERPOLATION

In this paper, we present the geometric Hermite interpolation for planar Pythagorean-hodograph cubics for some general Hermite data. curve ro(t) = r(t) + dn(t) at a xed distance d from a given polynomial curve r(t), in the direction of its unit normal vector n(t), is an important problem in computer aided geometric design. The oset ro(t) is not, in general, a rational curve. Thus we need some approximation schemes to deal with osets of a polynomial curves. However, if the speed function of a given polynomial curve is a polynomial, then an oset curve can be expressed in a rational parametrization. For this reason, the Pythagorean-hodograph (PH) curves, whose speed function is a polynomial, were introduced by Farouki and Sakkalis ((6)). Since then, there have been vast researches on this class of curves by themselves and others ((1), (2), (3), (4), (5), and (11)). A PH curve r(t) = (x(t);y(t)) means a special polyno- mial curve which is characterized by the algebraic property that its hodograph r 0 (t) = (x 0 (t);y 0 (t)) satises the Pythagorean condition, that is, x 02 (t) +y 02 (t) = 2 (t) for some polynomial (t). So, the arc-length function and the osets

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.

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R 3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.

Geometric Hermite interpolation by monotone helical quintics

Monotone helical curves are polynomial helices whose unit tangent maintains a fixed sense of rotation about an axis. We investigate the interpolation of geometric Hermite data consisting of end points, tangents, and curvatures by monotone helical quintics. Based on the Hopf map model for spatial Pythagorean-hodograph curves, which subsume polynomial helices, we show that the geometric Hermite interpolation can be determined by solving a certain univariate polynomial equation of degree twelve.

High-order approximation of implicit surfaces by [formula omitted] triangular spline surfaces

In this paper, we present a method for the approximation of implicit surface by G 1 triangular spline surface. Compared with the polygonization technique, the presented method employs piecewise polynomials of high degree, achieves G 1 continuity and is capable of interpolating positions, normals, and normal curvatures at vertices of an underlying base mesh. To satisfy vertex enclosure constraints, we develop a scheme to construct a C 2 consistent boundary curves network which is based on the geometric Hermite interpolation of normal curvatures. By carefully choosing the degrees of scalar weight functions, boundary Bézier curves and triangular Bézier patches, we propose a local and singularity free algorithm for constructing a G 1 triangular spline surface of arbitrary topology. Our method achieves high precision at low computational cost, and only involves local and linear solvers which leads to a straightforward implementation. Analyses of freedom and solvability are provided, and numerical experiments demonstrate the high performance of algorithms and the visual quality of results.

Curves with chord length parameterization

Motivated by the recent work (Farin, G., 2006. Rational quadratic circles are parameterized by chord length, Computer Aided Geometric Design 23, 722–724), we identify a family of curves that can be parameterized by chord length. The α- and ( α , β ) -schemes are presented for characterizing planar and spatial curves respectively. Rational chord-length parameterizations are thoroughly investigated. In particular, the low-degree rational curves such as cubics and quartics are studied and applied to geometric Hermite interpolation. The results advise that this new class of curves, subsuming straight lines and circular arcs, have several obvious advantages over general polynomial and rational curves.

Geometric Hermite interpolation by cubic [formula omitted] splines

In this paper, geometric Hermite interpolation by planar cubic G 1 splines is studied. Three data points and three tangent directions are interpolated per polynomial segment. Sufficient conditions for the existence of such a G 1 spline are determined that cover most of the cases encountered in practical applications. The existence requirements are based only upon geometric properties of data and can easily be verified in advance. The optimal approximation order 6 is confirmed, too.

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

It is shown that, depending upon the orientation of the end tangents $\t_0, \t_1$ relative to the end point displacement vector $\Delta\p=\p_1-\p_0$, the problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points $\p_0,\ldots,\p_n$ in $\mathbb{R}^3$, compatible with a $G^1$ piecewise--PH--cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape--preservation properties of the resulting curves, is illustrated by a selection of computed examples.

Geometric Hermite Interpolation for Space Curves by B-Spline

This paper considers the space GC 2 Hermite interpolation by cubic B-spline curve which is based on de Boor's idea for constructing the planar GC 2 Hermite interpolation. In addition to position and tangent direction, the