•We presents a Hermite interpolation scheme for G2 boundary data and arc length constraint using Pythagorean hodograph (PH) curves of degree 7.•The interpolation scheme is completely local. Each ...spline segment is defined as a PH biarc curve of degree 7.•In this way the solution of the G2 continuity equations can be derived in a closed form, depending on four free parameters.•By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents.
In this paper we address the problem of constructing G2 planar Pythagorean–hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree 7, which results in having a closed form solution of the G2 interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is 5.
In this paper the problem of constructing spatial G2 continuous Pythagorean-hodograph (PH) spline curves, that interpolate points and frame data, and in addition have the prescribed arc-length, is ...addressed. The interpolation scheme is completely local and can be directly applied for motion design applications. Each spline segment is defined as a PH biarc curve of degree 7 satisfying super-smoothness conditions at the biarc’s joint point. The biarc is expressed in a closed form with additional free parameters, where one of them is determined by the length constraint. The selection of the remaining free parameters is suggested, that allows the existence of the solution of the length interpolation equation for any prescribed length and any ratio between norms of boundary tangents. By the proposed automatic procedure for computing the frame and velocity quaternions from the first and second order derivative vectors, the paper presents a direct generalization of the construction done for planar curves to spatial ones. Several numerical examples are provided to illustrate the proposed method and to show its good performance, also when a spline construction in considered.
•We propose the Gauss–Legendre polynomials.•We also propose the Gauss–Legendre curve as the barycentric combination of the control points with the weights given by the Gauss–Legendre polynomials.•We ...discuss the shape control of polynomial curves using the Gauss–Legendre curves and analyze basic properties of the Gauss–Legendre polynomials.
The Gauss–Legendre (GL) polygon was recently introduced for the shape control of Pythagorean hodograph curves. In this paper, we consider the GL polygon of general polynomial curves. The GL polygon with n+1 control points determines a polynomial curve of degree n as a barycentric combination of the control points. We identify the weight functions of this barycentric combination and define the GL polynomials, which form a basis of the polynomial space like the Bernstein polynomial basis. We investigate various properties of the GL polynomials such as the partition of unity property, symmetry, endpoint interpolation, and the critical values in comparison with the Bernstein polynomials. We also present the definite integral and higher derivatives of the GL polynomials. We then discuss the shape control of polynomial curves using the GL polygon. We claim that the design process of high degree polynomial curves using the GL polygon is much easier and more predictable than if the curve is given in the Bernstein–Bézier form. This is supported by some neat illustrative examples.
•A novel approach to constructing polynomial minimal surfaces is presented.•Minimal surfaces arise from Pythagorean triples of complex polynomials.•They are Pythagorean-normal surfaces with ...isothermal parameterization.•Generalization to minimal surfaces in non-isothermal parameterization is given.•Cubic, quartic and quintic examples are presented and visualized.
A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.
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.
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.
In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈0,1 using the complex representation, it is convenient to invoke a translation/rotation/scaling ...transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.
•The construction of quintic PH curves with two prescribed control-polygon legs is formulated.•For given control-polygon legs, only the solution of a quadratic or quartic equation is required.•Certain families of quintic PH curves incorporating desired geometrical properties are identified.•Computed examples are used to illustrate the implementation and performance of the method.
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.
Left: The four canonical–form quintic PH curves r(t) determined by the two end derivatives di=2.0+1.5i, df=1.5–4.5i. Right: the corresponding pre–image polynomials w(t), showing the footpoints of the origin on them.
•The presence or absence of inflections on planar quintic PH curves is characterized.•The space of planar PH quintics is partitioned by curves that possess points of infinite curvature.•Proximity of the pre-image parabola to the origin of the hodograph plane incurs tight loops.•These observations provide insight into the behavior of planar PH quintic Hermite interpolants.
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 R2,1 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.
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