Pythagorean-hodograph Curves Research Articles

Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

On G2 approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

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.

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.

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.

Optimal interpolation with spatial rational Pythagorean hodograph curves

Using a residuum approach, we provide a complete description of the space of the rational spatial curves of given tangent directions. The rational Pythagorean hodograph curves are obtained as a special case when the norm of the direction field is a perfect square. The basis for the curve space is given explicitly. Consequently a number of interpolation problems (G1, C1, C2, C1/G2) in this space become linear, cusp avoidance can be encoded by linear inequalities, and optimization problems like minimal energy or optimal length are quadratic and can be solved efficiently via quadratic programming. We outline the interpolation/optimization strategy and demonstrate it on several examples.

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

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.

Gauss–Legendre polynomial basis for the shape control of polynomial curves

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.

Partial fraction decomposition for rational Pythagorean hodograph curves

All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces defined by fixing the denominator polynomial and describe the construction of canonical bases for them. We also show (as an analogy to the fraction decomposition of rational functions) that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator. Our results give insight into the structure of these spaces, clarify the role of their polynomial and truly rational (non-polynomial) curves, and suggest applications to interpolation problems.

On Control Polygons of Planar Sextic Pythagorean Hodograph Curves

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives’ factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified.

B-Spline Pythagorean Hodograph Curves in Clifford Algebras

B-Spline Pythagorean Hodograph Curves in Clifford Algebras

Global Optimization Method to Comprise Rotation-Minimizing Euler-Rodrigues Frames of Pythagorean-Hodograph Curve

Global Optimization Method to Comprise Rotation-Minimizing Euler-Rodrigues Frames of Pythagorean-Hodograph Curve

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.

Rational framing motions and spatial rational Pythagorean hodograph curves

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing literature.

Polynomial curves with projections to PH curves

Despite the fact that the orthogonal projection of a spatial Pythagorean hodograph (PH) curve into the plane is not a planar PH curve in general, we can find special cases such that the PH property is preserved when the curve is projected. In Farouki et al. (2021) the authors studied how to generate spatial PH curves with planar PH projections. Their approach and presented results motivated us to continue and extend this investigation. We study geometric conditions under which a spatial curve is projected to a PH curve. For this purpose, we introduced a suitable geometric characterization of the curves with PH property via intersection multiplicity of the associated curves described by the hodograph mapping with the absolute conic. As a consequence we will show that a generic polynomial curve of degree higher than five possesses no parallel projection to a PH curve. On contrary, for a spatial cubic there are finitely many ways how to orthogonally project it to a planar PH cubic. And the same holds for oblique parallel projections of spatial quintics. Hence these cases are examined in more detail.

Parametric PH Curves Model Based Kinematic Control of the Shape of Mobile Soft Manipulators in Unstructured Environment

Mobile soft continuum manipulator (MSCM) is more and more used in various applications of everyday life, such as logistics, farming, medical diagnosis, medical therapy, bakery, human collaboration, etc. However, the control of its shape remains a major challenge, especially when it is navigating in an unstructured environment. In this contribution, a kinematic-model-based shape control is designed in the space configuration for an MSCM, allowing its autonomous navigation in the presence of obstacles. For that purpose, the shape kinematic modeling is realized using a parametric spatial Pythagorean hodograph (PH) curve with a predefined length, whereas the artificial potential field algorithm acts on the control points of the PH curve for shape adaptation in presence of obstacles. The novelty of the proposed work resides in the use of parametric curve formalism to model the shape of an infinite degree of freedom mobile soft manipulator robot and its kinematic control by optimizing its bending potential energy during its motion. The shape optimization based on a minimal bending energy during the collision-free path is respecting a sliding mode strategy applied to the PH curve control points. Experimental results are obtained using a class of MSCM called RobotinoXT. These experiments show the advantages of using a reduced kinematic model based on PH curve to control the MSCM shape in dynamic motion.

Shape Reconstruction of Soft Manipulators Using Vision and IMU Feedback

In recent times, soft manipulators have gardened immense interest given their dexterous abilities. A critical aspect for their feedback control involves the reconstruction of the manipulator shape. The research, for the first time, presents shape reconstruction of a soft manipulator through sensor fusion of information available from Inertial Measurement Units (IMUs) and visual tracking. The manipulator is modeled using multi-segment continuous curvature Pythagorean Hodograph (PH) curves. PH curves are a class of continuous curvature curves with an analytical expression for the hodograph (slope). The shape reconstruction is formulated as an optimization problem that minimizes bending energy of the curve with a length constraint and the information from IMUs and/or visual markers. The paper experimentally investigates the robustness of shape reconstruction for scenarios when position of all visual markers, or slope at all the knots (placement of sensors) are known. Occlusion of manipulator segments is frequent, hence, this scenario is simulated by fusing information of available slopes (IMUs) at all knots and position (vision) at some knots. The experiments are performed on a planar tensegrity manipulator with IMUs feedback and visual tracking. The robustness study indicates reliability of these models for real world applications. Additionally, the proposed sensor fusion algorithm provides promising results where, for most cases, the shape estimates benefit from additional position information. Finally, the low dimensionality of the optimization problem argues for extension of the approach for real-time applications.

On the construction of polynomial minimal surfaces with Pythagorean normals

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.

Identifying Pythagorean-Hodograph Curves Closest to Prescribed Planar Bézier Curves

The problem of identifying the planar Pythagorean-hodograph curve that is “closest” to a given Bézier curve, and has the same end points (or end points and tangents), is considered. The “closeness” measure employed in this context is the root-mean-square magnitude of the differences between pairs of corresponding control points for the two curves. The methodology is developed in the context of quintic PH curves, although it readily generalizes to PH curves of higher degree. Using the complex representation for planar curves, it is shown that this problem can be reduced to the minimization of a quartic penalty function in certain real variables, subject to two quadratic constraints, which can be efficiently solved by the Lagrange multiplier method. By expressing the penalty function and constraints in terms of variables that identify a complex pre-image polynomial, the closest solution is guaranteed to be a PH curve. Several computed examples are used to illustrate implementation of the optimization methodology and typical approximation results that can be obtained.