We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first ...method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schröcker and Šír 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples.
•Three methods to compute rational curves with rational arc length function.•Compact quaternionic formulations, extending formulas for rational PH curves.•Comparison and critical discussion of merits of each approach.•Extensive examples relating our findings to existing literature.
In this paper, the problem of interpolation of two points, two corresponding tangent directions and curvatures, and the arc length sampled from a circular arc (circular arc data) is considered. ...Planar Pythagorean–hodograph (PH) curves of degree seven are used since they possess enough free parameters and are capable of interpolating the arc length in an easy way. A general approach using the complex representation of PH curves is presented first and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6 and a detailed analysis of the existence of admissible solutions is provided. In the case of several solutions, some criteria for selecting the most appropriate one are described and an asymptotic analysis is given. Numerical examples are included which confirm theoretical results.
•The interpolation of G2 data and an arc lengths is stated in general.•A detailed analysis for circular arc data is provided.•Four real solutions of the problem are confirmed.•An asymptotic analysis together with asymptotic approximation order is given.•A selection of appropriate solution is suggested and everal numerical examples are shown.
•Construction of G1 continuous PH cubic biarcs is done.•Thorough analysis of system of nonlinear equations is given.•An algorithm enabling an easy implementation is provided.
The interpolation of two ...points and two tangent directions by planar parametric cubic curves with prescribed arc lengths is considered. It is well known that this problem is highly nonlinear if standard cubic curves are used. However, if Pythagorean-hodograph (PH) curves are considered, the problem simplifies due to their distinguished property that the arc length is a polynomial function of its coefficients. Since a single segment of a PH cubic curve does not provide enough free parameters, the so called PH cubic biarcs are used. A detailed and thorough analysis of the resulting system of nonlinear equations is provided and closed form solutions are given for any possible configuration of given data. The lookup table of the solutions is constructed enabling an easy implementation of the described method. Some quantities arising from geometric properties of the resulting curves are suggested in order to select the most appropriate one and the bending energy is confirmed as the most promising selection criterion. Several numerical examples are presented which confirm theoretical results. Finally, an example of approximation of an analytic curve by G1 PH cubic biarc spline curve is presented and the approximation order is numerically established.
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.
In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties ...and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown.
•Generalized plane offsets have good regularity properties.•Generalized plane offsets to rational Pythagorean-hodograph curves are rational.•A bound on the offset distance prevents self-intersections on generalized offsets.•Any rationally parameterized curve by a support function is Pythagorean-hodograph.•A family of rational curves of constant width and Zindler curves is generated.
The shape information is the key to realize real-time, accurate and effective control of continuum robots. Traditional image processing method and start-end-points fitting method are combined. IMU ...and camera are used to capture the end position and attitude information of continuum robot, and do not rely on the acquisition of the global image to solve the problem of limited vision in traditional image processing method. In addition, a new start-end-points fitting method is proposed, which is Pythagorean Hodograph-Bézier(PH-B) fitting method. The new method simplifies the solving steps of control points of fitting curve The control points can be obtained only through the geometric relationship, and do not need to rely on optimization algorithm for iterative solution. This method has good shape fitting accuracy and applicability in the curvature range from 0°to 90° through the simulation analysis. A variety of experiments are designed to verify the results. The experimental results show that the maximum shape reconstruction error by the proposed method is about 2mm. This proves the applicability of the PH-B method in shape reconstruction of continuum robots.
Recently, Bizzarri et al. (2021) discussed the so-called Pythagorean-hodograph curves of Tschirnhaus type, a generalization to higher degrees of Tschirnhausen cubic. We recall that these curves in ...Bézier form coincide with the typical curves introduced by Mineur et al. (1998), as well as with a classical family of sinusoidal spirals. Therefore, they all enjoy the same properties, such as the rational character of their offsets or the existence of only one curve (up to similarities) for each degree. By elucidating this connection among curves of Tschirnhaus type, typical curves, and sinusoidal spirals, we rederive several relevant results found by Bizzarri et al. (2021).
•We elucidate the connection between curves of Tschirnhaus type and previous models.•They coincide with the typical Bézier curves introduced by Mineur et al. (1998).•They are also segments of sinusoidal spirals belonging to a classical family.•Several properties are direct consequences of this connection or easily rederived.
•We propose a new selection scheme for a spatial PH quintic interpolation problem.•The new scheme identifies extremal preimages whose control points are in the extremal position to other control ...points.•For a non-degenerate data set, these extremal preimages determine four PH quintic 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).
•We propose a topological criterion to select the best PH curves with a planar Gauss-Legendre control polygon.•We classify planar PH curves of degree 2n+1 into 2n subclasses by defining the types of ...PH curves.•By exploiting the types, we present a lower bound of the topological index of a PH curve.•We prove the uniqueness of the best PH curve with a given Gauss-Legendre control polygon.•We also show the existence of the best PH curves for cubic and quintic PH curves.
Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bézier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bézier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the “best” PH curves, which is the main topic of this paper.
To resolve the problem, we first classify PH curves of degree 2n+1 into 2n subclasses by defining the types of PH curves, and propose the absolute hodograph winding number as a topological index to characterize the topological behavior of PH curves in shape. We present a lower bound of the topological index of a PH curve which is given solely by its type, and prove the uniqueness of the best PH curve by exploiting it. The existence theorems are also proved for cubic and quintic PH curves. Finally, we propose a selection rule of the best PH curve only based on its type.
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