We solve the problem of G2∕C1 Hermite interpolation (i.e. interpolation of prescribed boundary points as well as first derivatives and curvatures at these points) by planar quintic Pythagorean ...Hodograph B-spline curves with one free interior knot which acts as a shape parameter. We present conditions on the data ensuring the existence of solutions. Finally, we illustrate the influence of the interior knot on the shape of the resulting interpolant and on the values of the absolute rotation index or the bending energy.
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.
•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.
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.
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.
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.
•ATPH curves with polynomial parametric speed are constructed by solving different geometric Hermite interpolation problems.•Interpolation to both spatial G1 Hermite data equipped with curvature ...values and spatial G2 Hermite data is considered.•The Hermite ATPH interpolants are obtained by solving simple linear systems.•The obtained Hermite ATPH interpolants have C0-continuous curvature at least.
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.
•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.
