We investigate the problem of constructing spatial C2 closed loops from a single polynomial curve segment r(t), t∈0,1 with a prescribed arc length S and continuity of the Frenet frame and curvature ...at the juncture point r(1)=r(0). Adopting canonical coordinates to fix the initial/final point and tangent, a closed-form solution for a two-parameter family of interpolants to the given data can be constructed in terms of degree 7 Pythagorean-hodograph (PH) space curves, and continuity of the torsion is also obtained when one of the parameters is set to zero. The geometrical properties of these closed-loop PH curves are elucidated, and certain symmetry properties and degenerate cases are identified. The two-parameter family of closed-loop C2 PH curves is also used to construct certain swept surfaces and tubular surfaces, and a selection of computed examples is included to illustrate the methodology.
•C2 closed loops of prescribed arc length are defined by a single polynomial curve.•Continuity of the Frenet frame and curvature of such loops is considered.•PH curves of degree 7 are used.•Geometrical properties of obtained loops are investigated.•Swept and tubular surfaces are constructed.
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.
•PH curves are characterized via intersection multiplicity of a suitable associated curve with the absolute conic.•Geometric conditions under which a spatial curve is projected to a PH curve are thoroughly investigated.•PH projections of generic polynomial curves, and especially of cubics and quintics are studied.
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.
•A methodology to identify the PH curve closest to a prescribed planar Bezier curve is developed.•The methodology focuses on quintic PH curves, and can accommodate G0 or G1 end conditions.•The identification of the closest PH curve is formulated as a constrained optimization problem.•Computed examples are used to illustrate the implementation and performance of the method.
Although the orthogonal projection of a spatial Pythagorean–hodograph (PH) curve on to a plane is not (in general) a planar PH curve, it is possible to construct spatial PH curves so as to ensure ...that their orthogonal projections on to planes of a prescribed orientation are planar PH curves. The construction employs an analysis of the root structure of the components of the quaternion polynomials that generate spatial PH curves, and it encompasses both helical and non–helical spatial PH curves. An initial characterization for orthogonal projections of spatial PH curves on to the coordinate planes provides the basis for a generalization to projections of arbitrary direction, based on unit quaternion rotation transformations of R3.
Examples of the special class of spatial PH curves (gray) that possess planar PH curves (blue) as their projections on to one of the coordinate planes.
•The conditions under which spatial PH curves admit planar PH projections are characterized.•Spatial PH curves with planar PH projections include helical and non-helical polynomial curves.•The product of sums of squares of polynomials must equal the perfect square of a polynomial.•The case of projections onto a coordinate plane is extended to projections onto arbitrary planes.
The PH B-spline interpolant of degree 7 together with the input C3 Hermite data.
•Cd Hermite interpolation by spatial PH B-spline curves is investigated.•The interpolants are obtained by using ...properties of B-spline basis functions and via solving quaternion equations.•All algorithms are purely symbolic and the main contribution lies in the unifying approach to the formulated problem.
Following the recent results of Albrecht et al. (2017, 2020), the problem of Hermite interpolation by clamped spatial Pythagorean hodograph (PH) B-spline curves is thoroughly investigated in this paper. The constructed interpolants are obtained by using beneficial properties of B-spline basis functions and via solving special quadratic and linear equations in quaternion algebra. All the designed procedures are purely symbolic and the main contribution lies in the unifying approach to the formulated problem. The results are confirmed by several computed examples.
•Singular cases of the planar and spatial PH quintic Hermite interpolation problems are identified.•Gaps in existing theory are addressed by a comprehensive treatment of these singular ...instances.•Cases that correspond to constraints on the derivative magnitudes or orientations are analyzed.•The specialization of spatial PH quintic Hermite interpolants to planar interpolants is identified.
A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.
We discuss special Pythagorean hodograph curves which can be considered, from construction point of view, as degree n generalizations of the famous Tschirnhaus cubic. It will be proved that for each ...n there exists only one curve of Tschirnhaus type up to similarities.
The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, is addressed. The solution employs planar quintic Pythagorean–hodograph (PH) ...curves with equal-magnitude end derivatives. By reduction to canonical form it is shown that, in this context, the problem can be expressed in terms of finding the real solutions to a system of three quadratic equations in three variables. This system admits further reduction to just a single univariate biquadratic equation, which always has positive roots. It is found that this construction of G1 Hermite interpolants of specified arc length admits two formal solutions — of which one has attractive shape properties, and the other must be discarded due to undesired looping behavior. The algorithm developed herein offers a simple and efficient closed-form solution to a fundamental constructive geometry problem that avoids the need for iterative numerical methods.
•An algorithm to construct interpolants to planar G1 Hermite data, with exact prescribed arc lengths, is presented.•The problem admits a closed-form solution, requiring little more than the solution of a quadratic equation.•There exist two formal solutions, the “good” solution corresponding to the least value for the absolute rotation index.•The algorithm accommodates the special cases of parallel or symmetric end tangents.
•The connection between geometric continuity of PH curves and their associated ER frames is examined.•Construction of interpolating G2 continuous PH (spline) curve of degree seven with G1 continuous ...ER frame is presented.•Asymptotic analysis is provided for data sampled from a smooth parametric curve and its general adapted frame.•The existence of a degree seven PH interpolant of optimal approximation order 6 is proven.
The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial PH curve, we consider the Euler-Rodrigues (ER) frame. Moreover G1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of Gk continuity of ER frames implies Gk+1 continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.
A unique feature of polynomial Pythagorean–hodograph (PH) curves is the ability to interpolate G1 Hermite data (end points and tangents) with a specified total arc length. Since their construction ...involves the solution of a set of non–linear equations with coefficients dependent on the specified data, the existence of such interpolants in all instances is non–obvious. A comprehensive analysis of the existence of solutions in the case of spatial PH quintics with end derivatives of equal magnitude is presented, establishing that a two–parameter family of interpolants exists for any prescribed end points, end tangents, and total arc length. The two free parameters may be exploited to optimize a suitable shape measure of the interpolants, such as the elastic bending energy.
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