Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames

Space curves defined by curvature–torsion relations and associated helices

The relationships between certain families of special curves, including the general helices, slant helices, rectifying curves, Salkowski curves, spherical curves, and centrodes, are analyzed. First, characterizations of proper slant helices and Salkowski curves are developed, and it is shown that, for any given proper slant helix with principal normal n, one may associate a unique general helix whose binormal b coincides with n. It is also shown that centrodes of Salkowski curves are proper slant helices. Moreover, with each unit–speed non–helical Frenet curve in the Euclidean space E3, one may associate a unique circular helix, and characterizations of the slant helices, rectifying curves, Salkowski curves, and spherical curves are presented in terms of their associated circular helices. Finally, these families of special curves are studied in the context of general polynomial/rational parameterizations, and it is observed that several of them are intimately related to the families of polynomial/rational Pythagorean–hodograph curves

Construction of rational curves with rational arc lengths by direct integration

A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r′(ξ)=(u2(ξ)−v2(ξ),2u(ξ)v(ξ))/w2(ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r′(ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples

Optimization of Corner Blending Curves

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner curves

Computing bisectors in a dynamic geometry environment

In this note, an approach combining dynamic geometry and automated deduction techniques is used to study the bisectors between points and curves. Usual teacher constructions for bisectors are discussed, showing that inherent limitations in dynamic geometry software impede their thorough study. We show that the interactive sketching of bisectors and an automatic treatment of the algebraic problem involved can give a reasonable knowledge about them. Since some cases are currently out of computational scope, despite the simplicity of the bisector problem, we sketch an alternative method for dealing with them

Model for performance prediction in multi-axis machining

This paper deals with a predictive model of kinematical performance in 5-axis milling within the context of High Speed Machining. Indeed, 5-axis high speed milling makes it possible to improve quality and productivity thanks to the degrees of freedom brought by the tool axis orientation. The tool axis orientation can be set efficiently in terms of productivity by considering kinematical constraints resulting from the set machine-tool/NC unit. Capacities of each axis as well as some NC unit functions can be expressed as limiting constraints. The proposed model relies on each axis displacement in the joint space of the machine-tool and predicts the most limiting axis for each trajectory segment. Thus, the calculation of the tool feedrate can be performed highlighting zones for which the programmed feedrate is not reached. This constitutes an indicator for trajectory optimization. The efficiency of the model is illustrated through examples. Finally, the model could be used for optimizing process planning

Mathematica tools for quaternionic polynomials

In this paper we revisit the ring of (left) one-sided quaternionic polynomials with special focus on its zero structure. This area of research has attracted the attention of several authors and therefore it is natural to develop computational tools for working in this setting. The main contribution of this paper is a Mathematica collection of functions QPolynomial for solving polynomial problems that we frequently find in applications.(undefined)info:eu-repo/semantics/publishedVersio