8 research outputs found
Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
AbstractA ādoubleā Pythagorean-hodograph (DPH) curve r(t) is characterized by the property that |rā²(t)| and |rā²(t)Ćrā³(t)| are both polynomials in the curve parameter t. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. As noted by Beltran and Monterde, the Hopf map representation of spatial PH curves appears better suited to the analysis of DPH curves than the quaternion form. A categorization of all DPH curve types up to degree 7 is developed using the Hopf map form, together with algorithms for their construction, and a selection of computed examples of (both helical and non-helical) DPH curves is included, to highlight their attractive features. For helical curves, a separate constructive approach proposed by Monterde, based upon the inverse stereographic projection of rational line/circle descriptions in the complex plane, is used to classify all types up to degree 7. Criteria to distinguish between the helical and non-helical DPH curves, in the context of the general construction procedures, are also discussed
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Mapping rational rotation-minimizing frames from polynomial curves on to rational curves
Given a polynomial space curve r(Ī¾) that has a rational rotationāminimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves rĖ(Ī¾) with the same rotationāminimizing frame as r(Ī¾) 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(Ī¾)=rā²(Ī¾)Ćrā³(Ī¾) and distances from the origin specified in terms of a rational function f(Ī¾) as f(Ī¾)/āu(Ī¾)ā. An explicit characterization of the rational curves rĖ(Ī¾) generated by a given RRMF curve r(Ī¾) 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(Ī¾), obviating the nonālinear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(Ī¾). Criteria for identifying lowādegree instances of the curves rĖ(Ī¾) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples