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1.
C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs
Bohumír Bastl, Michal Bizzarri, Marjetka Knez, Miroslav Lávička, Kristýna Michálkova, Zbiněk Šír, Vito Vitrih, Emil Žagar, 2014, original scientific article

Abstract: In this paper the ▫$C^1$▫ Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates ▫$C^1$▫ data at one point and they are then joined together with a ▫$C^1$▫ continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally ▫$C^1$▫ continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.
Keywords: mathematics, parametric curve, PH curve, Pythagorean-hodograph, Hermite interpolation, biarc, cubic curve
Published in RUP: 03.04.2017; Views: 2028; Downloads: 39
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2.
Pythagorean-hodograph cycloidal curves
Jernej Kozak, Marjetka Knez, Mladen Rogina, Vito Vitrih, 2015, original scientific article

Abstract: In the paper, Pythagorean-hodograph cycloidal curves as an extension of PH cubics are introduced. Their properties are examined and a constructive geometric characterization is established. Further, PHC curves are applied in the Hermite interpolation, with closed form solutions been determined. The asymptotic approximation order analysis carried out indicates clearly which interpolatory curve solution should be selected in practice. This makes the curves introduced here a useful practical tool, in particular in algorithms that guide CNC machines.
Keywords: pitagorejski hodograf, C-krivulje, trigonometrične funkcije, karakterizacija, Hermiteova interpolacija, asimptotični red aproksimacije, Pythagorean-hodograph, C-curves, trigonometric functions, characterization, Hermite interpolation, asymptotic approximation order
Published in RUP: 08.08.2016; Views: 2598; Downloads: 198
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3.
Parametric curves with Pythagorean binormals
Jernej Kozak, Marjetka Knez, Vito Vitrih, 2015, original scientific article

Abstract: In this paper, a class of rational spatial curves that have a rational binormal is introduced . Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing osculating frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly simplifies if the curve is a polynomial one. Further, polynomial PB curves of the degree % 7 and rational PB curves of the degree % 6 that possess rational rotation-minimizing osculating frames are derived, and it is shown that no lower degree curves, constructed from quadratic quaternion polynomials, with such a property exist.
Keywords: pitagorejski hodograf, pitagorejska binormala, racionalna krivulja, dualne koordinate, rotacijsko minimizirajoče ogrodje, pitagorejska binormala, racionalna krivulja, dualne koordinate, rotacijsko minimizirajoče ogrodje, Pythagorean-hodograph, Pythagorean-binormal, rational curve, dual coordinates, rotation-minimizing frame, osculating frame
Published in RUP: 15.10.2015; Views: 2573; Downloads: 123
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4.
Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Marjetka Knez, Vito Vitrih, 2012, original scientific article

Abstract: The paper presents an interpolation scheme for ▫$G^{1}$▫ Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler-Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controlling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler-Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.
Keywords: Pythagorean-hodograph, Euler-Rodrigues frame, rotation minimizing frame, motion design, quaternion, Hermite interpolation
Published in RUP: 15.10.2013; Views: 2739; Downloads: 98
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