11. C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcsBohumír Bastl, Michal Bizzarri, Marjetka Knez, Miroslav Lávička, Kristýna Michálkova, Zbiněk Šír, Vito Vitrih, Emil Žagar, 2014, izvirni znanstveni članek Opis: 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.
Ključne besede: mathematics, parametric curve, PH curve, Pythagorean-hodograph, Hermite interpolation, biarc, cubic curve
Objavljeno v RUP: 03.04.2017; Ogledov: 2133; Prenosov: 40
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12. Hermite interpolation by rational G [sup] k motions of low degreeGašper Jaklič, Bert Jüttler, Marjetka Knez, Vito Vitrih, Emil Žagar, 2013, izvirni znanstveni članek Opis: Interpolation by rational spline motions is an important issue in robotics and related fields. In this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation. This enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion. A general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed, i.e., geometric continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six. In both cases sufficient conditions on the given Hermite data are found which guarantee the uniqueness of the solution. If the given data do not fulfill the solvability conditions, a method to perturb them slightly is described. Numerical examples are presented which confirm the theoretical results and provide an evidence that the obtained motions have nice shapes.
Ključne besede: mathematics, numerical analysis, motion design, geometric interpolation, rational spline motion, geometric continuity
Objavljeno v RUP: 03.04.2017; Ogledov: 2206; Prenosov: 39
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13. Pythagorean-hodograph cycloidal curvesJernej Kozak, Marjetka Knez, Mladen Rogina, Vito Vitrih, 2015, izvirni znanstveni članek Opis: 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.
Ključne besede: pitagorejski hodograf, C-krivulje, trigonometrične funkcije, karakterizacija, Hermiteova interpolacija, asimptotični red aproksimacije, Pythagorean-hodograph, C-curves, trigonometric functions, characterization, Hermite interpolation, asymptotic approximation order
Objavljeno v RUP: 08.08.2016; Ogledov: 2754; Prenosov: 198
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14. Parametric curves with Pythagorean binormalsJernej Kozak, Marjetka Knez, Vito Vitrih, 2015, izvirni znanstveni članek Opis: 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.
Ključne besede: 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
Objavljeno v RUP: 15.10.2015; Ogledov: 2696; Prenosov: 124
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15. Construction of G[sup]3 rational motion of degree eightKarla Ferjančič, Marjetka Knez, Vito Vitrih, 2015, izvirni znanstveni članek Opis: The paper presents a construction of a rigid body motion with point trajectories being rational spline curves of degree eight joining together with ▫$G^3$▫ smoothness. The motion is determined through interpolation of positions and derivative data up to order three in the geometric sense. Nonlinearity in the spherical part of construction results in a single univariate quartic equation which yields solutions in a closed form. Sufficient conditions on the regions for the curvature data are derived, implying the existence of a real admissible solution. The algorithm how to choose appropriate data is proposed too. The theoretical results are substantiated with numerical examples.
Ključne besede: motion design, geometric interpolation, rational spline motion, geometric continuity
Objavljeno v RUP: 15.10.2015; Ogledov: 2888; Prenosov: 124
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16. Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curvesMarjetka Knez, Vito Vitrih, 2012, izvirni znanstveni članek Opis: 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.
Ključne besede: Pythagorean-hodograph, Euler-Rodrigues frame, rotation minimizing frame, motion design, quaternion, Hermite interpolation
Objavljeno v RUP: 15.10.2013; Ogledov: 2872; Prenosov: 98
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