31. Parametric curves with Pythagorean binormalsJernej 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: 2656; Downloads: 124
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32. Construction of G[sup]3 rational motion of degree eightKarla Ferjančič, Marjetka Knez, Vito Vitrih, 2015, original scientific article Abstract: 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.
Keywords: motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 15.10.2015; Views: 2836; Downloads: 124
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37. Lattices on tetrahedral partitionsVito Vitrih, 2008, published scientific conference contribution Abstract: In this paper, four-pencil lattices on tetrahedral partitions are studied. Theexplicit representation of a lattice, based upon barycentric coordinates, enables us to extend the lattice from a single tetrahedron to a tetrahedral partition. It is shown that the number of degrees of freedom is equal to the number of vertices of the tetrahedral partition. The proof is based on a lattice split approach.
Keywords: mreža, tetraeder, interpolacija
Published in RUP: 15.10.2013; Views: 3296; Downloads: 134
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40. Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curvesMarjetka 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: 2825; Downloads: 98
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