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1.
Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Marjetka Krajnc, 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.
Found in: osebi
Keywords: Pythagorean-hodograph, Euler-Rodrigues frame, rotation minimizing frame, motion design, quaternion, Hermite interpolation
Published: 15.10.2013; Views: 1147; Downloads: 55
URL Full text (0,00 KB)

2.
Construction of G[sup]3 rational motion of degree eight
Marjetka Krajnc, Vito Vitrih, Karla Ferjančič, 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.
Found in: osebi
Keywords: motion design, geometric interpolation, rational spline motion, geometric continuity
Published: 15.10.2015; Views: 940; Downloads: 61
URL Full text (0,00 KB)

3.
Parametric curves with Pythagorean binormals
Marjetka Krajnc, Vito Vitrih, Jernej Kozak, 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.
Found in: osebi
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: 15.10.2015; Views: 1126; Downloads: 60
URL Full text (0,00 KB)

4.
Three-pencil lattice on triangulations
Vito Vitrih, Gašper Jaklič, Emil Žagar, Marjetka Krajnc, Jernej Kozak, 2007, published scientific conference contribution

Abstract: In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S 3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.
Found in: osebi
Keywords: numerical analysis, lattice, barycentric coordinates, triangulations, interpolation
Published: 03.04.2017; Views: 690; Downloads: 51
URL Full text (0,00 KB)

5.
Barycentric coordinates for Lagrange interpolation over lattices on a simplex
Gašper Jaklič, Marjetka Krajnc, Jernej Kozak, Emil Žagar, Vito Vitrih, 2008, published scientific conference contribution

Abstract: In this paper, a ▫$(d+1)$▫-pencil lattice on a simplex in ▫${\mathbb{R}}^d$▫ is studied. The lattice points are explicitly given in barycentric coordinates. This enables the construction and the efficient evaluation of the Lagrange interpolating polynomial over a lattice on a simplex. Also, the barycentric representation, based on shape parameters, turns out to be appropriate for the lattice extension from a simplex to a simplicial partition.
Found in: osebi
Keywords: numerical analysis, lattice, barycentric coordinates, simplex, interpolation
Published: 03.04.2017; Views: 757; Downloads: 87
URL Full text (0,00 KB)

6.
On geometric Lagrange interpolation by quadratic parametric patches
Emil Žagar, Marjetka Krajnc, Gašper Jaklič, Vito Vitrih, Jernej Kozak, 2008, original scientific article

Abstract: In the paper, the geometric Lagrange interpolation by quadratic parametric patches is considered. The freedom of parameterization is used to raise the number of interpolated points from the usual 6 up to 10, i.e., the number of points commonly interpolated by a cubic patch. At least asymptotically, the existence of a quadratic geometric interpolant is confirmed for data taken on a parametric surface with locally nonzero Gaussian curvature and interpolation points based upon a three-pencil lattice. Also, the asymptotic approximation order 4 is established.
Found in: osebi
Keywords: numerična analiza, interpolacija, aproksimacija, parametrična ploskev, numerical analysis, interpolation, approximation, parametric surface
Published: 03.04.2017; Views: 611; Downloads: 88
URL Full text (0,00 KB)

7.
Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curves
Emil Žagar, Vito Vitrih, Marjetka Krajnc, Gašper Jaklič, Jernej Kozak, 2008, original scientific article

Abstract: In this paper, the geometric Lagrange interpolation of four points by planar cubic Pythagorean-hodograph (PH) curves is studied. It is shown that such an interpolatory curve exists provided that the data polygon, formed by the interpolation points, is convex, and satisfies an additional restriction on its angles. The approximation order is $4$. This gives rise to a conjecture that a PH curve of degree ▫$n$▫ can, under some natural restrictions on data points, interpolate up to ▫$n+1$▫ points.
Found in: osebi
Keywords: numerical analysis, planar curve, PH curve, geometric interpolation, Lagrange interpolation
Published: 03.04.2017; Views: 661; Downloads: 86
URL Full text (0,00 KB)

8.
Lattices on simplicial partitions
Emil Žagar, Marjetka Krajnc, Vito Vitrih, Jernej Kozak, Gašper Jaklič, 2010, published scientific conference contribution

Abstract: In this paper, a ▫$(d+1)$▫-pencil lattices on a simplex in ▫${\mathbb{R}}^d$▫ are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of thisfact leads to an efficient computer algorithm for the design of a lattice.
Found in: osebi
Keywords: numerical analysis, lattice, barycentric coordinates, simplicial partition
Published: 03.04.2017; Views: 710; Downloads: 93
URL Full text (0,00 KB)

9.
An approach to geometric interpolation by Pythagorean-hodograph curves
Vito Vitrih, Marjetka Krajnc, Emil Žagar, Jernej Kozak, Gašper Jaklič, 2012, original scientific article

Abstract: The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree ▫$n$▫ is studied independently of the dimension ▫$d \ge 2$▫. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.
Found in: osebi
Keywords: mathematics, parametric curve, PH curve, geometric interpolation, Lagrange interpolation, Hermite interpolation, cubic curves, homotopy
Published: 03.04.2017; Views: 651; Downloads: 39
URL Full text (0,00 KB)

10.
High order parametric polynomial approximation of quadrics in R [sup] d
Gašper Jaklič, Vito Vitrih, Emil Žagar, Jernej Kozak, Marjetka Krajnc, 2012, original scientific article

Abstract: In this paper an approximation of implicitly defined quadrics in ▫${\mathbb R}^d$▫ by parametric polynomial hypersurfaces is considered. The construction of the approximants provides the polynomial hypersurface in a closed form, and it is based on the minimization of the error term arising from the implicit equation of a quadric. It is shown that this approach also minimizes the normal distance between the quadric and the polynomial hypersurface. Furthermore, the asymptotic analysis confirms that the distance decreases at least exponentially as the polynomial degree grows. Numerical experiments for spatial quadrics illustrate the obtained theoretical results.
Found in: osebi
Keywords: mathematics, quadric hypersurface, conic section, polynomial approximation, approximation order, normal distance
Published: 03.04.2017; Views: 626; Downloads: 9
URL Full text (0,00 KB)

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