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1.
Three-pencil lattice on triangulations
Gašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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.
Keywords: numerical analysis, lattice, barycentric coordinates, triangulations, interpolation
Published in RUP: 03.04.2017; Views: 2123; Downloads: 84
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2.
Barycentric coordinates for Lagrange interpolation over lattices on a simplex
Gašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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.
Keywords: numerical analysis, lattice, barycentric coordinates, simplex, interpolation
Published in RUP: 03.04.2017; Views: 2234; Downloads: 139
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3.
On geometric Lagrange interpolation by quadratic parametric patches
Gašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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.
Keywords: numerična analiza, interpolacija, aproksimacija, parametrična ploskev, numerical analysis, interpolation, approximation, parametric surface
Published in RUP: 03.04.2017; Views: 2253; Downloads: 138
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4.
Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curves
Gašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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.
Keywords: numerical analysis, planar curve, PH curve, geometric interpolation, Lagrange interpolation
Published in RUP: 03.04.2017; Views: 2155; Downloads: 131
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5.
Lattices on simplicial partitions
Gašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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.
Keywords: numerical analysis, lattice, barycentric coordinates, simplicial partition
Published in RUP: 03.04.2017; Views: 2142; Downloads: 135
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6.
Lagrange geometric interpolation by rational spatial cubic Bézier curves
Gašper Jaklič, Jernej Kozak, Vito Vitrih, Emil Žagar, 2012, original scientific article

Abstract: V članku obravnavamo Lagrangeovo geometrijsko interpolacijo s prostorskimi racionalnimi kubičnimi Bézierovimi krivuljami. Pokažemo, da pod določenimi naravnimi omejitvami obstaja enolična rešitev problema. še več, rešitev je podana v preprosti zaključeni obliki in je zato zanimiva za praktične aplikacije. Asimptotična analiza potrdi pričakovani red aproksimacije, namreč 6. Numerični primeri nakažejo možnost uporabe te metode pri obetavni geometrijski nelinearni subdivizijski shemi.
Keywords: numerična analiza, geometrijska Lagrageova interpolacija, racionalna Bézierova krivulja, prostorska krivulja, asimptotična analiza, subdivizija, numerical analysis, geometric Lagrange interpolation, rational Bézier curve, spatial curve, asymptotic analysis, subdivision
Published in RUP: 03.04.2017; Views: 2416; Downloads: 86
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7.
Hermite interpolation by rational G [sup] k motions of low degree
Gašper Jaklič, Bert Jüttler, Marjetka Knez, Vito Vitrih, Emil Žagar, 2013, original scientific article

Abstract: 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.
Keywords: mathematics, numerical analysis, motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 03.04.2017; Views: 2155; Downloads: 39
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