1. Three-pencil lattice on triangulationsGašper 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: 2233; Downloads: 84 Link to full text |
2. Barycentric coordinates for Lagrange interpolation over lattices on a simplexGašper 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: 2344; Downloads: 139 Link to full text |
3. On geometric Lagrange interpolation by quadratic parametric patchesGašper 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: 2349; Downloads: 138 Link to full text |
4. Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curvesGašper 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: 2275; Downloads: 131 Link to full text |
5. Lattices on simplicial partitionsGašper 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: 2246; Downloads: 135 Link to full text |
6. An approach to geometric interpolation by Pythagorean-hodograph curvesGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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. Keywords: mathematics, parametric curve, PH curve, geometric interpolation, Lagrange interpolation, Hermite interpolation, cubic curves, homotopy Published in RUP: 03.04.2017; Views: 2327; Downloads: 71 Link to full text |
7. High order parametric polynomial approximation of quadrics in R [sup] dGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 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. Keywords: mathematics, quadric hypersurface, conic section, polynomial approximation, approximation order, normal distance Published in RUP: 03.04.2017; Views: 2230; Downloads: 32 Link to full text |
8. Lagrange geometric interpolation by rational spatial cubic Bézier curvesGašper 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: 2532; Downloads: 86 Link to full text |
9. High order parametric polynomial approximation of conic sectionsGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 2013, original scientific article Abstract: V članku je obravnavana parametrična polinomska aproksimacija stožnic, ki ohranja obliko. Pristop je osnovan na parametrični aproksimaciji implicitno definiranih ravninskih krivulj. Polinomski aproksimanti so zapisani v zaključeni obliki in ponujajo najvišji možen red aproksimacije. Keywords: matematika, stožnica, parametrična krivulja, implicitna krivulja, aproksimacija, mathematics, conic section, parametric curve, implicit curve, approximation Published in RUP: 03.04.2017; Views: 2179; Downloads: 82 Link to full text |
10. 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, 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: 2225; Downloads: 40 Link to full text |