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1.
Energy minimizing mountain ascent
Gašper Jaklič, Tadej Kanduč, Selena Praprotnik, Emil Žagar, 2012, original scientific article

Abstract: In this article, an optimal mountain ascent is studied as a particular problemof a human walking over a rugged terrain. First, an approximation of the terrain is constructed using particular smooth splines-macro-elements. Then a functional measuring the energy consumption along boundary curves of a macro-element is defined. Finally, the corresponding discrete problem of finding the optimal path on a mesh of curves is applied. Numerical results on real-life data indicate that computed paths are a good approximation of hiking paths in nature.
Found in: osebi
Keywords: mountain ascent, macro-element, minimization, energy, curve on a surface
Published: 15.10.2013; Views: 1433; Downloads: 50
URL Full text (0,00 KB)

2.
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: 768; Downloads: 55
URL Full text (0,00 KB)

3.
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: 831; Downloads: 96
URL Full text (0,00 KB)

4.
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: 665; Downloads: 96
URL Full text (0,00 KB)

5.
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: 731; Downloads: 94
URL Full text (0,00 KB)

6.
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: 778; Downloads: 101
URL Full text (0,00 KB)

7.
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: 715; Downloads: 42
URL Full text (0,00 KB)

8.
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: 675; Downloads: 9
URL Full text (0,00 KB)

9.
Lagrange geometric interpolation by rational spatial cubic Bézier curves
Emil Žagar, Jernej Kozak, Gašper Jaklič, Vito Vitrih, 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.
Found in: osebi
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: 03.04.2017; Views: 781; Downloads: 55
URL Full text (0,00 KB)

10.
High order parametric polynomial approximation of conic sections
Gašper Jaklič, Vito Vitrih, Emil Žagar, Jernej Kozak, Marjetka Krajnc, 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.
Found in: osebi
Keywords: matematika, stožnica, parametrična krivulja, implicitna krivulja, aproksimacija, mathematics, conic section, parametric curve, implicit curve, approximation
Published: 03.04.2017; Views: 635; Downloads: 56
URL Full text (0,00 KB)

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