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1.
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: ključnih besedah
Summary of found: ...data up to order three in the geometric sense. Nonlinearity in the spherical part of... ...▫$G^3$▫ smoothness. The motion is determined through interpolation of positions and derivative data up to...
Keywords: motion design, geometric interpolation, rational spline motion, geometric continuity
Published: 15.10.2015; Views: 1015; Downloads: 67
URL Full text (0,00 KB)

2.
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: ključnih besedah
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)

3.
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: ključnih besedah
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)

4.
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: ključnih besedah
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)

5.
Hermite interpolation by rational G [sup] k motions of low degree
Bert Jüttler, Marjetka Krajnc, Gašper Jaklič, Emil Žagar, Vito Vitrih, 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.
Found in: ključnih besedah
Summary of found: ...design is described by using techniques of geometric interpolation. This enables us to reduce the...
Keywords: mathematics, numerical analysis, motion design, geometric interpolation, rational spline motion, geometric continuity
Published: 03.04.2017; Views: 749; Downloads: 7
URL Full text (0,00 KB)

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