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1.
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: 2251; Downloads: 131
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2.
An approach to geometric interpolation by Pythagorean-hodograph curves
Gašper 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: 2312; Downloads: 71
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3.
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: 2504; Downloads: 86
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