20.500.12556/RUL-187631 Uporaba Bézierovih krivulj pri načrtovanju železniške proge Use of Bézier curves in railway alignment design V delu obravnavamo uporabo polinomskih in racionalnih Bézierovih krivulj pri opisu horizontalne geometrije železniških prog. Predstavimo njihove osnovne lastnosti, odvode in ukrivljenost ter pogoje parametrične in geometrijske zveznosti. Pokažemo, da lahko ravni odsek opišemo s polinomsko Bézierovo krivuljo, krožni lok pa eksaktno z racionalno kvadratno Bézierovo krivuljo. Osrednji del naloge je namenjen konstrukciji kvintične Bézierove prehodnice med premo in krožnim lokom. Geometrijske zahteve za začetni in končni položaj, tangentno smer ter ukrivljenost prevedemo v pogoje za Bézierove kontrolne točke. Proste oblikovne parametre določimo z numerično minimizacijo odstopanja ukrivljenosti Bézierove krivulje od ukrivljenosti klotoide. Kakovost dobljene aproksimacije ocenimo z geometrijsko primerjavo ter primerjavo ukrivljenosti na konkretnem primeru. Rezultati kažejo, da lahko primerno skonstruirana in optimizirana kvintična Bézierova krivulja zelo natančno aproksimira klotoido tako po geometrijski obliki kot tudi po poteku ukrivljenosti. Pristop razširimo tudi na prehod med dvema krožnima lokoma z različnima polmeroma, vključno s primerom S-krivine. This thesis considers the use of polynomial and rational Bézier curves in the description of horizontal railway alignment geometry. Their basic properties, derivatives and curvature are presented, together with the conditions of parametric and geometric continuity. It is shown that a straight segment can be represented by a polynomial Bézier curve, while a circular arc can be represented exactly by a rational quadratic Bézier curve. The main part of the thesis is devoted to the construction of a quintic Bézier transition curve between a straight segment and a circular arc. The geometric requirements for the initial and final positions, tangent directions and curvatures are translated into conditions on the Bézier control points. The remaining shape parameters are determined by numerically minimizing the curvature deviation of the Bézier curve from the corresponding clothoid. The quality of the resulting approximation is evaluated through a geometric comparison and an analysis of the curvature profiles. The results show that a suitably constructed and optimized quintic Bézier curve can approximate a clothoid very accurately, both in its geometric shape and in its curvature behaviour. The approach is also extended to a transition between two circular arcs of different radii, including the case of a reverse curve. Bézierove krivulje racionalne Bézierove krivulje ukrivljenost železniška proga prehodnica klotoida geometrijska zveznost numerična optimizacija Bézier curves rational Bézier curves curvature railway alignment transition curve clothoid geometric continuity numerical optimization true false false Slovenski jezik Angleški jezik Diplomsko delo/naloga 2026-09-12 08:15:42 2026-09-12 08:15:45 2026-09-14 03:56:24 0000-00-00 00:00:00 2026 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 1970-01-01 163683 24128.pdf 24128.pdf 1 F7375E4E24457A9AEC49B7FC6344D0DC f6ad414f53be0532fc55fe0e6ab1971a375f811b9526a621d65d7c4dff5f6f2b 5c850d0e-ae71-11f1-8bc5-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=247331 Fakulteta za matematiko in fiziko 0 0 0