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.
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