Revisit of the railway transition curve, this time generated directly from smoothness requirements.

The task is to design the transition curve for a railway track changing from curvature kappa_0 to a curvature kappa_0 +Delta kappa with limited rate of curvature change and limited rate of transitioning to that rate of change (which means limiting kappa''. Given these limits, the curve has to be as short as possible. If we hadn't the limitation of kappa'', the solution would be a clothoide (curve of linear kappa(s)).
Background: In curves, the outer rail is superelevated in order to align the roll angle with the resultant of centrifugal force and weight for minimum transverse forces. For a given velocity of the train, kappa'_max limits the angular velocity and kappa''_max limits the angular acceleration of this adjustment
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We first generate the curvature as function of position (arc length) and then integrate the the curve using the Runge-Kutta method.

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