A prior method for designing transition curves for railroad tracks and
other vehicle guideways begin with a choice of a roll function
representing a functional form for variation of the track or guideway
roll or cant angle as a function of distance and requires the curvature
of the transition shape to keep the components of centripetal and
gravitational acceleration in the plane of the track or guideway equal at
each point along the shape and integrates the equation expressing that
equality as part of a procedure for determining the resulting transition
curve shape. That method is supplemented by a method of defining basic
roll functions in terms of Gegenbauer orthogonal polynomials, including
roll functions which generate simple spirals as well as more complex
shapes (referred to as bends, jogs, and wiggles). Roll functions for the
various shapes are defined as weighted sums of the basic roll functions,
and can generate transition curve shapes that have good dynamic
characteristics and that are more general than the shapes that can be
constructed using the prior method. A resulting generalized spiral can be
used to compensate for inadequate offset when a spiral needs to be
lengthened for operation at higher speed or to realign an existing spiral
whose shape has become so different from its original design shape that
restoration to that shape would be impractical.