A piecewise defined planar curve, consisting of straight line segments, circular arcs and clothoid segments, which is twice continuously differentiable with respect to arc length is called a clothoid spline, as circles and straight lines can be considered limiting forms of clothoids [see E. Mehlum, Non-linear splines in Computer Aided Geometric Design, Eds. R. E. Barnhill and R. F. Riesenfield, Academic Press, New York, 173-207 (1974)]. Such splines are used in the route design of the centre lines of highways and railways [see K. G. Baass, Transportation Forum 1, 47-52 (1984)].
The authors study the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvature and unit tangents at the two points. Conditions are given for the existence and uniqueness of the clothoid spline transition spirals, and algorithms for finding them are outlined [cf. D. S. Meek and R. S. D. Thomas, Comput. Aided Geom. Des. 8, No. 2, 163-174 (1991)].