A guided clothoid spline

A method is described whereby a G* planar curve consisting of clothoid (Comu spiral) segments, circular arcs and straight line segments is constructed. The curve passes through given points, is expressed parametrically. and has the attractive feature that the arc length and curvature are piecewise linear functions of the parameter. The given points through which the curve passes must be restricted somewhat so that a unique clothoid e.tists between each pair of points. The clothoid is a spiral defined parametrically in terms of Fresnel integrals as where the scaling factor aB is positive, the parameter t is nonnegative, and the Fresnel integrals are [Abramowitz and Stegun '65, p. 3001 C(t) = k'cos $rr~' du, and S(t) = /'sin &ru2 du. 0 A recent paper [Heald 'X5] gives convenient rational approximations to the Fresnel integrals. The clothoid (1) is in the first quadrant, starts at the origin with f = 0, and approaches the limiting point (fvB, fnB) as I approaches infinity (see Fig. 1). Some useful formulae for the clothoid are that the angle between the tangent vector and the X-axis is $nt', the curvature is t/B, and an element of arc length is ds = TB dt. Below, t is restricted to a range which keeps the angle between the tangent vectors at the endpoints to less than 'II in each clothoid segment. In this paper, a method is described whereby a G2 planar curve consisting of clothoid (also called Comu spirals, [Comu '741) segments, circular arcs, and straight line segments is constructed. Such a curve will be called a clothoid spline as circles and straight lines can be considered limiting forms of clothoids. The clothoid spline is used in the route design of *