Clothoid spline transition spirals

Abstract

Highway and railway designers use clothoid splines (planar <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{G^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> curves consisting of straight line segments, circular arcs, and clothoid segments) as center lines in route location. This paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures 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.