The Highway Spiral for Combining Curves of Different Radii

The highway spiral connecting a tangent and a circular arc is rigorously defined mathematically. When two adjacent circular arcs must be connected by a spiral, mathematical exactness is replaced by the approximations of the theory of the osculating circle. The theory is satisfactory for flat spirals but has distinct failings for sharper spirals. In this paper rigorous equations for the highway spiral connecting circular arcs of different radii are derived, and the extent of the approximations involved in the theory of the osculating circle are shown. Three graphs based on these equations permit rapid and accurate determination of the corrections to be applied to values obtained from the theory. The engineer with a working knowlege of the theory should find that the determination and application of these corrections are straightforward and simple. The derivations themselves are cumbersome and extensive. They are given in outline form but in sufficient detail to permit checking.