Mathematical Examination of the Highway Transition Spiral

The growing importance of superhighways demands that increased attention be given to the inherent problems of construction and design. One of the most important of these problems is that of horizontal curvature and the attendant superelevation. A rational solution of this problem is the adaptation of a transition spiral, whose curvature varies inversely as the arc length. From considerations of speed, safety, and comfort it is found that two cases arise—that in which a central piece of circular arc is required, and that in which the curve is transitional throughout. The second case is desirable on grounds of relative simplicity of calculations, formulas, and field work. The question then naturally arises: “When, under certain given limiting conditions, can a curve be transitional throughout?” An answer to this question (and to other office and field problems related to the spiral) necessitates an examination of the mathematics of the transition spiral. Most existing mathematical treatments seem to suffer shortcomings due to lack of proof, or to failure to consider comfort and safety, or to loss of generality, or to confusion between exact and approximate relations, or to failure to be concise. Therefore, it is felt that an exposition of the curve and its properties obviating these faults and conducted primarily as a basis for examining questions, such as the foregoing, is eminently desirable. Such an attempt is offered in this paper.