Motivation

Common to the design of railway or road building (especially for highway exits), as well as those crazy loops in many roller coasters, is the solution of differential equations in two or three dimensions that address the effect of curvature and centripetal acceleration on moving bodies. In the 1970s, Werner Stengel studied and applied several models to attack this problem and, among the many solutions he found, one struck as particularly brilliant—the employment of clothoid loops (based on sections of Cornu's spiral). The first looping coaster designed with this paradigm was constructed in 1976 in the Baja Ridge area of Six Flags Magic Mountain, in Valencia, California, USA. It was coined the Great American Revolution, and it featured the very first vertical loop (together with two corkscrews, for a total of three inversions).

The tricky part of the design was based on a system of differential equations, whose solution depended on the integration of Fresnel-type sine...