Shapes of coil ends in racetrack layout for superconducting magnets

Racetrack coils have received considerable attention for Nb$_{3}$Sn magnets, both built using the React-and-Wind and Wind-and-React techniques. The geometry usually consists of a series of straight parts connected with circular arcs. Therefore, at the interface between these sections, a finite change in curvature is imposed on the cable. Alternative transition curves are analyzed here, with a particular focus on the total strain energy and the minimum / maximum radii of curvature. The study is presented in dimensionless form and the various alternatives are detailed in mathematical terms, so to be used for drafting or simulations. Extensions for the design of flared ends are also briefly discussed.