Series representations for the rectification of a superhelix

A superhelix is a curve that is helically coiled around a helix. Despite its importance in relation to the deformation modeling of various shapes, the superhelix is greatly overlooked, in part owing to its complexity and in part due to the lack of an analytical formula for its arc length. Deriving an exact analytical formula is not simple, because one needs to integrate a function without a closed-form integral solution to determine the arc length of a superhelix. In this study, we present a method by which to obtain the integral of the function that has no closed form integral by employing the series expansion approach of Maclaurin, as originally used to express the exact perimeter of an ellipse as an infinite sum. Our final expression of the arc length of a superhelix takes the form of two separate infinite sums, from which the one that converges is chosen to be applied, depending on the range of the geometric variables of the curve.