The Differential Geometry of Twisted Coil Windings

The techniques of differential geometry have been applied to the problem of predicting the shape of thick twisted coil windings as successive turns and layers of turns are applied to a winding form. The explicit expressions for the required Christoffel symbols for parallel surfaces are derived in terms of the starting surface parameterization. Expressions for geodesic windings on a particular surface, called the rectifying developable, and the family of surfaces parallel to it are derived. The advantages of the rectifying developable from the point of view of coil fabrication are discussed.