Construction of a spiral with given boundary conditions by inversion of the involute of a circle

To construct a curve with a monotonic curvature (spiral), and given tangents and curvatures at the ends, the author proposed the following method. From given boundary conditions, the values of two inverse invariants are determined. Then, on some base spiral (initially, a logarithmic spiral was chosen), an arc with the same invariant values is sought for. A linear-fractional map of the found arc solves the problem. It seems that choosing the involute of a circle as the base spiral yields the simplest solution, which we present here.