A construction is given for a planar rational Pythagorean hodograph spiral, which interpolates any two-point G Hermite data that a spiral can match. When the curvature at one of the points is zero, the construction gives the unique interpolant that is an involute of a rational Pythagorean hodograph curve of the form cubic over linear. Otherwise, the spiral comprises an involute of a Tschirnhausen cubic together with at most two circular arcs. The construction is by explicit formulas in the first case, and requires the solution of a quadratic equation in the second case.
Goodman, T. N. T., Meek, D. S., & Walton, D. J. (2009). An involute spiral that matches G2 Hermite data in the plane. Computer Aided Geometric Design, 26(7), 733-756. https://doi.org/10.1016/j.cagd.2009.03.009