An involute spiral that matches G2 Hermite data in the plane

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 the 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 constructions are explicit apart from solving two linear equations in the first case, and at most one quadratic equation in the second case.