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(2) 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. (C) 2009 Elsevier B.V. All rights reserved.