G2 Pythagorean hodograph quintic transition between two circles with shape control

In the highway and rail route designs, or a car-like robot path planning it is often desirable to have a method of joining a circle to a circle with an S-shaped or a broken back C-shaped spiral transition. This paper describes a transition between two such circles. It is shown that a single Pythagorean hodograph quintic curve can be used for blending or for a transition curve preserving G2 continuity with local shape control parameter and flexible constraints. Provision of a shape control parameter provides freedom to modify the shape in a stable manner.