Reachable regions for spiral segments and applications in geometric modelling

We obtain a G² fair curve by adding a spiral segment to one end of an existing curve. The added segments are cubic and Pythagorean hodograph quintic spirals. We derive the more explicit reachable regions for the end points of the added spirals. Spiral segments are useful in the design of fair curves. The advantage of our technique is its more flexible control of the curvature and inflection points of a designed curve.