G2 curve design with a pair of Pythagorean Hodograph quintic spiral segments

When designing curves, it is often desirable to join two points, at which G2 Hermite data are given, by a low degree parametric polynomial curve which has no extraneous curvature extrema. Such curves are referred to as being fair. The join can be accomplished by constructing the curve from a pair of polynomial spiral segments. The purpose may be practical, e.g., in highway design, or aesthetic, e.g., in the computer aided design of consumer products. A Pythagorean hodograph curve is polynomial and has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. A technique for composing a fair curve from a pair of Pythagorean hodograph quintic spiral segments is examined and presented.