This paper theoretically describes how to compose a single
Pythagorean hodograph (PH) quintic B\'ezier spiral segment, between
two circles with one circle inside the other. A spiral is free of
local curvature extrema, making spiral design an interesting
mathematical problem with importance for both physical and aesthetic
applications. The curvature of a spiral varies monotonically with
arc-length. A polynomial curve with a PH has the properties that its
arc-length is a polynomial of its parameter, and its offset is a
rational algebraic expression. A quintic is the lowest degree PH
curve that may have an inflection point and that inflection point
allows a segment of it to be joined to a straight line segment while
preserving continuity of curvature, continuity of tangent direction,
and continuity of position. A PH quintic spiral allows the design of
fair curves in a NURBS based CAD system. It is also suitable for
applications such as highway design in which the clothoid has
traditionally been used. We simplify and complete the analysis on
earlier results on PH quintic spiral segments which are proposed as
transition curve elements, and examine techniques for curve design
using the new results.