Abstract
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.
