Circle to Circle Transition with a Single Ph Quintic Spiral

This paper theoretically describes how to compose a single Pythagorean hodograph (PH) quintic Bézier 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.