A Pythagorean hodograph quintic spiral

A polynomial curve with a Pythagorean hodograph has the properties that its are-length is a polynomial of its parameter, and its offset is a rational algebraic expression. A quintic is the lowest degree Pythagorean hodograph 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 position, and continuity of tangential direction. The curvature of a spiral varies monotonically with are-length. Spiral segments are useful in the design of fair curves. A Pythagorean hodograph quintic spiral is presented which 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. Copyright (C) 1996 Elsevier Science Ltd.