A polynomial Pythagorean-hodograph (PH) curve has the property that its parametric speed — i.e., the

derivative of the arc length with respect to the curve parameter — is a polynomial rather than the square

root of a polynomial. Many computational advantages derive from this property and are useful in offsets, path

planning, geometric design and similar applications.

In this talk, a geometric characterization for planar polynomial PH curves is presented. It is based on a

variant of the dual representation of planar curves, where a curve may be regarded as the envelope of its tangent

lines. The approach used here is illustrated with many examples.

A comparison is made with the state-of-art method : three–stage procedure that transforms any differentiable

plane curve r(t) into a PH curve R(t) through the use of the conformal map z → z^2 . In this framework, the

Pythagorean structure of the hodograph R'(t) is achieved through the complex variable model. The a priori

implementation is done through an algebraic model.

In the technique presented here, the Pythagorean property of the hodograph is achieved by a suitable

geometric model. Notorious results for cubic PH curves and quintic PH curves are generalized. This geometric

characterization provides an alternative three–stage procedure of generating plane polynomial PH curves. This

work contributes to a different explanation of the theory and the applied algorithms for planar PH curves. It

can be developed in various other related topics.