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.