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\title{Geodesics in the space of Pythagorean--Hodograph curves}
\author{Juan Monterde\\
Dpt. de Geometria i Topologia, Universitat de Val\`encia,\\
Avd. Vicent Andr\'es Estell\'es, 1, E-46100-Burjassot (Val\`encia), Spain\\
monterde{@}uv.es}

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It is well known that the map $z\to \int z^2$ is a key point in the study of planar PH curves. Even more, when natural metrics are introduced, this map is an isometry as it was previously  noticed in a different research area: the study of the planar shape space (\cite{mich}) where  $z\to \int z^2$ is called ``the basic mapping''. We will show a kind of uniqueness of this basic mapping under natural conditions. In the $3$-dimensional case it is well known that the Hopf map plays now the role of the map $z\to z^2$. We will show that, again, the isometry condition is preserved. Both isometries allow to compute the distance between two PH curves or the geodesic joining them  in an explicit way.


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\begin{thebibliography}{99}
\bibitem{mich} Younes, L., Michor, P. W., Shah, J., Mumford, D., {\it A metric on shape space with explicit geodesics},
preprint,  Rend. Lincei  Mat. Appl.  {\bf 9} (2008) 25--57.
arXiv:0706.4299.
\end{thebibliography}
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