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In this announcement we consider the following problem. Let $n,m\geq 1$,$U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to thefollowing Whitney distortion extension problems: (a) Let $\phi:U\to \mathbbR^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and therestriction of $\phi$ to $E$ is an almost isometry with small distortion, howto decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almostisometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agreeswith $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to\mathbb R^n$ away from $E$. (b) Let $\phi:U\to \mathbb R^n$ be $C^{\infty}$map. If $E\subset U$ is compact (with some geometry) and the restriction of$\phi$ to $E$ is an almost isometry with small distortion, how to decide whenthere exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry$\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with$\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to\mathbb R^n$ away from $E$. Our results complement those of [14,15,20] wherethere, $E$ is a finite set. In this case, the problem above is also a problemof interpolation and alignment of data in $\mathbb R^n$.

On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and  $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment  of data in $\mathbb R^n$