On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $\Bbb R^D$

In this paper, we study the following problem. Let $D\geq 2$, $S\subset \mathbb R^D$ be finite and let $\phi:S\to \mathbb R^D$ with $\phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $\phi$ can be extended to a function $\Phi:\mathbb R^D\to \mathbb R^D$ which is a smooth small distortion on $\mathbb R^D$. The work in this paper appears in the research memoir [14]