# 3. Finding maps to local models

We now suppose that we're given a model metric space $(Y,y)$. For each $x \in X$, we'd like to find the "best" ways of pulling back functions on a neighborhood of $y$ in $Y$ to neighborhood of $x$ in $X$.

As discussed in the previous section, this should be essentially the same thing as finding a "nice map" from $(V,y)\to (U,x)$, where $V$ is a neighborhood of $y$ and $U$ is a neighborhood of $x$.

Since we are working with in a metric category, we'd like these maps to be nearly "geometry preserving." It therefore seems that we should look for maps that are as close to an isometry as possible.

It also seems that bigger charts are tend to be better, since this allows us to cover $X$ more efficiently.

We can therefore look at the following function

$Fit: X\times \mathbb{R}\to \mathbb{R}^+$

that maps $(x,r)$ to the best distortion constants for a map from $B_r(x)$ into $Y$ for whatever measure of distortion is being used.

We may also want the freedom to rescale $Y$, so we can replace the above with a different measure of fit given by minimizing distortion over all maps to rescalings $d*Y$ of $Y$.