  Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless
manifold with dimension $m$ and unit reach. We show how to construct a function
$\varphi: \mathbb{R}^d \rightarrow \mathbb{R}^{d-m}$ from a uniform
$(\varepsilon,\kappa)$-sample $P$ of $\cal M$ that offers several guarantees.
Let $Z_\varphi$ denote the zero set of $\varphi$. Let $\widehat{{\cal M}}$
denote the set of points at distance $\varepsilon$ or less from $\cal M$. There
exists $\varepsilon_0 \in (0,1)$ that decreases as $d$ increases such that if
$\varepsilon \leq \varepsilon_0$, the following guarantees hold. First,
$Z_\varphi \cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in
the sense that $Z_\varphi \cap \widehat{\cal M}$ is homeomorphic to $\cal M$,
the Hausdorff distance between $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$
is $O(m^{5/2}\varepsilon^{2})$, and the normal spaces at nearby points in
$Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ make an angle
$O(m^2\sqrt{\kappa\varepsilon})$. Second, $\varphi$ has local support; in
particular, the value of $\varphi$ at a point is affected only by sample points
in $P$ that lie within a distance of $O(m\varepsilon)$. Third, we give a
projection operator that only uses sample points in $P$ at distance
$O(m\varepsilon)$ from the initial point. The projection operator maps any
initial point near $P$ onto $Z_\varphi \cap \widehat{\cal M}$ in the limit by
repeated applications.
