  A very important question in geometric measure theory is how geometric
features of a set translate into analytic information about it. In 1960, E. R.
Reifenberg proved that if a set is well approximated by planes at every point
and at every scale, then the set is a bi-H\"older image of a plane. It is known
today that Carleson-type conditions on these approximating planes guarantee a
bi-Lipschitz parameterization of the set. In this paper, we consider an
$n$-Ahlfors regular rectifiable set $M \subset \mathbb{R}^{n+d}$ that satisfies
a Poincar\'{e}-type inequality involving the tangential derivative. Then, we
show that a Carleson-type condition on the oscillations of the tangent planes
of $M$ guarantees that $M$ is contained in a bi-Lipschitz image of an
$n$-plane. We also explore the Poincar\'e-type inequality considered here and
show that it is in fact equivalent to other Poincar\'e-type inequalities
considered on general metric measure spaces.
