  Let $\mathcal T$ be the Teichm\"{u}ller space of marked genus $g$, $n$
punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented
Teichm\"{u}ller space} of marked Riemann surfaces with nodes, \cite{Abdegn,
Bersdeg}. Provided with the WP metric $\Tbar$ is a complete CAT(0) metric
space, \cite{DW2, Wlcomp, Yam2}. An invariant of a marked hyperbolic structure
is the length $\ell_{\alpha}$ of the geodesic $\alpha$ in a free homotopy
class. A basic feature of Teichm\"{u}ller theory is the interplay of
two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the
behavior of geodesic-length functions. Our goal is to develop the understanding
of the intrinsic local WP geometry through a study of the gradient and Hessian
of geodesic-length functions. Considerations include expansions for the WP
pairing of gradients, expansions for the Hessian and covariant derivative,
comparability models for the WP metric, as well as the behavior of WP geodesics
including a description of the Alexandrov tangent cone at the augmentation.
Approximations and applications for geodesics close to the augmentation are
developed. An application for fixed points of group actions is described.
Bounding configurations and functions on the hyperbolic plane is basic to our
approach. Considerations include analyzing the orbit of a discrete group of
isometries and bounding sums of the inverse square exponential-distance.
