  We construct new coordinates for the Teichm\"uller space Teich of a punctured
torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the
representation of Teich as a space of marked Kleinian groups $G_\mu$ that
depend holomorphically on a parameter $\mu$ varying in a simply connected
domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold
$\bold{H}^3/G_\mu$; they reflect exactly the visual patterns one sees in the
limit sets of the groups $G_\mu$; and they are directly computable from the
generators of $G_\mu$.

Geometric Topology;Complex Variables