  Escape from a potential well through an index-1 saddle can be widely found in
some important physical systems. Knowing the criteria and phase space geometry
that govern escape events plays an important role in making use of such
phenomenon, particularly when realistic frictional or dissipative forces are
present. We aim to extend the study the escape dynamics around the saddle from
two degrees of freedom to three degrees of freedom, presenting both a
methodology and phase space structures. Both the ideal conservative system and
a perturbed, dissipative system are considered. We define the five-dimensional
transition region, $\mathcal{T}_h$, as the set of initial conditions of a given
initial energy $h$ for which the trajectories will escape from one side of the
saddle to another. Invariant manifold arguments demonstrate that in the
six-dimensional phase space, the boundary of the transition region, $\partial
\mathcal{T}_h$, is topologically a four-dimensional hyper-cylinder in the
conservative system, and a four-dimensional hyper-sphere in the dissipative
system. The transition region $\mathcal{T}_h$ can be constructed by a solid
three-dimensional ellipsoid (solid three-dimensional cylinder) in the
three-dimensional configuration space, where at each point, there is a cone of
velocity -- the velocity directions leading to transition are given by cones,
with velocity magnitude given by the initial energy and the direction by two
spherical angles with given limits. To illustrate our analysis, we consider an
example system which has two potential minima connected by an index 1 saddle.
