  We introduce a geometric transition between two homogeneous three-dimensional
geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path
of three-dimensional hyperbolic structures that collapse down onto a hyperbolic
plane, we describe a method for constructing a natural continuation of this
path into AdS structures. In particular, when hyperbolic cone manifolds
collapse, the AdS manifolds generated on the "other side" of the transition
have tachyon singularities. The method involves the study of a new transitional
geometry called half-pipe geometry. We demonstrate these methods in the case
when the manifold is the unit tangent bundle of the (2,m,m) triangle orbifold
for m at least 5.
