  We unveil an alluring alternative to parametric search that applies to both
the non-geodesic and geodesic Fr\'echet optimization problems. This randomized
approach is based on a variant of red-blue intersections and is appealing due
to its elegance and practical efficiency when compared to parametric search. We
present the first algorithm for the geodesic Fr\'echet distance between two
polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic
Fr\'echet decision problem is solved almost as fast as its non-geodesic sibling
and requires $O(N^{2\log k)$ time and $O(k+N)$ space after $O(k)$
preprocessing, where $N$ is the larger of the complexities of $A$ and $B$ and
$k$ is the complexity of $P$. The geodesic Fr\'echet optimization problem is
solved by a randomized approach in $O(k+N^{2\log kN\log N)$ expected time and
$O(k+N^{2)$ space. This runtime is only a logarithmic factor larger than the
standard non-geodesic Fr\'echet algorithm (Alt and Godau 1995). Results are
also presented for the geodesic Fr\'echet distance in a polygonal domain with
obstacles and the geodesic Hausdorff distance for sets of points or sets of
line segments inside a simple polygon $P$.
