TABLE OF CONTENTS In this section, we consider the case in which 
is a set of k pairwise-disjoint simple polygons with a total of n vertices. We describe a method for maintaining a pseudo-triangulation of 
by combining our algorithm for convex polygons with the algorithm by Basch et al. [2] for two simple polygons.
For each 
, define the relative geodesic cycle C( P) of P to be the shortest cycle in 
with the same homotopy type as ?P. The region enclosed by C( P) is called the relative convex hull and is denoted by P. We decompose 
into two parts 
and 
. We compute pseudo-triangulations 
and 
separately. These two triangulations together give the mixed pseudo-triangulation of 
.
First, we consider 
. Since the interiors of all the relative convex hulls are pairwise disjoint, we can compute a pseudo-triangulation of each of them separately. We triangulate P each as in [2]. Following [2], we define the pinned relative geodesic cycle of P, with respect to a pinning point set B (a subset of the vertices of P), to be the shortest cycle in 
that passes through the vertices in B and has the same homotopy type as ?P. The pinned relative convex hull of P with respect to B is the region enclosed...
