  The main ob jective of this research is to find the different types of
elliptic triangulations for planar discs and spheres. We begin in Chapter 1
with the mandatory introduction. In the second chapter we define and study the
notion of a patch, that is, a triangulation of a planar disc. By introducing a
suitable notion of degree for the vertex points, we focus on those patches with
points of degrees less tah or equal to 6. Such patches are called el liptic. We
show that the elliptic patches with precisely three points of degree four,
denoted by (0, 3, 0), can be classified. The number of vertex points of these
patches are calculated, and we also describe their triangulation structures.
  From the classification of patches of the type (0, 3, 0), we describe and
find the number of vertex points for three other elliptic patches of types (0,
2, 2), (0, 1, 4), (0, 0, 6). We also describe an enlargement method for
constructing patches (which we call the generic construction method) and apply
this method to derive formulas for certain patches. In the third chapter we
describe some configurations for constructing elliptic spherical
triangulations. These are the mutant configu- ration, the productive
configuration and the self-reproductive configuration. We also describe the
face-fullering and edge-fullering methods as well as the glueing of patches
method for constructing triangulations and patches. We show that there are only
19 possible types of elliptic triangulations for spheres and determine the
existence (as well as nonexistence) of all these types except for a small
number of cases.
