  This paper is the first of two in which relations between least area and
normal surfaces in 3-manifolds, are established. The two papers focus on
surfaces embedded in either Euclidean or hyperbolic 3-manifolds. In this paper
we introduce a relaxed version on normal surfaces, termed quasi-normal, and it
is shown that under appropriate conditions, every embedded least area surface
is quasi-normal with respect to a fine enough triangulation of the manifold. In
addition, it is shown that the intersection of such a surface with the
tetrahedra of a fine enough triangulation, even when not as simple as in the
case of normal surfaces, are also well behaved.

Geometric Topology