  A knotted surface in the 4-sphere may be described by means of a hyperbolic
diagram that captures the 0-section of a special Morse function, called a
hyperbolic decomposition. We show that every hyperbolic decomposition of a
knotted surface K defines a projection of K onto a 2-sphere, whose set of
critical values is the hyperbolic diagram of K. We apply such projections,
called flattenings, to define three invariants of knotted surfaces: the width,
the trunk and the partition number. The basic properties of flattenings and
their derived invariants are obtained. Our construction is used to study
flattenings of satellite 2-knots.
