Given a set of points in three dimensions, we study the maximum numbers
of quadrics spanned by subsets of points in in various ways. We prove
that the set of empty or enclosing ellipsoids has complexity.
The same bound applies to empty general cylinders, while for empty circular
cylinders a gap remains between the lower bound and the
upper bound. We also take interest in pairs of empty homothetic
ellipsoids, with complexity , while the specialized versions
yield for pairs of general homothetic cylinders, and
and for pairs of parallel circular cylinders,
respectively. This implies that the number of combinatorially distinct
Delaunay triangulations obtained by orthogonal projections of on a
two-dimensional plane is and . A tight bound
holds for shape Delaunay triangulations, where the Euclidean
metric is replaced by convex elliptic metrics. Our lower bounds are derived
from a generic geometric construction and its variants. The upper bounds
result from tailored linearization schemes, in conjunction with a new result
on convex polytopes which is of independent interest: In even dimensions , the convex hull of a set of points, where one half lies in a subspace
of odd dimension
, and the second half is the
(multi-dimensional) projection of the first half on another subspace of
dimension , has complexity only
.