  A great sphere fibration is a sphere bundle with total space $S^n$ and fibers
which are great $k$-spheres. Given a smooth great sphere fibration, the central
projection to any tangent hyperplane yields a \emph{nondegenerate} fibration of
$\mathbb{R}^n$ by pairwise skew, affine copies of $\mathbb{R}^k$ (though not
all nondegenerate fibrations can arise in this way). Here we study the topology
and geometry of nondegenerate fibrations, we show that every nondegenerate
fibration satisfies a notion of Continuity at Infinity, and we prove several
classification results. These results allow us to determine, in certain
dimensions, precisely which nondegenerate fibrations correspond to great sphere
fibrations via the central projection. We use this correspondence to reprove a
number of recent results about sphere fibrations in the simpler, more explicit
setting of nondegenerate fibrations. For example, we show that every germ of a
nondegenerate fibration extends to a global fibration, and we study the
relationship between nondegenerate line fibrations and contact structures in
odd-dimensional Euclidean space. We conclude with a number of partial results,
in hopes that the continued study of nondegenerate fibrations, together with
their correspondence with sphere fibrations, will yield new insights towards
the unsolved classification problems for sphere fibrations.
