  We introduce the notion of hyperbolic equivalence for quadric bundles and
quadratic forms on vector bundles and show that hyperbolic equivalent quadric
bundles share many important properties: they have the same Brauer data;
moreover, if they have the same dimension over the base, they are birational
over the base and have equal classes in the Grothendieck ring of varieties.
  Furthermore, when the base is a projective space we show that two quadratic
forms are hyperbolic equivalent if and only if their cokernel sheaves are
isomorphic up to twist, their fibers over a fixed point of the base are Witt
equivalent, and, in some cases, certain quadratic forms on intermediate
cohomology groups of the underlying vector bundles are Witt equivalent. For
this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic
equivalent to a quadratic form whose underlying vector bundle has many
cohomology vanishings; this class of bundles, called VLC bundles in the paper,
is interesting by itself.
