  For a set of maps of based spaces $S$ we construct a version of Weiss'
orthogonal calculus which depends only on the $S$-local homotopy type of the
functor involved. We show that $S$-local homogeneous functors of degree $n$ are
equivalent to levelwise $S$-local spectra with an action of the orthogonal
group $O(n)$ via a zigzag of Quillen equivalences between appropriate model
categories. Our theory specialises to homological localizations and
nullifications at a based space. We give a variety of applications including a
reformulation of the Telescope Conjecture in terms of our local orthogonal
calculus and a calculus version of Postnikov sections. Our results also apply
when considering the orthogonal calculus for functors which take values in
spectra.
