Binary systems where the axis of rotation (spin) of one or both components is
tilted w.r.t. the axis of revolution are called misaligned binary systems...
angle of misalignment, obliquity, has been measured for a handful of stars and
extrasolar planets to date. Here we present a mathematical framework for a
complete and rigorous treatment of misalignment and introduce an extension to
the public PHOEBE code that implements this framework. We discuss misalignment
for the Roche geometry and introduce methods for computing stellar shapes,
equilibrium (generalized Lagrange) points of the potential and minimal
requirements for lobe existence. Efficient parametrization of misalignment is
proposed in the plane-of-sky coordinates and implementation details in PHOEBE
are given alongside the proof-of-concept toy model, comparison with a known
misaligned binary DI Her, and comparison with a misaligned planetary system
Kepler-13. We provide important mathematical details of the model in the
Appendix. This paper accompanies the release of PHOEBE 2.1, which will be
available soon from its website http://phoebe-project.org.