  We formulate a method for computing Stokes flow past a highly deformed sphere
with arbitrarily defined surface velocity. The fundamental ingredient is an
explicit extrapolation operator extending a velocity field from the surface of
a sphere, which is expressed in terms of a complete set of basis Stokes fields
for the pressure and velocity derived from scalar and vector spherical
harmonics. We present a matrix algebra packaging suitable for numerical
computation to arbitrary order in the deformation amplitude (deviation from
sphericity). The hydrodynamic force and torque on a deformed sphere with
arbitrary surface velocity are expressed in terms of basis field amplitudes,
and for the classic problem of a rotating and translating rigid body, we
compute explicitly the first order in deformation corrections to the flow field
as well as the hydrodynamic force and torque.
