  We examine the non-equilibrium radiative heat transfer between a plate and
finite cylinders and cones, making the first accurate theoretical predictions
for the total heat transfer and the spatial heat flux profile for
three-dimensional compact objects including corners or tips. We find
qualitatively different scaling laws for conical shapes at small separations,
and in contrast to a flat/slightly-curved object, a sharp cone exhibits a local
\emph{minimum} in the spatially resolved heat flux directly below the tip. The
method we develop, in which a scattering-theory formulation of thermal transfer
is combined with a boundary-element method for computing scattering matrices,
can be applied to three-dimensional objects of arbitrary shape.
