  For any scalar-valued bivariate function that is locally Lipschitz continuous
and directionally differentiable, it is shown that a subgradient may always be
constructed from the function's directional derivatives in the four compass
directions, arranged in a so-called "compass difference". When the original
function is nonconvex, the obtained subgradient is an element of Clarke's
generalized gradient, but the result appears to be novel even for convex
functions. The function is not required to be represented in any particular
form, and no further assumptions are required, though the result is
strengthened when the function is additionally L-smooth in the sense of
Nesterov. For certain optimal-value functions and certain parametric solutions
of differential equation systems, these new results appear to provide the only
known way to compute a subgradient. These results also imply that centered
finite differences will converge to a subgradient for bivariate nonsmooth
functions. As a dual result, we find that any compact convex set in two
dimensions contains the midpoint of its interval hull. Examples are included
for illustration, and it is demonstrated that these results do not extend
directly to functions of more than two variables or sets in higher dimensions.
