  Particles bound to an interface interact because they deform its shape. The
stresses that result are fully encoded in the geometry and described by a
divergence-free surface stress tensor. This stress tensor can be used to
express the force on a particle as a line integral along any conveniently
chosen closed contour that surrounds the particle. The resulting expression is
exact (i.e., free of any "smallness" assumptions) and independent of the chosen
surface parametrization. Additional surface degrees of freedom, such as vector
fields describing lipid tilt, are readily included in this formalism. As an
illustration, we derive the exact force for several important surface
Hamiltonians in various symmetric two-particle configurations in terms of the
midplane geometry; its sign is evident in certain interesting limits.
Specializing to the linear regime, where the shape can be analytically
determined, these general expressions yield force-distance relations, several
of which have originally been derived by using an energy based approach.
