that pierces only one surface, is zero.) If the objects are now spatially separated the integral $\int_{1/\Lambda}^\infty
\frac{ds}{s^5} \langle\Theta(sy,x)\rangle_y$ is finite and well behaved for $\Lambda\to0$, so we can safely set $\Lambda=0$. One is easily convinced that this is indeed the correct expression by considering a simple geometry (such as two parallel flat slabs) and requesting that the Casimir force does not change if one object is instead thought of as being made of two adjacent bodies. The counting weights are dictated by the convention for the “zero energy” configuration.

For each loop $y$, the weight $\Theta$, as a function of the re-scaling factor $s$, is piece-wise constant. The $s$-integral hence can easily be performed analytically. Rather than being only a convenient simplification that saves computing time, this property plays a crucial role for the efficient simultaneous computation of multibody forces, cf. Section \[sec:autodiff\].

Loop Generation {#subsubsec:generation}
---------------

When trying to evaluate Equation \[eq:loopenergy.final\], one naturally would try to discretize the loop as consisting of a finite number of straight sections. Taking the procedure literally, the presence of complicated curved geometries would mandate computationally fairly expensive ray-surface intersection checks. In many cases, a better investment of the computational effort may 