Useful geometric boundary integrals

Here I give a couple of geometrical results in billiards which found no other place to hide. In arbitrary $d$ we have

$$\oint_\Gamma \!\! d{\mathbf s} \,r_n \; = \; 2 {\mathsf{V}},$$ (I.24)

where as usual ${\mathsf{V}}$ is the billiard volume. For $d=2$ we have

$$\oint \! ds \, r_t = 0,$$ (I.25)

$$\oint \! ds \, \alpha = 2 \pi,$$ (I.26)

where the local curvature $\alpha(s)$ is defined in Section I.2.