next up previous
Next: Appendix J: Helmholtz basis Up: Appendix I: Scaling expansion Previous: Tension matrix expansion

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}},
\end{displaymath} (I.24)

where as usual ${\mathsf{V}}$ is the billiard volume. For $d=2$ we have
$\displaystyle \oint \! ds \, r_t$ $\textstyle =$ $\displaystyle 0,$ (I.25)
$\displaystyle \oint \! ds \, \alpha$ $\textstyle =$ $\displaystyle 2 \pi,$ (I.26)

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

Alex Barnett 2001-10-03