Unfolding a reflected interval trajectory

Reflecting an interval repeatedly across its endpoints maps a reflected path to a free path in an image interval. A return trajectory from $q\in (0,L)$ ends at $q+2rL$ after an even number of wall reflections, or $-q+2rL$ after an odd number. An infinite square well has reflection amplitude $-1$, so the corresponding Dirichlet heat kernel on an interval is the difference of direct and mirrored free kernels. Individual straight paths illustrate the geometry; the path integral sums all time-sliced paths with the same endpoints.