Behaviour of the whispering gallery rays in a vicinity of a point where curvature of the boundary vanishes

Behavior of whispering gallery rays in a vicinity of the boundary point $S=0$ with zero curvature is invertigated in two cases. In the first one boundary curvature $K$ is zero in this point, but $\frac{dK}{dS}\mid_{S=0}\ne0$. In the second one $K=\frac{dK}{dS}=0$ in the point $S=0$, but $\frac{d^2K}{dS^2}\ne0$. Results are obtained by the help of a computer and presented on the pictures.