billiard table 和 Lissajous曲線的關係

hbghlyj

Multiple Points on Lissajous's Curves in Two and Three Dimensions FIG. 2. 第4页
和lissajous曲線的關係：
\begin{aligned}
& x=\cos \frac{\pi}{\alpha}\left(t+t_1\right), \\
& y=\cos \frac{\pi}{\beta}\left(t+t_2\right),
\end{aligned}作代换$\begin{aligned}
u&=\frac{\alpha}{\pi} \cos ^{-1} x, \\
v&=\frac{\beta}{\pi} \cos ^{-1} y,
\end{aligned}$后变成 billiard table 轨迹

hbghlyj

mathcurve.com關於lissajous曲線的結論：

In the case where the curve can be described in both directions, then there are $\dfrac{(p-1)(q-1)}2$ double points.

和billiard table 从角点开始45°发球的結論相同。