The domino curve
Solution 1:
Let $\alpha_k$ and $\beta_k$ be the angles shown below ($\alpha_1=\pi/2$). From the sine rule one gets: $\sin\beta_k=d\sin\alpha_k$. In addition we also have $\alpha_{k+1}=\alpha_k+\beta_k$, which leads to a recursion for $\sin\alpha_k$: $$ \sin\alpha_{k+1}=\sin\alpha_k \left(\sqrt{1-d^2\sin^2\alpha_k}-d\sqrt{1-\sin^2\alpha_k}\right) $$ The coordinates $(p_k,q_k)$ of the vertex of $\beta_k$ can then be expressed as a function of $\sin\alpha_k$: $$ q_k=\sin\alpha_{k+1},\quad p_k=kd+\cos\alpha_{k+1}=kd-\sqrt{1-\sin^2\alpha_{k+1}}. $$
