Given $a,b,r$, I would like to find the roots of $f$ on $\mathbb{R}_+^2$:
$$f(x,y)=\sin(a\,x)\sin(b\,y)-r\,\sin(b\,x)\sin(a\,y)$$
As you can see below, the roots of $f$ are curves (in red), parametrized by $x$ and $y$ (for arbitrary $a,b,r$).
Can I find the equations of the red curves?
Numerically, a possibility is to find an approximate root $(x_0,y_0)$, then numerically identify $\delta$ such that $f(x_0+\varepsilon,y_0+\delta)=o(\varepsilon^{p+1})$ for a given $\varepsilon$, using a Taylor expansion of order $p$. I am wondering if there would be a more elegant solution, and possibly closed-form.