## On the curve generated by plotting one sine against another

A circle? A line? Actually, it depends. It depends upon parameters like the frequency or the phase; when these change, really interesting things happen. What I mean by “against another” is that the first sine function will be the $x$-coordinate while the other, the $y$-coordinate; then, if you studied engineering you might say “$y$ vs. $x$” but if you studied biology you might be used to say “$x$ vs. $y$” (personal observation). We end up then with a curve, say $\gamma(t) = (x(t), y(t))$, where

$x(t) = A_1\sin(\omega_1 t + \phi_1)$

and

$y(t) = A_2\sin(\omega_2 t + \phi_2)$.

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