Hightlight all the self-intersections of a Lissajous figure

I will start with the most general Lissajous figure, which has a parametric equation as follows:

$$ x(t) = f_a(at+\phi_a) \\ y(t) = f_b(bt+\phi_b) \\ t \in [0,2\pi) $$

Where:

$$ f_a, f_b \in \{\sin, \cos\} \\ a, b \in \mathbb{N} \\ \phi_a, \phi_b \in [0, 2\pi) $$

Note that this curve is periodic, so we can "shift" the range of $t$ without changing the curve. Consider one such shift, given by $t \to t'-\frac{\phi_b}{b}$. This gives the new parameterization:

$$ x(t') = f_a\left(a\left(t'-\frac{\phi_b}{b}\right)+\phi_a\right) =f_a\left(at'+\left(\phi_a-\frac{a}{b}\phi_b\right)\right) \\ y(t') = f_b\left(b\left(t'-\frac{\phi_b}{b}\right)+\phi_b\right) = f_b(bt') \\ t \in [0,2\pi) $$

This shows that we can eliminate one of the phase parameters and still describe the entire set of Lissajous figures. At this point we can also notice that $\sin$ and $\cos$ are related by a phase shift of $\pi/2$, so we can absorb the choice of function into the phase parameter, giving us the simplified equation:

$$ x(t) = \cos(at+\phi) \\ y(t) = \cos(bt) \\ t \in [0,2\pi) $$

Finally, take $d=\gcd(a,b)$. This means there exist integers $a'$ and $b'$ such that $a=a'd$ and $b=b'd$, so that:

$$ x(t) = \cos(a'(dt)+\phi) \\ y(t) = \cos(b'(dt)) \\ $$

This is the same curve as the one described by $a'$ and $b'$, except covered $d$ times. Therefore we can describe the entire set of Lissajous figures if we restrict ourselves to cases where $a$ and $b$ are coprime, i.e. $\gcd(a,b)=1$.

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Now consider a self-crossing of the Lissajous figure. Such a point occurs when two distinct values of $t$ give the same $x,y$ coordinates:

$$ x(t_1)=x(t_2) \to \cos(at_1+\phi) = \cos(at_2+\phi) \\ y(t_1)=y(t_2) \to \cos(bt_1) = \cos(bt_2) $$

The cosine function is periodic, so one way for two cosines to be equal is for their arguments to differ by a factor of $2\pi$:

$$ \alpha=\beta+2\pi n\to \cos\alpha=\cos\beta $$

However, the cosine function is also even, so it is also possible for one argument to be the negative of the other (plus a factor of $2\pi$):

$$ \alpha=-\beta+2\pi n\to \cos\alpha=\cos\beta $$

Note that at all of the self-crossings, one branch of the curve is moving in an "upwards" diagonal direction and the other is moving in a "downwards" diagonal direction. This means that one of $x$ or $y$ must obey the "negated" equality, so we have one of:

$$ at_1+\phi = at_2+\phi+2\pi n \\ bt_1 = -(bt_2)+2\pi m \\ \textrm{or} \\ at_1+\phi = -(at_2+\phi) + 2\pi n\\ bt_1 = bt_2 + 2\pi m $$

These two systems of equations are linear in $t_1$ and $t_2$ so they each have a unique solution:

$$ t_1 = \left(\frac{n}{a}+\frac{m}{b}\right)\pi - \frac{\phi}{a} \\ t_2 = \left(\frac{n}{a}-\frac{m}{b}\right)\pi - \frac{\phi}{a} \\ \textrm{or} \\ t_1 = \left(\frac{n}{a}+\frac{m}{b}\right)\pi \\ t_2 = \left(-\frac{n}{a}+\frac{m}{b}\right)\pi $$

This gives the following sets of intersection points:

$$ (x,y) = \left(\cos\left(n\pi + \frac{a}{b}m\pi\right), \cos\left(\frac{b}{a}n\pi + m\pi - \frac{b}{a}\phi)\right)\right) \\ 0\le n\le a-1,\quad1 \le m \le b-1\\ \textrm{and} \\ (x,y) = \left(\cos\left(n\pi + \frac{a}{b}m\pi + \phi\right), \cos\left(\frac{b}{a}n\pi + m\pi\right)\right) \\ 1\le n\le a-1,\quad0 \le m \le b-1 $$

Plotting these points in Mathematica:

Manipulate[ ParametricPlot[{Cos[a t + ϕ], Cos[b t]}, {t, 0, 2 Pi}, PlotLabel -> {a, b}, AspectRatio -> Automatic, Axes -> False, Epilog -> {Red, PointSize[Large], Table[Point[{Cos[(n + a/b m)Pi], Cos[(b/a n + m)Pi - b/a ϕ]}], {n, a}, {m, b - 1}], Table[Point[{Cos[(n + a/b m)Pi + ϕ], Cos[(b/a n + m)Pi]}], {n, a - 1}, {m, b}] } ], {{a, 5}, 2, 20, 1}, {{b, 4}, Select[Range[a], CoprimeQ[a, #] &]}, {{ϕ, Pi/10}, 0, 2 Pi} ]