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These patterns are epicycloids, and the construction is attributed to Cremona, see for example this webpage.

The question is about the envelope of a system of lines: one draws a line through the points $(\cos t, \sin t)$ and $(\cos nt, \sin nt)$ for every $t$. If $F(x,y,t)=0$ is an equation of this line, then the envelope is found from resolving the system of equations $$ F(x,y,t)=0, \quad \frac{\partial F}{\partial t}(x,y,t)=0 $$ with respect to $t$. The result is $$ \gamma(t) = \frac{1}{n+1} \begin{pmatrix} \cos nt + n\cos t\\ \sin nt + n\sin t \end{pmatrix} $$ which parametrizes the trajectory of a point on a circle of radius $\frac{1}{n+1}$ rolling on a circle of radius $\frac{n-1}{n+1}$.

This can also be proved geometrically by looking at the instantaneous motion of the point (it is rotated about the point of contact of the two circles).

The following does not answer the question about the number of cusps, but might be useful.

Consider a continuous version of your construction: draw a line through the points $(\cos t, \sin t)$ and $(\cos nt, \sin nt)$ for every $t$ and look at its envelope. If $F(x,y,t)=0$ is an equation of this line, then the envelope is found from resolving the system of equations $$ F(x,y,t)=0, \quad \frac{\partial F}{\partial t}(x,y,t)=0 $$ with respect to $t$. The result is $$ \gamma(t) = \frac{1}{n+1} \begin{pmatrix} \cos nt + n\cos t\\ \sin nt + n\sin t \end{pmatrix} $$ which parametrizes the trajectory of a point on a circle of radius $\frac{1}{n+1}$ rolling on a circle of radius $\frac{n-1}{n+1}$.

(That this trajectory has the $t$-$nt$ lines as tangents can also be proved geometrically by looking at the instantaneous motion of the point: it rotates about the point of contact between the two circles.)

The curve is called epicycloid, and the construction is attributed to Cremona, see for example this webpage.

It has $n-1$ cusp, which is different from your result. Probably the reason is that you consider a discrete set of lines. By the way, your pictures look similar to those for epicycloids for rational non-integer ratio of the circles radii, see the Wikipedia page.

Also I found a page discussing Mathematica codes for Cremona construction.

These patterns are epicycloids, and the construction is attributed to Cremona, see for example this webpage.

The question is about the envelope of a system of lines: one draws a line through the points $(\cos t, \sin t)$ and $(\cos nt, \sin nt)$ for every $t$. If $F(x,y,t)=0$ is an equation of this line, then the envelope is found from resolving the system of equations $$ F(x,y,t)=0, \quad \frac{\partial F}{\partial t}(x,y,t)=0 $$ with respect to $t$. The result is $$ \gamma(t) = \frac{1}{n+1} \begin{pmatrix} \cos nt + n\cos t\\ \sin nt + n\sin t \end{pmatrix} $$ which parametrizes the trajectory of a point on a circle of radius $\frac{1}{n+1}$ rolling on a circle of radius $\frac{n}{n+1}$.

This can also be proved geometrically by looking at the instantaneous motion of the point (it is rotated about the point of contact of the two circles).

These patterns are epicycloids, and the construction is attributed to Cremona, see for example this webpage.

The question is about the envelope of a system of lines: one draws a line through the points $(\cos t, \sin t)$ and $(\cos nt, \sin nt)$ for every $t$. If $F(x,y,t)=0$ is an equation of this line, then the envelope is found from resolving the system of equations $$ F(x,y,t)=0, \quad \frac{\partial F}{\partial t}(x,y,t)=0 $$ with respect to $t$. The result is $$ \gamma(t) = \frac{1}{n+1} \begin{pmatrix} \cos nt + n\cos t\\ \sin nt + n\sin t \end{pmatrix} $$ which parametrizes the trajectory of a point on a circle of radius $\frac{1}{n+1}$ rolling on a circle of radius $\frac{n-1}{n+1}$.

This can also be proved geometrically by looking at the instantaneous motion of the point (it is rotated about the point of contact of the two circles).

The envelopes of these families of lines are epicycloids, and the construction is attributed to Cremona, see for example this webpage.

These patterns are epicycloids, and the construction is attributed to Cremona, see for example this webpage.

The question is about the envelope of a system of lines: one draws a line through the points $(\cos t, \sin t)$ and $(\cos nt, \sin nt)$ for every $t$. If $F(x,y,t)=0$ is an equation of this line, then the envelope is found from resolving the system of equations $$ F(x,y,t)=0, \quad \frac{\partial F}{\partial t}(x,y,t)=0 $$ with respect to $t$. The result is $$ \gamma(t) = \frac{1}{n+1} \begin{pmatrix} \cos nt + n\cos t\\ \sin nt + n\sin t \end{pmatrix} $$ which parametrizes the trajectory of a point on a circle of radius $\frac{1}{n+1}$ rolling on a circle of radius $\frac{n}{n+1}$.

This can also be proved geometrically by looking at the instantaneous motion of the point (it is rotated about the point of contact of the two circles).