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This was a very tricky question! I made an animated Desmos graph to investigate it. If you look at the graph, you will see that I have 4 variables. The first variable, $a$, represents the current time. Click this to animate the hypercycloid. The second variable, $S$, represents the speed of the outer circle. After playing around with this animation, I discovered that, in general, the number of "petals" on a hypercycloid is equal to $S - 1$. The next two variables represent the radii of the inner and outer circle respectively.

The equation that I used to graph the entire hypercycloid is as follows:

$$ \left(R_1\cos (t) + R_2\cos (St)+R_2\cos (t),\space R_1\sin (t)+R_2\sin (St)+R_2\sin (t)\right) $$

As you can see, there are three main components. $ R_1\cos (t) $ is the $x$-component of the inner circle at time $t$. $ R_2\cos (t)$ is the $x$-component of the outer circle at time $t$. When you add these two together, you get the center of the correct second outer circle. The remaining middle part, $ R_2\cos (St) $ is where the magic happens. If you replace $t$ with $a$, you will get the equation for the point $P$ that the book mentions on p. 252. By using $t$ instead of $a$, however, we get the point's entire path—which is the hypercycloid that we are looking for!

Note that I only explained the $x$-component portion of my equation, but the exact same logic applies to the $y$-coordinate.

There seems to be almost no information about hypercycloids online compared to hypocycloids. Even the spell-check on my computer thinks that hypercycloid isn't a word, so it was tough to do research! Hopefully this is clear enough to answer your question. I would suggest messing around with my animation for a better understanding.

This was a very tricky question! I made an animated Desmos graph to investigate it. If you look at the graph, you will see that I have 4 variables. The first variable, $a$, represents the current time. Click this to animate the hypercycloid. The second variable, $S$, represents the speed of the outer circle. After playing around with this animation, I discovered that, in general, the number of "petals" on a hypercycloid is equal to $S - 1$. The next two variables represent the radii of the inner and outer circle respectively.

The equation that I used to graph the entire hypercycloid is as follows:

$$ \left(R_1\cos (t) + R_2\cos (St)+R_2\cos (t),\space R_1\sin (t)+R_2\sin (St)+R_2\sin (t)\right) $$

As you can see, there are three main components. $ R_1\cos (t) $ is the $x$-component of the inner circle at time $t$. $ R_2\cos (t)$ is the $x$-component of the outer circle at time $t$. When you add these two together, you get the center of the correct second outer circle. The remaining middle part, $ R_2\cos (St) $ is where the magic happens. If you replace $t$ with $a$, you will get the equation for the point $P$ that the book mentions on p. 252. By using $t$ instead of $a$, however, we get the point's entire path—which is the hypercycloid that we are looking for!

Note that I only explained the $x$-component portion of my equation, but the exact same logic applies to the $y$-coordinate.

There seems to be almost no information about hypercycloids online compared to hypocycloids. Even the spell-check on my computer thinks that hypercycloid isn't a word, so it was tough to do research! Hopefully this is clear enough to answer your question. I would suggest messing around with my animation for a better understanding.