Parametrizing Position of a Spyrograph

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To parametrize the position of a spyrograph, the user needs to understand the relationship between the number of gears and the resulting path. The problem involves a spyrograph completing eight rotations with specific gear counts: 63 on the outer gear and 72 on the inner one. The user initially struggled with incorporating trigonometric functions, specifically sine and cosine, into the parametrization. After some experimentation, they derived the equations x=cos(t) + cos(8t) and y=sin(t) + sin(8t) for the spyrograph's path. Further clarification on how to connect the visual design to these equations is requested.
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Homework Statement


I'm supposed to parametrize the position of a spyrograph.

Say you have a spyrograph and you make a design like this:
Spirograph+3.jpg


In my problem, the spyrograph had to come around 8 times in order to complete its path.
There are 63 gears on the gear that went around and 72 gears on the spyrograph. I can try to find a picture if anyone doesn't understand what I'm talking about or doesn't know what a spyrograph is.



Homework Equations


sin and cos could be used, and possibly the equation for finding lengths of arcs when you know the radius and angle:
arc length=radius(theta in radians)


The Attempt at a Solution



I tried putting it in terms of theta, but I don't know why or how to do that since they're not circles. Unless maybe the whole thing is the circle, but still, why would cos and sin come into play? And how does the amount of gears come into play?

I'd very much appreciate your help please! And sorry if this is is more Calc than PreCalc.
 
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Hi again! I found out using guess and check that the equation to the picture attached to this reply is x=cost + cos8t and y=sint + sin8t. Could someone please tell me how to get from the picture to that equation? Please!
 

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Thank you so much! It makes more sense now.
 

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