Parametrizing Position of a Spyrograph

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Homework Help Overview

The original poster attempts to parametrize the position of a spyrograph, describing a specific design that completes its path after 8 rotations, involving gears with different counts. The context involves understanding the geometric and mathematical principles behind the motion of the spyrograph.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster explores the use of trigonometric functions and arc length equations but expresses uncertainty about how to apply them to the spyrograph's design. They question the role of the gears in the parametrization process. Another participant shares a potential equation derived from visual observation, prompting further inquiry into its derivation.

Discussion Status

The discussion is ongoing, with some participants providing insights that clarify aspects of the problem. The original poster has received a suggested equation but seeks further understanding of how it relates to the visual representation of the spyrograph.

Contextual Notes

There is mention of potential confusion regarding whether the problem is more aligned with calculus or precalculus concepts, indicating a possible gap in foundational knowledge or assumptions about the mathematical tools required.

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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.
 
Physics news on Phys.org
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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  • Screen shot 2010-09-27 at 4.39.53 PM.jpg
    Screen shot 2010-09-27 at 4.39.53 PM.jpg
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Thank you so much! It makes more sense now.
 

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