LAB 3: THE SPIROGRAPH
In this lab, we use the Spirograph (a childs toy from Hasbro) to investigate parametric curves. Suppose you begin with a circle of radius R centered at the origin, C. Take another circle of radius r, label it C, as shown in Figure 1. A felt pen is connected and it rotates about C. The pen leaves a trace- a sample trace is given in to C, Figure. Here are the lab questions. Your write up should be a discussion that incorporates the answers- Dont just list them! See the grading rubric for more information. (1) Come up with parametric equations, x(t) and y(t), for the path of the pen in the Spirograph. (2) Come up with some nice patterns! You should try dierent end values of t. (3) The parametric curve (x(t), y(t)) is said to be closed (or the functions are periodic) if there is a T so that (x(t), y(t)) = (x(T ), y(T )). Can you make a proposition about what might need to be true about the relative circumferences of the circles? Give a specic example. (4) Give the general formula for the arc length of the pen in the spirograph. Give a numerical value for the arc length of a closed curve (from your previous example). (5) In the Conclusions section, you can bring in some of the mathematical names of these curves. You might do a little internet research to see if there is anything interesting about them. Is there such a thing as an extension to three dimensions, for example? Some Maple examples: A sample parametric curve in two dimensions. The numpoints option forces Maple to interpolate through more data points, resulting in a smoother plot (the default is 50). x:=(t*cos(t)-sin(t))/(1+t^2); y:=cos(2*t); plot([x,y,t=0.50],numpoints=1000); You can also use
Figure 1. The set up for the spirograph.

Figure 2. Two examples.