Tough Question in Calculus

1. Sep 14, 2006

Duhoc

Let's say that someone is drawing a circle with a compass. As that circle is being drawn a second compass is attached to the first such that the needle leg is attached to the pencil of the first. Only instead of a needle it is a small wheel. As the first compass inscribes its circle the second compass is tracing out the circumference of the first and drawing a circle at twice the rate of rotation of the first. An analogy would be someone at the edge of a merry-go-round that is rotating above a piece of paper. The rider endeavors to trace out a circle on the paper as the merry-go-round rotates at twice the rate of rotation of the merry-go-round. Is there an exponential relation between the relative rates of rotation and the area incribed under the curve created by the second compass? Additionally, if a third compass rode on the path inscribed by the second attempting to sketch a circle at some ratio of the rate of rotation of the first and second could that area be expressed? (Please private message as well as post.)

Thank you,
Duhoc

2. Sep 15, 2006

HallsofIvy

Staff Emeritus
Doesn't look all that hard. If you take (0,0) as the center of the circle described by the compasses and $\omega$ as the "rate of rotation", assuming radius R1, then the circle itself is described by the equations $(R_1cos(\omega t), R_1 sin(\omega t))$. A second circle with center at that point, radius R2 and "rate of rotation" $2\omega$ is given, relative to that point, by $(R_2cos(2\omega t), R_2 sin(2\omega t))$.

That point, relative to the original coordinate system is just the sum:
$(R_1 cos(\omega t)+ R_2cos(2\omega t), R_1 sin(\omega t)+ R_2 sin(2\omega t))$

3. Sep 15, 2006

Duhoc

Are you sure?

Remember that the second circle is not a circle. It is rotating around the circumference of circle 1 as it attempts to inscribe a circle forming a curve. I am seeking an expression for the area under that curve relative to the rotation rates of the circle and the would-be circle. And I would like to know, is that area in some way related to the relative rates of rotation of the two compasses. Again, to clarify, compass two has a wheel that is riding along the track of the circumference of circle one as it is trying to inscribe circle 2 at twice the rate of rotation of circle one. This is exactly analagous to an elbow joint rotating in relation to a shoulder joint.

4. Sep 16, 2006

HallsofIvy

Staff Emeritus
Yes, I understand all that. The equations I gave are correct.