Equations of curves

1. Mar 9, 2008

wilhelmsir

[SOLVED] equations of curves

a circle of diameter A rolls without slipping along the outer circumference of a stationary circle of the same diameter. use polar coordinates to derive the equation of a curve described by some fixed point on the rolling circle. "can anyone help me out in this?"thanks 4 yo kindness

2. Mar 9, 2008

HallsofIvy

Staff Emeritus
Assume the stationary circle has center at (0, 0). Let $\theta$ be the angle the center of the rolling circle makes with the x-axis. Then the center of the rolling circle is distance 2A from the origin and has $x= 2A cos(\theta)$, $y= 2A sin(\theta)$ Now let $\phi$ be the angle the line from the center of the rolling circle to the given point. Then the given point has coordinates $x= A cos(\phi)+ 2A sin(\theta)$ and $y= A sin(\phi)+ 2A sin(\theta)$.

Of course, $\theta$ and $\phi$ are not independent. Draw a few pictures and right triangles to see how they are related. Then you can replace one by the other to get a single parameter.

3. Mar 13, 2008

wilhelmsir

thanks a lot HallsofIvy