# Parametric Equations of Epicycloid

The curve generated by tracing the path of a chosen point on the circumference of a circle which rolls without slipping around a fixed circle is called an

**epicycloid**. Use the applet to explore these curves for different values of*a*and*b*(the radii of the circles).To find parametric equations for an epicycloid, check the "show auxiliary objects" box.
Assume
(a) the radius of the fixed circle is

*a*(b) the radius of the rolling circle is*b*Let`∠AOB=t`

an `∠OAP=s`

. Note that because of the rolling, the two orange arcs have the same length, so *at=bs*. Follow the following steps to come up with equations for the*x*and*y*coordinates of*P*in terms of the parameter*t*. (a) Express ∠OAB in terms of*t*. (b) Express ∠ DAP in terms of*s*and*t*. (c)*x=OC=OB+BC*. You should be able to express*OB*and*BC*in terms of*t*and/or (*s*by looking at right triangles*OBA*and*ADP*. Then since*at=bs*, you should be able to express*x*in terms of just*t*(and of course*a*and*b*). (d)*y=CP=AB-AD*, so you should be able to express*y*in terms of*t*,*a*, and*b*. Once you have your equations, enter them into the input boxes and click the ``Graph parametric equations'' button to verify your answers.## epicycloids

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