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.