Epicycloid
An epicycloid is traced by a fixed point on a circle of radius r rolling around the outside of a circle of radius R. Use the slider to adjust the ratio R/r – this controls the shape of the curve. If R/r is an integer, the curve will have R/r cusps. The epicycloid is a special case of the epitrochoid.
Special cases are a cardioid when R/r = 1, a nephroid when R/r = 2, and a ranunculoid when R/r = 5. See if you can figure out the connection with non-integer values of R/r and the number of cusps.
Move the slider to adjust the value of t to see the curve traced out - you can also click the play button in the lower left to animate.
To change the viewing window, hold the shift button and left-click to drag the graph or use the scroll wheel to zoom in/out. You can also adjust the axes by holding the shift button and left-click on the axis you want to change.