GraphingCalculator 3.5;
Window 44 322 742 962;
PaneDivider 264;
FontSizes 14 12 9;
Slider -1.5 1.5;
SliderSteps 101;
SliderControlValue 40;
SliderVariable b;
T 0 31.4159;
2D.BottomLeft -2.453125 -3.671875;
Parameter b = -2;
Color 3;
Grain 1;
Expr vector(x,y)=vector([1+b]*cos(t)+h*b*cos([[1+b]*t/b]),[1+b]*sin(t)+h*b*sin([[1+b]*t/b]));
Text "1 is the radius of the fixed circle. b is the radius of the rolling circle. h is the distance of the tracing point to rolling circle's center. If b is positive, this formula gives epicycloid. If b is negative, this formula gives the hypocycloid. Let a:=1, b:=p/q where p, q are coprime. Then the formula has a period of 2*π*p. If b is irrational, the formula is nonperiodic.
Try to adjust b from a narrow range around 0, and huge as 100 and think about the general behavior of the curve. Also note the range of t.
XahLee.org, 2004.";