Epicycloids and Hypocycloids
Negative frequencies let the circle role inside
Modify the curve
frequency, -12 ... +12:
stick, 0 ... 8:
The curves are defined by these equations:
c(t) = RR * [ cos(t), sin(t) ] +
sticklength * [ cos(frequency*t), sin(frequency*t)];
The curve and ists tangents are obtained
from a rolling construction.
If sticklength = RR/|frequency| (stick = 1), the curves are
called hypocycloids - which roll inside - and epicycloids - which roll outside.
For other sticklengths these curves are called hypotrochoids resp. epitrochoids.
Circle rolling inside a bigger circle, tracing out a rose curve.
The random dots help to show the rotating plane centered on the contact point of 2 rolling curves.
The significance is that the contact point to the tracing point is the normal of the curve, and this is true for any curve rolling on another curve.