The fraction on the upper coners are the two signed parameters b that generates the same curve. Those that can only be generated as hypocycloids are colored red. Mathematica Notebook for This Page . The beautiful double generation theorem of these curves was first noticed by Daniel Bernoulli in 1725.
Astronomers find forms of the cycloidal curves in various coronas.
They also occur as caustics . The measuring of curve length was given by Newton in his Principia. Description Epicycloid and hypocycloid both describe a family of curves. Epicycloid is a special case of epitrochoid , and hypocycloid is a special case of hypotrochoid . When the rolling is internal, we have hypocycloid.