Cycloids

I've always been fascinated with cycloids.  It's a shape that is best illustrated not by a formula but by a particular type of motion.  A cycloid is the shape that is traced out by a fixed point on a circle as the circle rolls (without slipping) along a line.

Here is an animation that shows a rolling circle generating a cycloid.

The cycloid has many interesting properties - including:
      •  the length of one arch of a cycloid is exactly 4 times the diameter of the circle that generated it
      •  the area under one arch of a cycloid is exactly  where r is the radius of the generating circle

There are other kinds of cycloids.  If the generating point is not on the rim of the circle but inside the circle (on a 'spoke' of the wheel) then the shape generated is a curtate cycloid.  And if the generating point is outside the circle (on an extension of a spoke of the wheel) then the shape generated is a prolate cycloid.

Another type of cycloid is created when a circle rolls inside a larger circle.  A point on the smaller circle will trace out a shape called a hypocycloid.  This is the shape which gives us a method for drawing a line by using two circles.

Hypocycloids

The shape of a hypocycloid is determined by the ratio of the radius of the larger fixed circle to the radius of the smaller 'rolling' circle.  For example the animation below shows the hypocycloid created when the radius of the larger circle is 4 times the radius of the smaller circle.  This type of hypocycloid has four sharp 'corners' called cusps.

A hypocycloid generated by rolling a smaller circle inside a large circle with a radius three times that of the smaller circle creates a hypocylcoid with three cusps - as shown below.

It seems that the number of cusps is equal to the ratio of the larger radius to the smaller radius provided that the ratio is an integer.

What do you think happens when the radius of the larger circle is two times that of the smaller circle?  Before you play the animation below and see the answer to this question - try and guess the answer given what you know about the 4-cusped hypocycloid and the 3-cusped hypocycloid shown above.  Try to visualize the answer - and then play the animation.  The radius of the larger black circle is two times the radius of the rolling blue circle.