Epicycloids

Roulettes – 2: Epicycloids

In the last post we saw how a cardioid can be generated by watching the locus of a point on as one circle rolls around another circle with the same radius. In the first post of this series Roulette Generators are explained. Here are the files for Winplot or Geometer’s Sketchpad. Use them to quickly see the graphs of these curves by adjusting one or two parameters.

The parametric equations of these curves are below. and S are parameters that are adjusted for each curve.

x\left( t \right)=(1+R)\cos \left( t \right)-S\cos \left( \tfrac{1}{R}t+t \right)

y\left( t \right)=\left( 1+R \right)\sin \left( t \right)-S\sin \left( \tfrac{1}{R}t+t \right)

A cardioid, a type of cycloid R=S=1;\quad 0\le t\le 2\pi

If the circles have the same radii and therefore the same circumferences, as the moving circle rolls around the fixed circle once, the point traces a path called a cardioid, and returns to its original orientation.

Let’s see what happens if we make the moving circle smaller. Make R= S = ½. Now the radius of the moving circle is one-half the radius of the fixed circle. The circumference is also half the circumference of the fixed circle. The smaller circle makes two rotations in going once around the larger circle before returning to the initial orientation.

Epicycloid

Epicycloid R = S = 0.5 R=S=0.5\quad 0\le t\le 2\pi

Now try other values keeping R = S. Make their values unit fractions, 1/n where n is a positive integer. (Note that \tfrac{1}{R}=n.) Here are some examples:

For R = S = 1/n there are n sections of the graph as t goes from 0 to 2\pi . The places where the cycles end are evenly spaced around the fixed circle and the locus has a cusp at these places.

What if we use other rational numbers? All of a sudden things are very different:

R=S=\tfrac{2}{3}\text{, }0\le t\le 2\pi

Now the circumference of the larger circle is not a multiple of the circumference of the smaller. To return the circle to its starting orientation we have to go around once more by letting t go from 0 to 4\pi . The first time around (from 0 to 2\pi ) is shown in orange and the rest of the path in blue.

R=S=\tfrac{2}{3},\quad 0\le t\le 4\pi

Investigation 1: Using the ratio of the radius of the fixed circle to the moving circle, determine how many times the moving circle must go around the fixed circle to draw a complete curve.
 

If  R=\frac{n}{d} is a rational number (reduced), then by increasing the maximum value of t to 2n\pi  the moving circle will return to its original position after n revolutions and after  that the curve will be retraced. This is the same regardless of whether \tfrac{n}{d} is greater than or less than 1.

We now turn our attention to the cusps; the places where the point “bounces off” the fixed circle.

Investigation 2: Using the ratio of the radius of the fixed circle to the moving circle, determine how many cusps the graph will have. 

 

For these curves, If R=\frac{n}{d} there are d places where the locus “bounces” of the fixed circle. These appear as cusps of the locus.

R=S=\tfrac{7}{5}\text{, }0\le t\le 14\pi

Finally, if R is not rational, the moving circle will never return to its original orientation and the locus will keep adding cusps as the circle continues to roll round the fixed circle

R=S=\sqrt{2},\quad 0\le t\le 100

Curves of this type, with the moving circle smaller or larger than the fixed circle and R = S, are called epicycloids. Epicycloids are a special case of Epitrochoids which will be the subject of the next post in this series.

Investigation 3: Determine the value of t for each cusp, of a graph with d cusps.  

Solution left as an exercise.

Investigation 4: Investigate the derivative at a cusp. 

Solution left as an exercise. This will be discussed in a later post.

 

References:

Cardioids: http://en.wikipedia.org/wiki/Cardioid

Epicycloids: http://en.wikipedia.org/wiki/Epicycloid

Epitrochoids: http://en.wikipedia.org/wiki/Epitrochoid

Corrections made July 5, 2014

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