Hypocycloids and Hypotrochoids

Roulettes – 4: Hypocycloids and Hypotrochoids

In our last few posts we investigated rouletts, the curves that are formed by the locus of points attached to a circle as it rolls around the outside of a fixed circle. Depending on the ratio of the radii (and therefore the circumferences) of the circles these curves are the cardiods (equal radii), epicycloids (moving circle’s radius is less than the fixed circle), and epitrochoids (the point is in the interior or exterior of the moving circle).

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)

We shall now consider the curves that result when the moving circle rolls around the inside of the fixed circle. These curves are called hypocycloids and hypotrochoids. To generate today’s curves make the radius, R, of the moving circle negative.

The first seems almost a special case. Let R =  – 0.5 and S = 0.5 (below left) and then let R = S = – 0.5 (below right). The results are segments, which as we shall see are actually degenerate ellipses.

R = S = - 0.5

R = S = – 0.5

R = - 0.5,  S = + 0.5

R = – 0.5, S = + 0.5

In the following I will keep S negative. This makes the starting point (t = 0) on the positive side of the x-axis. If S is positive the starting point is to the left of the origin. The resulting curves are the same shapes by oriented differently (rotated a quarter-turn).

If R = – 1/2 the curves are ellipses. If S < R < 0 then ellipse stays inside the fixed circle (below left); if  R < S < 0 the ellipse extends outside the fixed circle (below right). If S = 0 the locus is a circle.

R = - 0.5,  S = - 0.3

R = – 0.5, S = – 0.3

R = - 0.5,  S = - .75

R = – 0.5, S = – .75

Next we consider the more general case for which the moving circle’s radius is not exactly half of the fixed circle’s radius.

If R < S < 0, the point is in the interior of the moving circle and the graph is a series of loops (below left). When R = S , the point is on the circle, there is a star-like figure (below right). These are both called hypocycloids.

When S < R < 0 the point is outside the circle and the “stars” form rounded ends and get larger. These are the hypotrochoids (below center).

R = - 0.6,  S = - 0.3

R = – 0.6, S = – 0.3

R = S = - 0.6

R = S = – 0.6

R = - 0.6,  S = - 1

R = – 0.6, S = – 1

The next video shows the progression from S = 0 (a circle) to S = –2 and back again. (S is the distance between the center of the moving circle to the blue point.)

R = - 0.6, - 2 < S < 0,  A = 6pi

R = – 0.6, – 2 < S < 0, t = 6pi

Exploration 6: When R = –0.5 segments and ellipses are formed. Discuss how these are not really different from the cases with different negative values of R.
Exploration 7: In the case where R = S star-like figures are formed. The points of the “star” are cusps. Find the number and location of these cusps in terms of R. (Hint: see the discussion of the cusps in the second post in this series.)
Exploration 8: (Calculus) Find and discuss the derivative at the cusps when R = S.

This will be discussed in the next post. 

References:

Hyposycloid: http://en.wikipedia.org/wiki/Hypocycloid

Hypotrochoid: http://en.wikipedia.org/wiki/Hypotrochoid

Epitrochoids

Roulettes – 3: Epitrochoids

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)

Before looking at Epitrochoids, consider the three kinds of cycloids. A cycloid is the locus of a point attached to a circle rolling along a line.If the point is on the circle a cycloid is generated.

Cycloid - A point on a circle rolling on a line.

Cycloid – A point on a circle rolling on a line.

If the point is in the interior of the circle a curtate cycloid is generated.

Curtate Cycloid

Curtate Cycloid. A point in the interior of a circle rolling on a line.

If the point is in the exterior of the circle a prolate cycloid is generated.

Prolate Cycloid

Prolate Cycloid. A point attached outside a circle (such as on the flange of a train wheel) rolling on a line.

Using our Roulette Generator we can produce similar curves called epitrochoids the locus of a point attached to one circle as it rolls around another circle. If the point is on the moving circle an epicycloid is generated. These were discussed in the preceding postR is the radius of the moving circle and S is the distance of the point whose locus is graphed from the center of the moving circle.

R = S = 1/3

R = S = 1/3

By changing the position of the point relative to the center (where S = 0) we can see a similarity with the cycloids.

If the point is in the interior of the moving circle (SR), then the curves look like this:

R3-d

R = 0.4 and S = 0.25. 0\le t\le 4\pi

A close inspection will show that this curve is similar to the curtate cycloid wrapped around a circle.

The next obvious question is what happens if S > R? Then the resulting curves have inner loops similar to those of the prolate cycloid.

R3-e

S = 0.84, R = 0.6 0\le t\le 6\pi

Finally, we can see the range of curves by changing the values of S. The next video shows the progression of shapes as S changes from 4 to -4 . Watch the orange point. S is the distance between the orange point and the center of the smaller moving circle(open point). The negative values amount to starting the moving circle on the opposite side of the fixed circle and gives the same curves in a different orientation.

R3-f

R=0.6,\,t=6\pi ,\,-4\le S\le 4

Investigation 5: What is the shape of the curve when S = 0?
Investigation 6: What shape does the curve approach as S approaches infinity?

Next post: hypocycloid – for those who like to be negative.

References:

Cycloid: http://en.wikipedia.org/wiki/Cycloid

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