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. R and S are parameters that are adjusted for each curve.
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.
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.
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).
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.)
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.