Roulettes and Art – 2

Continuing with our discussion of ways to produce interesting designs with the Roulette Generator (RG) for Winplot, here are some hints on making more detailed designs using more than one function.

Hint #5: Adding more graphs to the first: You may add other graphs by selecting the roulette and/or velocity equations from the Inventory (CTRL+I) and clicking “dupl”  to duplicate the equation. Then click the duplicate and then “edit” and change the sign in front of the S in both equations. This will let you graph S and –S at the same time. This will give you two congruent graphs with the first rotated 2\pi /n from the first. (From the previous posts: if \left| R \right|=\tfrac{n}{d} then there are d dips, loops, or cusps in n full revolutions). Both graphs will have the same R value.  

You may also change the color of all the graphs.

Here is a graph with that idea. The values are in the caption.

R = -0.322, S = -0.3 (blue) and S = +0.3 (green) and their derivatives (orange and   xxx  15 Revolutions

R = -0.322, S = -0.3 (blue) and S = +0.3 (green) and their derivatives (orange and gray),
15 Revolutions

Hint #6: Plotting Density (PD) (Winplot RG only) also affects the final drawing. Graphers work by calculating a number of points and joining them with straight segments. The default plotting density is 1. Usually this results in a nice graph with smooth looking curves because the segments are very short. If your graph looks like a bunch of segments, select the equation in the Inventory, click “edit”, and increase the plotting density (to 10 or 100 or more) and the curves will no longer look like segments. Of course, you may want them to look like segments. The graph below shows how PD works. The graph on the left has a plotting density of 10. The center graph is a detail of the first with the same PD. The plot on the right is the same as the center graph but with a plotting density of 100.

Hint #7: In addition to PD, Winplot is very sensitive to image size and zooming in and out. Once you have a graph you like, experiment with zooming in and our (page-up and page-down keys) or dragging the corners of the frame. You will see a lot of different graphs.

Your turn: Try making this graph below with S=\pm \tfrac{1}{3}, R = 0.00667 and slightly less than a full revolution. Make the image size (under the file tab) 12.3 x 12.3 (the units are cm.), or 465 x 465 pixels (type @ after the number to use pixels). Amazing!


Every slight change makes a whole new design. Start with your own values. I would like to see what you and your students come up with. When you get the perfect one, e-mail it to me as a .jpg file and I will post it. (Please include the SR, and other data.


Roulettes and Art – 1

For the last few post, we have been exploring roulettes using the roulette generator (RG) for either Winplot or Geometer’s Sketchpad. These files use the equations

 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)

 The derivative is given by the equations

{x}'\left( t \right)=-\left( 1+R \right)\sin \left( t \right)+S\left( \tfrac{1}{R}+1 \right)\sin \left( \tfrac{1}{R}t+t \right)

{y}'\left( t \right)=\left( 1+R \right)\cos \left( t \right)-S\left( \tfrac{1}{R}+1 \right)\cos \left( \tfrac{1}{R}t+t \right)

Notice that the derivative is also a form of roulette.

To generate various roulettes by changing the values of R and S as explained in the first post in this series.

When my friend Audrey Weeks finished making the Sketchpad RG for me she sent these three designs that she made using the generator. Suddenly, we were into art!

The designs may make a nice project for students studying parametric curves and help them learn a little more about the curves and their graphs. Here are some hints about how to use the Winplot RG to make these designs. (Most of the hints will also do for the Sketchpad RG except for changing colors, and generating several curves at once.)

 Hint 1: Begin by opening the RG file and saving it with a different name so you can makes the changes and still have the original available. In the new file,

  • Open the Inventory (CRTL+I) and delete everything except the one marked “Roulette” and the one marked “Velocity Graph” by selecting each and clicking the “delete” button.  This will remove the circles and other lines from the final drawings.
  • Then make a duplicate of the two remaining files by selecting them and clicking “dupl.” For the duplicates, click “edit” and change the sign preceding the S. This will let you draw graphs for S and –S at the same time. More on this below.
  • Click CTRL+G and turn the axes off.

 Hint 2: Number of revolutions: We learned in the first post in this series that from the number R expressed as a reduced fraction \left| R \right|=\tfrac{n}{d}, that d is the number of dips, loops, or cusps in the graph and n full revolutions will draw the entire graph (i. e. after n revolutions the same graph will be redrawn).

When using a large value of d the parts of the graph overlap each other and add to the design. So, use a large d to get a “fancier” graph.

If you use n revolutions your graph will have a number of rotational symmetries offset by 2\pi /n radians.

But you also get nice designs by using less than n revolutions. This draws part of the graph and can also make a pleasing design. See the captions to the figures in this post to get an idea of how this works.

Hint #3: Color: You can change the color by selecting the equation in the Inventory list and clicking “edit” and then “color.”  The background color can be changed by clicking “Misc” in the top bar and then “Background.”

Hint #4: Dips, loops and cusps: These are controlled by the S slider. If \left| S \right|=\left| R \right| there will be cusps, if \left| S \right|<\left| R \right| dips, and if \left| S \right|>\left| R \right| loops. (Of course, you could have your students discover this on their own.) Experiment with this to make other designs.

So, let’s try one.  First graph: I chose R = –0.321 and S = 0.440. Since 321/1000 does not reduce, there will be 1000 loops in 321 revolutions. But I graphed only about 6.5 revolutions (A = 40.212). Second Graph: includes the derivative of the first made by using the velocity equation in the Inventory.

These were just a few hints to get you started. In the next post we’ll look at some much more fancy designs. Meanwhile try some of your own and post them as comments. (Please include the R, S, and revolution values you used.).

Preview of the next post Roulettes and Art – 2:


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. 





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:


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.


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.


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.





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 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.





Corrections made July 5, 2014

Rolling Circles

A few weeks ago I covered some trigonometry classes for another teacher. They were studying polar and parametric graphs and the common curves limaçon, rose curves, cardioids, etc. I got to thinking about these curves. the next few posts will discuss what I learned. To help me see what was happening I made a Winplot animation. My “Roulette Generator” (RG) is a rather simple setup and turned out to be very well suited to study a variety of curves: cardioids, epicycloids, epitrochoids, hypochoids, and hypotrochoids to name a few. They are all examples of roulettes – curves generated by a point on a curve as it moves around anther curve. I considered only cases where both curves are circles. Calculus will make it appearance after the first few posts in this series. I hope you will find information here that will let you make a good project or investigation for calculus or precalculus students. The RG can be set to graph any number of situations as I will discuss. So here goes.

Roulettes – 1: Equations and the Roulette Generator.

In this post I will discuss the derivation of the parametric equations used to make the animations and some notes on how to use the RG. Later posts will discuss some of the various curves that result. I began with a simple cardioid.


Cardioid R = S = 1

A cardioid is defined as the locus of a point on a circle as it rolls without slipping around another circle with the same radius. 

The setup described here will allow us to change the equations using sliders, for this and a number of related curves. The two circles with the same radii are shown in the figure below. The circle with center at C rolls counterclockwise around the circle with center at the origin, O. The point D traces the cardioid. The blue curve from F to D is the beginning of the cardioid. Cardioid setup pix The equations of the circles are: The circle centered at the origin with radius 1:

x\left( t \right)=\cos \left( t \right)\text{ and }y\left( t \right)=\sin \left( t \right)

The moving circle with center at C and radius R:

x\left( t \right)=(1+R)\cos (t)+R\cos (t) and

y\left( t \right)=(1+R)\sin \left( t \right)+R\sin \left( t \right).

The equation of the locus: In the figure above, the locus of the point marked D, as the moving circle rolls counterclockwise around the fixed circle, will be the path of the curves. A small portion of the curve is shown in blue running from F to D. In our investigations we will eventually want to place the moving point inside, on, or outside the moving circle. To do this we will use S as the distance from the center of the moving circle to the point we are watching. For the moment and for the simple cardioid, we will assume they are the same: R=S=1.

The ratio of the radius of the fixed circle to the radius of the moving circle is 1/R,  and will be of interest. The ratio can be adjusted by changing R. We can, therefore, keep the fixed circle’s radius constant and equal to one.

We will use vectors to write the locus. Before writing the individual vectors consider first that the arc length on both circles is always the same for all the curves and combinations of their radii:

\text{arc }EF=t=\text{arc }DE=R\left( \measuredangle DCE \right).

Therefore, \measuredangle DCE=\tfrac{1}{R}t. Then  \measuredangle BCD=\tfrac{\pi }{2}-\left( \tfrac{1}{R}t+t \right) Then the locus of D has the vector equation:


\overrightarrow{OD}=\left\langle (1+R)\cos (t),0 \right\rangle +\left\langle 0,\left( 1+R \right)\sin \left( t \right) \right\rangle +

\left\langle -S\sin \left( \tfrac{\pi }{2}-(\tfrac{1}{R}t+t) \right),0 \right\rangle +\left\langle 0,-S\cos \left( \tfrac{\pi }{2}-(\tfrac{1}{R}t+t) \right) \right\rangle

Notice that \left( \tfrac{1}{R}t+t \right) is the complement of \left( \tfrac{\pi }{2}-\left( \tfrac{1}{R}t+t \right) \right), so that \sin \left( \tfrac{\pi }{2}-(\tfrac{1}{R}t+t) \right)=\cos \left( \tfrac{1}{R}t+t \right) and  \cos \left( \tfrac{\pi }{2}-(\tfrac{1}{R}t+t) \right)=\sin \left( \tfrac{1}{R}t+t \right). The parametric equations of the path are

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)

You may enter these equations on any graphing calculator by entering specific values of R and S. You will have to re-type them for each different curve. With the RG you can make the changes easily with sliders.

The roulette generators: I used Winplot a free graphing program I’ve used for years. Here are the links so you can download Winplot or Winplot for Macs. the Winplot file containing the generator is here: Winplot Roulette Generator. [Sorry, this is no longer available here and WordPress will not accept Winplot files for download. Please contact me at and I’ll send you a copy.] The Winplot equations are discussed here: Notes on the Roulette Generator.

A Geometer’s Sketchpad version may be downloaded Geometer’s Sketchpad Roulette Generator. You will need Geometer’s Sketchpad to run this on a computer or the “Sketch Explorer” app for iPads and other tablets. A big ‘Thanks” to Audrey Weeks who was kind enough to make this for us. Audrey is the author of the Algebra in Motion and Calculus in Motion software. Audrey is my “go to” person when I have math questions.

Other graphers with sliders such as Geogebra and TI-Nspire will probably work as well. For those who wish to adapt this to some other graphing program there are some syntax consideration to making the one circle roll around the other and showing the path as the circle rolls. If you make your own generator on one of these other platforms, please send it along and I’ll post it and give you credit.

Experiment with the R and S sliders. In the next several posts well will do this and learn about various other curves.

Next: Epicycloids




Algebra in Motion

Calculus in Motion