Spirotechnics!

September 7, 2011

Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Project

1 The Beginning The general consensus of our group began with one thought: are awesome. Period. This simple claim was the motivation for pursuing this topic for our nal project. But once motivated the question became, what exactly do we do with Spirographs? Fortunately, we had some inspiration via an exercise assigned to us in class: Exercise 1.1.16 (The Asteroid)  along with Example 1.1.15 that preceded it. The asteroid exercise merely had us use some trigonometric identities to trans- form a parametrization, but the example before it described a specic example of a : one in which the inner had a radius one-fourth the radius of the circle in which it was rotating. We pondered: why should somebody go and spend 20 dollars on a product to make these fanciful curves when in a few hours you could program something yourself? Not having a good answer to this question  and not having 20 dollars  we set out to try and generalize the asteroid example. From our rst trial runs, we recognized the advantage of programming over the real-world product: there are physical limitations to a real spirograph that are easily overcome in a program. For example, when you take a circle of smaller radius inside of a larger circle, the gears that link the require a specic direction of rotation, whereas in a program we were able to produce patterns as if the circles were frictionless surfaces passing one another. In essence, we created our own Spirograph universe where physical limitations were not a hindrance. Once we had this program in hand, we again began to wonder: what do we do with this thing? More important: what can we do with it? The program was indeed a fun concept, but turning it into a nal project was going to take a bit longer to gure out. Through brainstorming and experimentationand with our program ready at our ngertips we discovered some neat properties. 1) If r= R , i some natural number, then the curve produced is a simple, i closed curve (simple meaning no self-intersections). 2) If r is rational but not of the form R/i, the image of the path is a closed curve that is not simple. 3) If r is irrational, the curve is not closed, meaning the image of the path as n goes to innity (n being the degrees of rotation relative to the large circle) is an annulus. But do not take us simply on our word. Let us continue to the paper where we will discuss our project by rst, dening what we mean by Spirographs, and then precede to examine the characteristics and properties of these quirky objects.

Abstract

Given two circles of arbitrary radii R and r, x one circle of radius R and roll the smaller circle along the rst. By doing this, we can create geometric structures called and or more commonly known (to many of

2 the children growing up in the 90s) as Spirographs. In this paper we discuss the characteristics and properties associated to Spirographs. Moreover, we will discuss the eects that rational versus irrational numbers have on the overall structures, as well as dene and present parameterizations of the curvatures of a given Spirograph.

Introduction to Curves

The Greeks dened a curve as the path traced out by a particle in motion. More precise, the continuous map a:I→ R2, where I is some interval on R, denes a curve in 2-space (a similar denition can be expanded for a:I →Rn). By thinking about curves in terms of time and this idea of particle's path, we can parameterize the curve a such that, for t in I, . a(t)= (a1(t),a2(t),a3(t)) where each component ai: I →R, is also a function. In addition, a is dier- entiable (i.e. smooth) if each of its coordinates are dierentiable. As we will soon discover, dierentiability is not always guaranteed in Spirographs. This in turn has an eect on how one calculates curvature of a given curve.

Roulettes

Spirographs fall under the category of curves. A roulette curve is the curve generated by tracing the path of a point, attached to a curve, as it rolls without slipping along another xed curve. For our project, we looked at circles formed by rolling a circle around another xed circle. These types of roulettes all have specic names based on the location of the xed point and whether the moving circle is on the inside or the outside of the xed circle. We have a mechanism to talk about theses creatures, namely, we can parametrize these curves. Once again, think back to the little wheels and circles we used to make spirogrpahs as children. The tools we used were gears with little teeth on them that allowed the gear (i.e. a circle) to roll along the xed circle without slipping. Convenient, right? But also limiting. Limiting because as the gear rolls along the outside of our circle, it is only able to turn in the same direction as it is moving around the xed circle. If we were moving along the inside, the gear would be turning in the opposite direction of the overall movement of the gear around the xed circle. Physically, the gears (which allow rolling without slipping) limit us to these directions of movement, but with the magic of modern mathematics we can create a parametrization that simulates moving in the opposite way. For example, moving along the outside of a circle, the gear could be rotating in a clockwise direction but moving along the xed circle in a counter clockwise direction.

3 and Hypotrochoids and Hippopota- muses....well actually not the latter

The curves we will generate are the hypocycloids and hypotrochoids. These curves are both formed when the moving circle is rotating around the inside of the xed circle. A is a plane curve generated by the trace of a xed point on a smaller circle which is rolling along the inside of a larger circle. Let r be the radius of the moving circle and R the radius of the xed circle. If R where i is some constant, then the curve produced is a simple closed curve r = i (simple meaning there are no self intersections). In fact, the resulting curve will have i cusps where the curve is not dierentiable. The cusp forms where the xed point on the smaller circle is in direct contact with the large circle. This makes sense because the circumference of the small circle is 2πR and the 2π = i circumference of the large circle is 2πR which means the small circle makes precisely i rotations to rotate around the inside of the large circle. These curves can be parametrized with the following equations: R−r x(t) = (R − r)cos(t) + rcos( r t) R−r y(t) = (R − r)sin(t) − rsin( r t) and will look something like this (depending on the specs)

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Only the other hand, we have the hypocycloid's sibling, the . A hypotrochoid is a plane curve formed in the same way as a hypocycloid except that the xed point is a distance of d away from the center of the moving circle (i.e. the point may lie on the inside or outside of the smaller circle, it does not have to be on the smaller circle's boundary). These curves are parametrized as follows: R−r x(t) = (R − r)cos(t) + dcos( r t) R−r y(t) = (R − r)sin(t) − dsin( r t) A few worthy things to note. If d < r, the spirograph will form cusps, but if d > r, we will get loops instead of cusps. Further, if R=2nd/(n+1) and r=(n- 1)d/(n+1) where n is some natural number, we get a rose. When R=2r, this forms an ellipse.

4 Although this is neat, for our project, we focused on playing with the hypocy- cloids. For these curves the natural direction of movement is such that the moving circle is rolling in a one direction, it will move around the xed circle in the opposite direction. This will create Spirographs with i cusps connected by smooth curves that are the opposite concavity to the edge of the xed circle, similar to a kapow! shape in comic books. As you can see here: i=3,5,8,10,20 and 50 respectively.

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-100 -100 -100 Now, we can parametrize our curve so that we are rotating our moving circle in the same direction as we are moving around the xed circle. When i is an integer, this will create a ower type shape, where the curve will periodically come to a cusp. There are i-2 petals for each ower. Below we can see what happens when i is varied. Below i=3,5,8,10,20, and 50 respectively.

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Epic and Epic and Epic Deltoids (be- cause shoulder strength is key to drawing spirographs): Our next focus is about what happens when we rotate our moving circle around the outside of the xed circle. The objects formed are called and epitrochoids. Like a hypocycloid, an is a plane curve generated by the trace of a xed point on the edge of a circle as it rolls along the outside of another xed circle. Merely the location of the smaller circle has changed. Using a similar setup as before, let r be the radius of the moving circle and R be the radius of the xed circle and i be a constant such that R . If i is a natural number, r = i the epicycloid has i cusps that are dierentiable. If i is rational such that p , i = q where p is in simplest terms, then there will be p cusps on the curve, but the q curve will no longer be simple, it will intersect itself. Regardless, the curve will be closed if i is rational, but if i is an , then the curve is not closed. It will form a dense subset in the shape of an annulus with outer radius R+2r and inner radius R. Theses curves can be parametrized in the following way: R+r x(t) = (R + r)cos(t) − rcos( r t) R+r y(t) = (R + r)sin(t) − rsin( r t) Onto our next candidate, epitrochoids are formed in a similar manner as epicycloids except that the xed point that traces out the curve is at a dis- tance d from the center on the moving circle (recall this same scenario with hypotrochoids). Their parameterizations are as follows: R+r x(t) = (R + r)cos(t) − dcos( r t) R+r y(t) = (R + r)sin(t) − dsin( r t) Again, we focused in primarily on the epicycloids for our project. Naturally (i.e. given the constraint of the direction the gears allow us to move) if the moving circle is rolling in a clockwise direction, it will be moving around the xed circle in a clockwise direction. This yields spirographs that are similar to the hypocycloids formed when moving in the direction opposite to the natural direction (i.e. when the rolling circle is moving in the same direction that it is revolving). However, in this case, the cusps are now rounded corners that are in fact dierentiable. Here we have i=3, 5, 8, 10, 20 and 50 respectively.

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-100 -100 -100 When the rolling circle moves in the opposite direction of rotation, we fur- thermore nd that this epicycloid is similar to the hypocycloid when moving in the natural direction (i.e. when the rolling circle is moving in the same direction as it is rotating.) Again, cusps are not formed, rather there are rounded corners that are dierentiable. Here i=3,5,8, 10, 20 and 50 respectively. 150

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7 A discussion on curvature

We think of curvature as the absolute value of the change in acceleration as we travel along our curve. That is to say, as we draw out our Spirographs, the curvature at any given point is the absolute value of how quickly we increase or decrease speed in one direction: the greater the increase in speed the greater the curvature and the opposite holds true for a decrease in speed. More formally, we can parameterize curvature for each of our discussed Spirographs. The curvature was calculated using the following formula:

|α0 × α”| κ = |α0|3

where αis our curve. This is the curvature function for the hypocycloid moving in the natural direction (i.e. rolling is in the opposite direction as rotation):

R t 2 r-R 2 2 r-R Sin 2 Abs 2 r r Out[7]= H L H L B F 2 2 3 2 Abs -r + R Cos t - Cos -r+R Bt + Abs r - R F Sin t + Sin -r+R t r r  This is the curvature functionH forL the hypocycloid moving oppositeH L the nat- J AH L I @ D A EME AH L I @ D A EME N ural direction (i.e. rolling is in same direction as rotation):

R t 2 r-R 2 R Cos t- 2 Abs 2 r r Out[14]= H L B F 2 2 3 2 Abs -r + R Cos t + Cos -r+R t B + Abs r - RF Sin t + Sin -r+R t r r  This is the curvature functionH forL the epicycloid moving inH theL natural di- J AH L I @ D A EME AH L I @ D A EME N rection (i.e. rolling is in the same direction as rotation):

R R 5 r2-2 r R+R2+ -r2+R2 Cos 2- t Abs r r Out[21]= J I M BJ N FN 2 2 3 2 Abs r + R Cos t + -r + R CosB -r+R t + Abs r + R Sin tF + -r + R Sin -r+R t r r  H L H L ThisJ is theAH curvatureL @ D H functionL A for theEE epicycloidAH L moving@ D H oppositeL A theEE naturalN direction (i.e. rolling is in the opposite direction as rotation):

R t 2 r3-r2 R+4 r R2-R3+ 2 r3-r2 R-2 r R2+R3 Cos Abs r r Out[28]= I M B F 2 2 3 2 Abs r + R Cos t + r - R CosB -r+R t + Abs r + R Sin t +F -r + R Sin -r+R t r r  H L H L To helpJ A usH reallyL @ D understandH L A theseEE equations,AH L @ letsD H plotL themA forEE severalN spirographs. This is where the program came in real handy, we needed only to change parameters on the slide bar at the top. First lets look at the hypocycloid moving in the natural direction.

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9 We can see that as i increases, the variance in curvature increases. We can also see the non-dierentiable cusps where there are vertical asymptotes on the graphs. Now consider moving in the direction opposite the natural direction.

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Again, we can see the increase in the increase in curvature (note the scale change) and can see the non-dierentiable cusps. Now lets look at the epicycloids. First we will move in the natural direction:

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Interestingly we can see the curvature functions here dip down to zero ap- proaching and leaving the corner. Though we know the corners are dieren- tiable, the curvature appears to be asymptotic at these points simply because the curvature is so great. Finally lets look at the epicycloid moving opposite to the natural direction.

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We nd a similar pattern here where the curvature goes to zero at points

13 approaching and leaving the corners, which are areas of nite, but large cur- vature. We can especially see that the curvature is not asymptotic in the last gure.

Summary

The programming associated with this project was an incredibly insightful part as well. Not only did having a program save us 20 dollars, it enabled the group to take an in depth approach to researching and working with Spirographs. Our results, although not ground breaking, did provide some intriguing observations regarding dierentiability of curves and the notion of dense subsets. The dis- cussion of the characteristics and properties associated to Spirographs generates both interesting images as well as presents possibilities for further investigations. For example, what occurs if you rotate within an ellipse? Or upon a closed Mo- bius strip? Our group feels it safe to assume that, not only is the creation of Spirographs a beautiful way to spend one's time, it also lends a great hand in understanding curvature of simple, closed curves.

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