Spirotechnics!

September 7, 2011

Amanda Zeringue, Michael Spannuth and Amanda ZeringueDi�erential Geometry Project

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The Beginning

The general consensus of our group began with one thought: Spirographs areawesome. Period. This simple claim was the motivation for pursuing this topicfor our �nal project. But once motivated the question became, what exactly dowe 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 speci�c exampleof a spirograph: one in which the inner circle had a radius one-fourth the radiusof the circle in which it was rotating.

We pondered: why should somebody go and spend 20 dollars on a productto make these fanciful curves when in a few hours you could program somethingyourself? 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 overthe real-world product: there are physical limitations to a real spirograph thatare easily overcome in a program. For example, when you take a circle of smallerradius inside of a larger circle, the gears that link the circles require a speci�cdirection of rotation, whereas in a program we were able to produce patterns as ifthe circles were frictionless surfaces passing one another. In essence, we createdour own Spirograph universe where physical limitations were not a hindrance.

Once we had this program in hand, we again began to wonder: what do wedo with this thing? More important: what can we do with it? The programwas indeed a fun concept, but turning it into a �nal project was going to take abit longer to �gure out. Through brainstorming and experimentation�and withour program ready at our �ngertips� we discovered some neat properties.

1) If r=Ri , i some natural number, then the curve produced is a simple,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 closedcurve that is not simple.

3) If r is irrational, the curve is not closed, meaning the image of the pathas n goes to in�nity (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 wherewe will discuss our project by �rst, de�ning what we mean by Spirographs,and then precede to examine the characteristics and properties of these quirkyobjects.

Abstract

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

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the children growing up in the 90s) as Spirographs. In this paper we discussthe characteristics and properties associated to Spirographs. Moreover, we willdiscuss the e�ects that rational versus irrational numbers have on the overallstructures, as well as de�ne and present parameterizations of the curvatures ofa given Spirograph.

Introduction to Curves

The Greeks de�ned a curve as the path traced out by a particle in motion. Moreprecise, the continuous map a: I→ R2, where I is some interval on R, de�nesa curve in 2-space (a similar de�nition can be expanded for a: I →Rn). Bythinking about curves in terms of time and this idea of particle's path, we canparameterize 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 di�er-entiable (i.e. smooth) if each of its coordinates are di�erentiable. As we willsoon discover, di�erentiability is not always guaranteed in Spirographs. This inturn has an e�ect on how one calculates curvature of a given curve.

Roulettes

Spirographs fall under the category of roulette curves. A roulette curve is thecurve generated by tracing the path of a point, attached to a curve, as it rollswithout slipping along another �xed curve. For our project, we looked at circlesformed by rolling a circle around another �xed circle. These types of roulettesall have speci�c names based on the location of the �xed point and whetherthe moving circle is on the inside or the outside of the �xed circle. We havea mechanism to talk about theses creatures, namely, we can parametrize thesecurves. Once again, think back to the little wheels and circles we used tomake spirogrpahs as children. The tools we used were gears with little teeth onthem that allowed the gear (i.e. a circle) to roll along the �xed circle withoutslipping. Convenient, right? But also limiting. Limiting because as the gearrolls along the outside of our circle, it is only able to turn in the same directionas 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 movementof the gear around the �xed circle. Physically, the gears (which allow rollingwithout slipping) limit us to these directions of movement, but with the magicof modern mathematics we can create a parametrization that simulates movingin the opposite way. For example, moving along the outside of a circle, the gearcould be rotating in a clockwise direction but moving along the �xed circle in acounter clockwise direction.

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Hypocycloids and Hypotrochoids and Hippopota-

muses....well actually not the latter

The curves we will generate are the hypocycloids and hypotrochoids. Thesecurves are both formed when the moving circle is rotating around the insideof the �xed circle. A hypocycloid 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. Ifr = Ri where i is some constant, then the curve produced is a simple closed curve(simple meaning there are no self intersections). In fact, the resulting curve willhave i cusps where the curve is not di�erentiable. The cusp forms where the�xed point on the smaller circle is in direct contact with the large circle. Thismakes sense because the circumference of the small circle is 2π = 2πRi and thecircumference of the large circle is 2πR which means the small circle makesprecisely i rotations to rotate around the inside of the large circle. These curvescan be parametrized with the following equations:

x(t) = (R− r)cos(t) + rcos(R−rr t)y(t) = (R− r)sin(t)− rsin(R−rr 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 hypotrochoid. Ahypotrochoid is a plane curve formed in the same way as a hypocycloid exceptthat 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 nothave to be on the smaller circle's boundary). These curves are parametrized asfollows:

x(t) = (R− r)cos(t) + dcos(R−rr t)y(t) = (R− r)sin(t)− dsin(R−rr 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, thisforms an ellipse.

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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 themoving circle is rolling in a one direction, it will move around the �xed circle inthe opposite direction. This will create Spirographs with i cusps connected bysmooth 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=3,5,8,10,20and 50 respectively.

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Now, we can parametrize our curve so that we are rotating our moving circlein 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 curvewill periodically come to a cusp. There are i-2 petals for each ��ower�. Below wecan see what happens when i is varied. Below i=3,5,8,10,20, and 50 respectively.

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Epic Cycloid and Epic Trochoids 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 aroundthe outside of the �xed circle. The objects formed are called epicycloids andepitrochoids.

Like a hypocycloid, an epicycloid is a plane curve generated by the trace ofa �xed point on the edge of a circle as it rolls along the outside of another �xedcircle. Merely the location of the smaller circle has changed. Using a similarsetup as before, let r be the radius of the moving circle and R be the radius ofthe �xed circle and i be a constant such that r = Ri . If i is a natural number,the epicycloid has i cusps that are di�erentiable. If i is rational such that i = pq ,

where pq is in simplest terms, then there will be p cusps on the curve, but thecurve will no longer be simple, it will intersect itself. Regardless, the curve willbe closed if i is rational, but if i is an irrational number, then the curve is notclosed. It will form a dense subset in the shape of an annulus with outer radiusR+2r and inner radius R.

Theses curves can be parametrized in the following way:x(t) = (R+ r)cos(t)− rcos(R+rr t)y(t) = (R+ r)sin(t)− rsin(R+rr 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 withhypotrochoids). Their parameterizations are as follows:

x(t) = (R+ r)cos(t)− dcos(R+rr t)y(t) = (R+ r)sin(t)− dsin(R+rr 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 movingaround the �xed circle in a clockwise direction. This yields spirographs that aresimilar to the hypocycloids formed when moving in the direction opposite to thenatural direction (i.e. when the rolling circle is moving in the same directionthat it is revolving). However, in this case, the cusps are now rounded cornersthat are in fact di�erentiable. Here we have i=3, 5, 8, 10, 20 and 50 respectively.

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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 inthe natural direction (i.e. when the rolling circle is moving in the same directionas it is rotating.) Again, cusps are not formed, rather there are rounded cornersthat are di�erentiable. Here i=3,5,8, 10, 20 and 50 respectively.

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A discussion on curvature

We think of curvature as the absolute value of the change in acceleration as wetravel along our curve. That is to say, as we draw out our Spirographs, thecurvature at any given point is the absolute value of how quickly we increase ordecrease speed in one direction: the greater the increase in speed the greater thecurvature 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:

κ =|α′ × α”||α′|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):

Out[7]=

2 AbsB Hr-RL2 H2 r-RL SinB R t

2 rF2

rF

JAbsAH-r + RL ICos@tD - CosA H-r+RL tr

EME2 + AbsAHr - RL ISin@tD + SinA H-r+RL tr

EME2N32

This is the curvature function for the hypocycloid moving opposite the nat-ural direction (i.e. rolling is in same direction as rotation):

Out[14]=

2 AbsB Hr-RL2 R CosBt- R t

2 rF2

rF

JAbsAH-r + RL ICos@tD + CosA H-r+RL tr

EME2 + AbsAHr - RL ISin@tD + SinA H-r+RL tr

EME2N32

This is the curvature function for the epicycloid moving in the natural di-rection (i.e. rolling is in the same direction as rotation):

Out[21]=

AbsB R J5 r2-2 r R+R2+I-r2+R2M CosBJ2- R

rN tFN

rF

JAbsAHr + RL Cos@tD + H-r + RL CosA H-r+RL tr

EE2 + AbsAHr + RL Sin@tD + H-r + RL SinA H-r+RL tr

EE2N32

This is the curvature function for the epicycloid moving opposite the naturaldirection (i.e. rolling is in the opposite direction as rotation):

Out[28]=

AbsB 2 r3-r2 R+4 r R2-R3+I2 r3-r2 R-2 r R2+R3M CosB R t

rF

rF

JAbsAHr + RL Cos@tD + Hr - RL CosA H-r+RL tr

EE2 + AbsAHr + RL Sin@tD + H-r + RL SinA H-r+RL tr

EE2N32

To help us really understand these equations, lets plot them for severalspirographs. This is where the program came in real handy, we needed only tochange parameters on the slide bar at the top. First lets look at the hypocycloidmoving in the natural direction.

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Out[8]=

number of revolutions 1

size of inner radius 25

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Out[12]=

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number of revolutions 27

size of inner radius 40.3

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We can see that as i increases, the variance in curvature increases. We canalso see the non-di�erentiable cusps where there are vertical asymptotes on thegraphs.

Now consider moving in the direction opposite the natural direction.

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number of revolutions 1

size of inner radius 20

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Again, we can see the increase in the increase in curvature (note the scalechange) and can see the non-di�erentiable 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 di�eren-tiable, the curvature appears to be asymptotic at these points simply becausethe 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

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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 partas well. Not only did having a program save us 20 dollars, it enabled the groupto take an in depth approach to researching and working with Spirographs. Ourresults, although not ground breaking, did provide some intriguing observationsregarding di�erentiability of curves and the notion of dense subsets. The dis-cussion of the characteristics and properties associated to Spirographs generatesboth 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 ofSpirographs a beautiful way to spend one's time, it also lends a great hand inunderstanding curvature of simple, closed curves.

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