Spirotechnics! September 7, Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Transcription

1 Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1

2 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome. Peiod. This simple claim was the motivation fo pusuing this topic fo ou nal poject. But once motivated the question became, what exactly do we do with Spiogaphs? Fotunately, we had some inspiation via an execise assigned to us in class: Execise (The Asteoid) along with Example that peceded it. The asteoid execise meely had us use some tigonometic identities to tansfom a paametization, but the example befoe it descibed a specic example of a spiogaph: one in which the inne cicle had a adius one-fouth the adius of the cicle in which it was otating. We pondeed: why should somebody go and spend 20 dollas on a poduct to make these fanciful cuves when in a few hous you could pogam something youself? Not having a good answe to this question and not having 20 dollas we set out to ty and genealize the asteoid example. Fom ou st tial uns, we ecognized the advantage of pogamming ove the eal-wold poduct: thee ae physical limitations to a eal spiogaph that ae easily ovecome in a pogam. Fo example, when you take a cicle of smalle adius inside of a lage cicle, the geas that link the cicles equie a specic diection of otation, wheeas in a pogam we wee able to poduce pattens as if the cicles wee fictionless sufaces passing one anothe. In essence, we ceated ou own Spiogaph univese whee physical limitations wee not a hindance. Once we had this pogam in hand, we again began to wonde: what do we do with this thing? Moe impotant: what can we do with it? The pogam was indeed a fun concept, but tuning it into a nal poject was going to take a bit longe to gue out. Though bainstoming and expeimentationand with ou pogam eady at ou ngetips we discoveed some neat popeties. 1) If = R i, i some natual numbe, then the cuve poduced is a simple, closed cuve (simple meaning no self-intesections). 2) If is ational but not of the fom R/i, the image of the path is a closed cuve that is not simple. 3) If is iational, the cuve is not closed, meaning the image of the path as n goes to innity (n being the degees of otation elative to the lage cicle) is an annulus. But do not take us simply on ou wod. Let us continue to the pape whee we will discuss ou poject by st, dening what we mean by Spiogaphs, and then pecede to examine the chaacteistics and popeties of these quiky objects. Abstact Given two cicles of abitay adii R and, x one cicle of adius R and oll the smalle cicle along the st. By doing this, we can ceate geometic stuctues called hypotochoids and epitochoidso moe commonly known (to many of 2

3 the childen gowing up in the 90s) as Spiogaphs. In this pape we discuss the chaacteistics and popeties associated to Spiogaphs. Moeove, we will discuss the eects that ational vesus iational numbes have on the oveall stuctues, as well as dene and pesent paameteizations of the cuvatues of a given Spiogaph. Intoduction to Cuves The Geeks dened a cuve as the path taced out by a paticle in motion. Moe pecise, the continuous map a: I R2, whee I is some inteval on R, denes a cuve in 2-space (a simila denition can be expanded fo a: I Rn). By thinking about cuves in tems of time and this idea of paticle's path, we can paameteize the cuve a such that, fo t in I, a(. = (a1(,a2(,a3() whee each component ai: I R, is also a function. In addition, a is dieentiable (i.e. smooth) if each of its coodinates ae dieentiable. As we will soon discove, dieentiability is not always guaanteed in Spiogaphs. This in tun has an eect on how one calculates cuvatue of a given cuve. Roulettes Spiogaphs fall unde the categoy of oulette cuves. A oulette cuve is the cuve geneated by tacing the path of a point, attached to a cuve, as it olls without slipping along anothe xed cuve. Fo ou poject, we looked at cicles fomed by olling a cicle aound anothe xed cicle. These types of oulettes all have specic names based on the location of the xed point and whethe the moving cicle is on the inside o the outside of the xed cicle. We have a mechanism to talk about theses ceatues, namely, we can paametize these cuves. Once again, think back to the little wheels and cicles we used to make spiogpahs as childen. The tools we used wee geas with little teeth on them that allowed the gea (i.e. a cicle) to oll along the xed cicle without slipping. Convenient, ight? But also limiting. Limiting because as the gea olls along the outside of ou cicle, it is only able to tun in the same diection as it is moving aound the xed cicle. If we wee moving along the inside, the gea would be tuning in the opposite diection of the oveall movement of the gea aound the xed cicle. Physically, the geas (which allow olling without slipping) limit us to these diections of movement, but with the magic of moden mathematics we can ceate a paametization that simulates moving in the opposite way. Fo example, moving along the outside of a cicle, the gea could be otating in a clockwise diection but moving along the xed cicle in a counte clockwise diection. 3

4 Hypocycloids and Hypotochoids and Hippopotamuses...well actually not the latte The cuves we will geneate ae the hypocycloids and hypotochoids. These cuves ae both fomed when the moving cicle is otating aound the inside of the xed cicle. A hypocycloid is a plane cuve geneated by the tace of a xed point on a smalle cicle which is olling along the inside of a lage cicle. Let be the adius of the moving cicle and R the adius of the xed cicle. If = R i whee i is some constant, then the cuve poduced is a simple closed cuve (simple meaning thee ae no self intesections). In fact, the esulting cuve will have i cusps whee the cuve is not dieentiable. The cusp foms whee the xed point on the smalle cicle is in diect contact with the lage cicle. This makes sense because the cicumfeence of the small cicle is 2π = 2πR i and the cicumfeence of the lage cicle is 2πR which means the small cicle makes pecisely i otations to otate aound the inside of the lage cicle. These cuves can be paametized with the following equations: x( = (R )cos( + cos( R y( = (R )sin( sin( R and will look something like this (depending on the specs) Only the othe hand, we have the hypocycloid's sibling, the hypotochoid. A hypotochoid is a plane cuve fomed in the same way as a hypocycloid except that the xed point is a distance of d away fom the cente of the moving cicle (i.e. the point may lie on the inside o outside of the smalle cicle, it does not have to be on the smalle cicle's bounday). These cuves ae paametized as follows: x( = (R )cos( + dcos( R y( = (R )sin( dsin( R A few wothy things to note. If d <, the spiogaph will fom cusps, but if d >, we will get loops instead of cusps. Futhe, if R=2nd/(n+1) and =(n- 1)d/(n+1) whee n is some natual numbe, we get a ose. When R=2, this foms an ellipse. 4

5 Although this is neat, fo ou poject, we focused on playing with the hypocycloids. Fo these cuves the natual diection of movement is such that the moving cicle is olling in a one diection, it will move aound the xed cicle in the opposite diection. This will ceate Spiogaphs with i cusps connected by smooth cuves that ae the opposite concavity to the edge of the xed cicle, simila to a kapow! shape in comic books. As you can see hee: i=3,5,8,10,20 and espectively. Now, we can paametize ou cuve so that we ae otating ou moving cicle in the same diection as we ae moving aound the xed cicle. When i is an intege, this will ceate a owe type shape, whee the cuve will peiodically come to a cusp. Thee ae i-2 petals fo each owe. Below we can see what happens when i is vaied. Below i=3,5,8,10,20, and espectively. 5

6 Epic Cycloid and Epic Tochoids and Epic Deltoids (because shoulde stength is key to dawing spiogaphs): Ou next focus is about what happens when we otate ou moving cicle aound the outside of the xed cicle. The objects fomed ae called epicycloids and epitochoids. Like a hypocycloid, an epicycloid is a plane cuve geneated by the tace of a xed point on the edge of a cicle as it olls along the outside of anothe xed cicle. Meely the location of the smalle cicle has changed. Using a simila setup as befoe, let be the adius of the moving cicle and R be the adius of the xed cicle and i be a constant such that = R i. If i is a natual numbe, the epicycloid has i cusps that ae dieentiable. If i is ational such that i = p q, whee p q is in simplest tems, then thee will be p cusps on the cuve, but the cuve will no longe be simple, it will intesect itself. Regadless, the cuve will be closed if i is ational, but if i is an iational numbe, then the cuve is not closed. It will fom a dense subset in the shape of an annulus with oute adius R+2 and inne adius R. Theses cuves can be paametized in the following way: x( = (R + )cos( cos( R+ y( = (R + )sin( sin( R+ Onto ou next candidate, epitochoids ae fomed in a simila manne as epicycloids except that the xed point that taces out the cuve is at a distance d fom the cente on the moving cicle (ecall this same scenaio with hypotochoids). Thei paameteizations ae as follows: x( = (R + )cos( dcos( R+ y( = (R + )sin( dsin( R+ Again, we focused in pimaily on the epicycloids fo ou poject. Natually (i.e. given the constaint of the diection the geas allow us to move) if the moving cicle is olling in a clockwise diection, it will be moving aound the xed cicle in a clockwise diection. This yields spiogaphs that ae simila to the hypocycloids fomed when moving in the diection opposite to the natual diection (i.e. when the olling cicle is moving in the same diection that it is evolving). Howeve, in this case, the cusps ae now ounded cones that ae in fact dieentiable. Hee we have i=3, 5, 8, 10, 20 and espectively. 6

7 1 1 1 When the olling cicle moves in the opposite diection of otation, we futhemoe nd that this epicycloid is simila to the hypocycloid when moving in the natual diection (i.e. when the olling cicle is moving in the same diection as it is otating.) Again, cusps ae not fomed, athe thee ae ounded cones that ae dieentiable. Hee i=3,5,8, 10, 20 and espectively

8 A discussion on cuvatue We think of cuvatue as the absolute value of the change in acceleation as we tavel along ou cuve. That is to say, as we daw out ou Spiogaphs, the cuvatue at any given point is the absolute value of how quickly we incease o decease speed in one diection: the geate the incease in speed the geate the cuvatue and the opposite holds tue fo a decease in speed. Moe fomally, we can paameteize cuvatue fo each of ou discussed Spiogaphs. The cuvatue was calculated using the following fomula: κ = α α α 3 whee αis ou cuve. This is the cuvatue function fo the hypocycloid moving in the natual diection (i.e. olling is in the opposite diection as otation): Out[7]= Abs R Cost Cos R t 2 Abs R2 2 R Sin R t Abs R Sint Sin R t 2 32 This is the cuvatue function fo the hypocycloid moving opposite the natual diection (i.e. olling is in same diection as otation): Out[14]= Abs R Cost Cos R t 2 Abs R2 R Cost R t Abs R Sint Sin R t 2 32 This is the cuvatue function fo the epicycloid moving in the natual diection (i.e. olling is in the same diection as otation): Out[21]= Abs R Cost R Cos R t Abs R RR 2 2 R 2 Cos2 R t 2 Abs R Sint R Sin R t 2 32 This is the cuvatue function fo the epicycloid moving opposite the natual diection (i.e. olling is in the opposite diection as otation): Out[28]= Abs R Cost R Cos R t Abs R4 R 2 R R2 R 2 R 3 Cos R t 2 Abs R Sint R Sin R t 2 32 To help us eally undestand these equations, lets plot them fo seveal spiogaphs. This is whee the pogam came in eal handy, we needed only to change paametes on the slide ba at the top. Fist lets look at the hypocycloid moving in the natual diection. 8

9 numbe of evolutions 1 size of inne adius 25 Out[8]= numbe of evolutions 1 size of inne adius 10 Out[12]= numbe of evolutions 27 size of inne adius 40.3 Out[7]= 0 1 9

10 We can see that as i inceases, the vaiance in cuvatue inceases. We can also see the non-dieentiable cusps whee thee ae vetical asymptotes on the gaphs. Now conside moving in the diection opposite the natual diection. numbe of evolutions 1 size of inne adius 20 Out[39]= numbe of evolutions 1 size of inne adius 10 Out[44]= 10

11 numbe of evolutions 27 size of inne adius 40.3 Out[6]= 0 1 Again, we can see the incease in the incease in cuvatue (note the scale change) and can see the non-dieentiable cusps. Now lets look at the epicycloids. Fist we will move in the natual diection: numbe of evolutions 1 size of oute adius 20 Out[70]= 11

12 numbe of evolutions 1 size of oute adius 10 Out[75]= numbe of evolutions 9 size of oute adius Out[78]= Inteestingly we can see the cuvatue functions hee dip down to zeo appoaching and leaving the cone. Though we know the cones ae dieentiable, the cuvatue appeas to be asymptotic at these points simply because the cuvatue is so geat. Finally lets look at the epicycloid moving opposite to the natual diection. 12

13 numbe of evolutions 1 size of oute adius 20 Out[99]= numbe of evolutions 1 size of oute adius 10 Out[104]= numbe of evolutions 10 size of oute adius Out[109]= We nd a simila patten hee whee the cuvatue goes to zeo at points 13

14 appoaching and leaving the cones, which ae aeas of nite, but lage cuvatue. We can especially see that the cuvatue is not asymptotic in the last gue. Summay The pogamming associated with this poject was an incedibly insightful pat as well. Not only did having a pogam save us 20 dollas, it enabled the goup to take an in depth appoach to eseaching and woking with Spiogaphs. Ou esults, although not gound beaking, did povide some intiguing obsevations egading dieentiability of cuves and the notion of dense subsets. The discussion of the chaacteistics and popeties associated to Spiogaphs geneates both inteesting images as well as pesents possibilities fo futhe investigations. Fo example, what occus if you otate within an ellipse? O upon a closed Mobius stip? Ou goup feels it safe to assume that, not only is the ceation of Spiogaphs a beautiful way to spend one's time, it also lends a geat hand in undestanding cuvatue of simple, closed cuves. 14