Spirotechnics!
September 7, 2011
Amanda Zeringue, Michael Spannuth and Amanda Zeringue
Dierential Geometry Project
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The Beginning
The general consensus of our group began with one thought: Spirographs 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 transform a parametrization, but the example before it described a specic example
of a spirograph: one in which the inner circle 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 circles 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 , 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 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 hypotrochoids and epitrochoidsor more commonly known (to many of
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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
a such that, for
.
a(t)= (a1(t),a2(t),a3(t))
where each component ai: I →R, is
parameterize the curve
t in I,
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 roulette 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.
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Hypocycloids and Hypotrochoids and Hippopotamuses....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 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. If
R
i 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 will
r=
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
2πR
i and the
which means the small circle makes
makes sense because the circumference of the small circle is
circumference of the large circle is
2πR
2π =
precisely i rotations to rotate around the inside of the large circle. These curves
can be parametrized with the following equations:
x(t) = (R − r)cos(t) + rcos( R−r
r t)
y(t) = (R − r)sin(t) − rsin( R−r
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 hypotrochoid. 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:
x(t) = (R − r)cos(t) + dcos( R−r
r t)
y(t) = (R − r)sin(t) − dsin( R−r
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=(n1)d/(n+1) where n is some natural number, we get a rose. When R=2r, this
forms an ellipse.
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Although this is neat, for our project, we focused on playing with the hypocycloids.
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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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 Cycloid and Epic Trochoids and Epic Deltoids (because 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 epicycloids and
epitrochoids.
Like a hypocycloid, an epicycloid 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
R
i . If i is a natural number,
p
the epicycloid has i cusps that are dierentiable. If i is rational such that i = ,
q
p
where
q is in simplest terms, then there will be p cusps on the curve, but the
curve will no longer be simple, it will intersect itself. Regardless, the curve will
the xed circle and i be a constant such that
r=
be closed if i is rational, but if i is an irrational number, 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:
x(t) = (R + r)cos(t) − rcos( R+r
r t)
y(t) = (R + r)sin(t) − rsin( R+r
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 distance d from the center on the moving circle (recall this same scenario with
hypotrochoids). Their parameterizations are as follows:
x(t) = (R + r)cos(t) − dcos( R+r
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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When the rolling circle moves in the opposite direction of rotation, we furthermore 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.
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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:
κ=
where
αis
|α0 × α”|
|α0 |3
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):
2 AbsB
Out[7]=
JAbsAH-r + RL ICos@tD - CosA
Hr-RL2 H2 r-RL SinB
Rt
2r
F
2
r
2
H-r+RL t
EME
r
F
+ AbsAHr - RL ISin@tD + SinA
2 32
H-r+RL t
EME N
r
This is the curvature function for the hypocycloid moving opposite the natural direction (i.e. rolling is in same direction as rotation):
2 AbsB
Out[14]=
JAbsAH-r + RL ICos@tD + CosA
Hr-RL2 R CosBtr
2
H-r+RL t
EME
r
Rt
2r
F
2
F
+ AbsAHr - RL ISin@tD + SinA
2 32
H-r+RL t
EME N
r
This is the curvature function for the epicycloid moving in the natural direction (i.e. rolling is in the same direction as rotation):
R J5 r2 -2 r R+R2 +I-r2 +R2 M CosBJ2- r N tFN
R
AbsB
Out[21]=
JAbsAHr + RL Cos@tD + H-r + RL
r
2
H-r+RL t
CosA
EE
r
F
+ AbsAHr + RL Sin@tD + H-r + RL SinA
2 32
H-r+RL t
EE N
r
This is the curvature function for the epicycloid moving opposite the natural
direction (i.e. rolling is in the opposite direction as rotation):
AbsB
Out[28]=
2 r3 -r2 R+4 r R2 -R3 +I2 r3 -r2 R-2 r R2 +R3 M CosB
JAbsAHr + RL Cos@tD + Hr - RL CosA
r
2
H-r+RL t
EE
r
Rt
r
F
F
+ AbsAHr + RL Sin@tD + H-r + RL SinA
2 32
H-r+RL t
EE N
r
To help us really understand these equations, lets plot them for several
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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number of revolutions
1
size of inner radius
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Out[8]=
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number of revolutions
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Out[12]=
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number of revolutions
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size of inner radius
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0.04
Out[7]=
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0
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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.
number of revolutions
1
size of inner radius
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Out[39]=
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Out[44]=
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number of revolutions
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size of inner radius
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Out[6]=
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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:
number of revolutions
1
size of outer radius
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0.05
0.04
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Out[70]=
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Out[78]=
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Interestingly we can see the curvature functions here dip down to zero approaching 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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number of revolutions
1
size of outer radius
20
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0.05
0.04
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Out[99]=
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Out[104]=
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number of revolutions
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size of outer radius
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Out[109]=
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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 curvature. 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 discussion 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 Mobius 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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