COMPUTATIO NAL DI VERSION S

Elliptical Wheels

Michael Eisenberg

Published online: 18 May 2013� Springer Science+Business Media Dordrecht 2013

As you’ve probably noticed, this special issue of Technology, Knowledge, and Learning is

devoted to bicycle-related topics; and in keeping with that theme, this installment of the

computational diversions column will focus on rolling wheels. Not just any sort of rolling

wheels, however; instead of looking at plain ordinary circular wheels, we’ll expand our

horizons to explore the wonderful world of elliptical wheels.

Actually, ‘wonderful’’ might not be the right word here; ‘‘dubious’’, ‘‘ill-advised’’, or

maybe even ‘‘disastrous’’ could be more appropriate. There’s a very good reason that wheels

are circular, and that reason was expertly articulated by the nonsense poet Gelett Burgess

(1866–1951). Burgess is today best remembered for his little verse about the purple cow,1

but many of his poems and drawings display a keen sense of geometric or physical absurdity

[perhaps not too surprising since–according to his Wikipedia (2013) entry–he was a grad-

uate of MIT]. The limerick that he wrote on the subject of elliptical wheels goes as follows:

Remarkable truly, is Art!

See—Elliptical wheels on a Cart!

It looks very fair

In the Picture up there;

But imagine the Ride when you start!

Before proceeding, you might want to ponder that notion: elliptical wheels on a cart. It’s

perhaps a bit difficult for the reader at this juncture to see just how intensely funny

Burgess’ idea is. The poem quoted above was retrieved from the book The Wit and Humor

of America (Wilder 1907, p. 1245), available as a Project Gutenberg e-book on the Web;

unfortunately, that book does not include accompanying pictures, but only text. In the case

of this particular poem, the accompanying pictures (also by Burgess) count for a great deal.

You can see the pictures in a different compilation, The Burgess Nonsense Book (Burgess

M. Eisenberg (&)University of Colorado, Boulder, CO, USAe-mail: duck@cs.colorado.edu; ijcml-diversions@ccl.northwestern.edu

1 ‘‘I never saw a purple cow, I never hope to see one; But I can tell you anyhow, I’d rather see than beone!’’ [1, pp. 24–25].

123

Tech Know Learn (2013) 18:95–101DOI 10.1007/s10758-013-9205-1

1901, p. 31), which I found among the scanned books on the Google site; but I don’t want

to risk copyright violation by inserting Burgess’ sketches in this column.

The pictures are, in fact, absolutely hilarious (at least in my view), and the reader is

heartily recommended to view them at the website referenced above, as an incentive to

what follows in this diversion column. The initial sketch (‘‘the Picture up there’’, in

Burgess’ phrase) looks calm enough. A little man–a typical blobby Burgess figure–is

seated in the driver’s position of a horse-drawn cart; the cart has two pairs of elliptical

wheels, one (shorter) pair in front and one (taller) pair at the rear. This is followed by seven

smaller rough sketches of the cart as it moves forward—a total nightmare of jolts and

bruises for both horse and man (‘‘imagine the Ride when you start!’’).

In the remainder of this column, I’m going to try and make a virtue out of copyright

restrictions, and explore how we can use a few geometric ideas to create diagrams of a

rolling elliptical wheel. In effect, this column is an attempt to recreate–via a bit of geo-

metric programming–the brilliant visual intuition of Burgess’ sketches. Maybe I can’t

insert Burgess’ original artwork here, but we can at least work out the key ideas for making

an accurate graphical representation of a rolling ellipse. Once we see the diagrammatic

disadvantages of the idea, we can return to a final consideration of Burgess’ limerick.

1 Step 1. Creating a Set of Points to Represent an Ellipse

Our first step in creating a diagram of a rolling ellipse is to represent (in some standard

fashion) the shape that we wish to depict. Our convention in this column will be to create

an initial set of points that approximate, to any desired precision, the perimeter of the

ellipse, centered at the origin; the major axis of the ellipse will be along the x-axis and the

minor axis will be along the y-axis. Here, in pseudocode, is the basic idea:

This particular example, as illustrated in Fig. 1, will create an ellipse whose left and

right extremes are the points (-2, 0) and (2, 0), and whose upper and lower extremes are

the points (1, 0) and (-1, 0).

An examination of this pseudocode will reveal that every point that we create does

indeed conform to the standard algebraic definition of (some particular) ellipse; in our

example, the points all satisfy the equation:

ðx2=4Þ þ ðy2=1Þ ¼ 1

96 M. Eisenberg

123

One final note: our points, as you can see from the pseudocode, are evenly spaced

around the ellipse not by their distance from each other, but by the angle of the vector to

each point from the origin.

2 Step 2. Drawing an Ellipse ‘‘on the Ground’’

Now that we have a basic elliptical shape to work with, our next job—really the key job in

this entire project—is determining how to draw the ellipse as it rolls into different ori-

entations ‘‘on the ground’’. For example, suppose we consider the x-axis to be our ‘‘ground

level’’: how do we draw successive versions of the ellipse to produce a faithful repre-

sentation of the shape as it rolls along the ground?

To anticipate: we’re trying to draw successive ellipses so that we can produce a drawing

like Fig. 2 below. In Fig. 2, we see a succession of drawings of the basic ellipse that we

created earlier, rolling from left to right. At the left position of the figure, our ellipse is

‘‘resting on’’ the point that was originally the point (2, 0) in our standard ellipse; that is, we

have positioned the standard shape so that its rightmost point is now sitting on the ground.

As the ellipse rolls toward the right of Fig. 2, the ‘‘ground contact point’’ of the ellipse

changes to correspond to a counterclockwise circuit around the original standard shape. For

example, the second shape (from the left) in Fig. 2 represents ‘‘rolling’’ the original shape

so that its 25th point [counting in a counterclockwise direction from the point (2, 0)] is now

the one resting on the ground. The third shape from the left in Fig. 2 represents a further

roll, so that the 50th point in our set is now resting on the ground.

Let’s consider, then, how to draw the leftmost ellipse in Fig. 2. We wish to ‘‘start’’ the

rolling ellipse, resting on its rightmost point, at a particular position along the x-axis (let’s say,

the origin). In order to do this, we have to apply a linear map to the set of points in our standard

ellipse: we have to (a) rotate the ellipse so that its rightmost point is now at the bottom of the

shape, and (b) move the ellipse so that the (new) bottom point is positioned at the origin.

Okay, how should we rotate the original ellipse so that the point (2, 0) is now at the

bottom of the ellipse? There are several ways of thinking about this, but it’ll prove handy to

imagine rotating the ellipse around the point (2, 0) itself; if we rotate the standard ellipse by

Fig. 1 Our standard ellipse is centered at the origin, with its major axis along the x-axis and minor axisalong the y-axis. In the example that we use for our experiments, we have an ellipse with a semi-major axisof 2 [the rightmost point is (2, 0)] and a semi-minor axis of 1. We will represent our ellipse by a set of points,as suggested by the dotted line representation of the perimeter in the figure (though the dots in the figure arenot identical to the points generated by our pseudocode)

Elliptical Wheels 97

123

90 degrees clockwise around the point (2, 0), we end up with a shape in which the point (2,

0) is unmoved, but the other points have been rotated above it. (See Fig. 3.)

Now that we’ve rotated the standard ellipse so that it is ‘‘sitting on’’ the point (2, 0), we

can translate the entire set of points by 2 units leftward. Once we’ve done that, our original

standard ellipse has been rotated-and-shifted so that it is sitting on its rightmost point,

which is now located at the origin.

Let’s pause now to consider the general pattern in what we’re doing. There’s a recipe

lurking here for placing our standard ellipse so that it is resting on any given point,

positioned anywhere along the x-axis. Here’s the recipe:

Let (x, y) be a given point in our standard ellipse. Suppose we want this point to be the

bottom-most point of a rolling ellipse, positioned at location (x’, 0) on the horizontal

axis. The way we do this is (a) first rotate the standard ellipse around the point (x, y) so

that the tangent line to the ellipse is horizontal; this in effect rotates the ellipse until it is

‘‘sitting on’’ the point (x, y). Then we (b) translate the entire set of points by (x0–x, -y)

so that the newly-rotated shape is placed at the right spot on the horizontal axis.

3 Step 3. Drawing a Sequence of Ellipses on the Ground

In Step 2 above, we saw how to draw our standard ellipse so that any chosen perimeter

point is now positioned as the ‘‘resting point’’ at the location (x, 0). What we now wish to

Fig. 2 The ellipse of Fig. 1, in a series of snapshots, rolling from left to right along a horizontal line. In thestarting (leftmost) position, the ellipse is ‘‘resting on’’ its original rightmost point, now in contact with theground. As the ellipse rolls, the contact point of the original shape changes; and the position of each newcontact point on the horizontal line moves along from left to right to reflect the rolling of the entire shape

Fig. 3 Rotating our set of ellipse points so that a particular chosen point becomes the bottom-most(‘‘contact’’) point. In this case, we wish the rightmost point (2, 0) of the original shape to become the contactpoint; so we rotate the ellipse around that point. The amount by which we rotate the ellipse in this case is 90degrees clockwise; this rotation is chosen so that the (vertical) tangent to the original ellipse at (2, 0) is nowturned so that it is horizontal

98 M. Eisenberg

123

do is to create a sequence of such rotated ellipses to produce a diagram such as Fig. 2. That

is, we want to draw our ellipse in a sequence of positions, from left to right, corresponding

to a ‘‘rolling’’ version of the figure.

I actually struggled a bit with this step, trying to find some nice closed algebraic formula

for shifting drawings rightward along the x-axis, corresponding to a rolling ellipse.

Eventually, I decided on a ‘‘brute force’’ solution: we can calculate how far the ellipse has

rolled along the ground by summing the distances between its perimeter points, starting

from the original bottom point. In other words, if the ellipse rolls from the left position in

Fig. 2 to the second-from-left position, the contact point should be shifted rightward by a

distance equal to the sum of distances between neighboring perimeter points (in this case,

starting from the 0th point and continuing up to the 25th point). This isn’t an exact value

for the distance rolled by the ellipse, but if we have a lot of points in our original

representation it’ll be close.

4 Summing up

So now, to summarize our entire plan:

1. To draw a rolling ellipse, we first create a ‘‘standard’’ version of the ellipse composed

of (a large number) of perimeter points, centered at the origin and with major axis

along the x-axis and minor axis along the y-axis.

2. Given a point (x, y) in our standard ellipse, we can rotate the shape so that (x, y) is at

the bottom of the shape. This makes (x, y) a ‘‘resting point’’ for the shape.

3. We can now translate our rotated ellipse so that the ‘‘resting point’’ is now positioned

anywhere along the ‘‘ground’’ (for simplicity’s sake, we’ll use the x-axis as our

‘‘ground’’).

4. We draw a sequence of rotated ellipses along the ground, positioning the sequence of

contact points so that they reflect rightward movement. The net rightward movement

of each contact point is determined by calculating (approximately) how much of the

ellipse perimeter has now been traversed as we rolled the shape from the original

orientation.

5 Rolling Ellipses

We can now put our plan into effect, to see what happens when an elliptical wheel rolls

along the ground. Figure 4 shows several examples, varying according to the ratio between

major and minor axes of the ellipse. At top, a circle is shown rolling from left to right (the

major/minor ratio is 1); at center, our Fig. 1 ellipse (with major/minor ratio of 2) is shown

rolling; and at bottom, an ellipse with major/minor axis of 3 is shown.

Studying Fig. 4 gives us a more exact sense of the problem to which Burgess was

originally referring. When a circle rolls along the ground, its center moves in a straight

line (from left to right); but when an ellipse rolls along the ground, the center of the

shape oscillates. Figure 5 repeats the snapshots of Fig. 4, but this time highlighting the

path of the center of each shape as it rolls. If you try to attach an elliptical wheel to a

cart (with the axle attached to the center of the wheel), you’ll get an awfully bumpy ride,

just as Burgess described. Indeed, the cart in the poem has an even worse problem, since

Elliptical Wheels 99

123

it has two pairs of elliptical wheels of different sizes; thus, the oscillations of the front

pair of wheels aren’t identical to those of the back pair of wheels, and so the tilt of the

overall cart keeps changing as the vehicle moves.

Fig. 4 Rolling ellipses. At top, snapshots of a rolling circle; center, snapshots of a rolling ellipse withmajor/minor axis ratio of 2:1; and at bottom snapshots of a more elongated rolling ellipse with major/minoraxis ratio of 3:1

Fig. 5 Following the path of the center of a rolling ellipse. The thick line shows successive positions of ourellipses as the contact point changes. At top, the center moves along a straight line; as the ellipse becomesmore elongated, the path of the center oscillates between greater extremes of height

100 M. Eisenberg

123

As is customary in this column, the purpose of our exploration here is only to provide a

few first steps in what might be future investigations for the reader. For example, the reader

might wish to see if there is a closed form expression for the path taken by the center of a

rolling ellipse. (The paths in Fig. 5 look to me rather like cycloids, but I’m not positive.)

One might try to produce an animation of the rolling ellipse to get a sense of velocities: for

instance, let’s assume that the successive contact points of the ellipse move from left to

right at a constant velocity. In that case, what is the velocity, as the rolling proceeds, of the

geometric center of the shape? (And there are other points of interest: what about the path

taken by the focal points of the ellipse? Or a path taken by a particular chosen point on the

perimeter?) The ambitious reader who wishes to pay special honor to the memory of the

talented Gelett Burgess might even create an animation of a cart with elliptical wheels: a

bumpy ride for the driver, but a fun one for the programmer. Ideas or implementations

for follow-up projects, as always, can be sent to this column at the email address:

ijcml-diversions@ccl.northwestern.edu.

References

Burgess, G. (1901). The Burgess Nonsense Book. New York: Frederick A. Stokes. (Available for viewing onthe Web via books.google.com).

Wikipedia entry for Gelett Burgess: http://en.wikipedia.org/wiki/Gelett_Burgess (Retrieved April 25, 2013).Wilder, M. P. (1907). The Wit and Humor of America (Vol. VII) New York: Funk and Wagnalls. Available

on the Web via Project Gutenberg (http://www.gutenberg.org/).

Elliptical Wheels 101

123