Roads and Wheels, Roulettes and Pedals
Fred Kuczmarski
Abstract. We revisit the idea of road-wheel pairs, first introduced 50 years ago by Gerson
Robison and later popularized by Stan Wagon and his square-wheeled tricycle. We show how
to generate such pairs geometrically: the road as a roulette curve and the wheel as a pedal
curve. Along the way we gain geometric insight into two theorems proved by Jakob Steiner
relating the area and arc length of a roulette to those of a corresponding pedal. Finally, we use
our results to generate parabolas, ellipses, and sine curves as roulettes.
1. INTRODUCTION. In an article in Crelle’s Journal in 1838 Jakob Steiner published two theorems relating the arc length and area of a roulette curve to those of a
corresponding pedal curve. In 1960, G. Robison [6] introduced the idea of rockers and
rollers: pairs of curves, a rocker and a roller, or a road and a wheel, such that the axle
of the wheel moves in a straight line while the wheel rolls on the road without slipping. In this article we give a geometric method of generating road-wheel pairs and
find a surprising connection with Steiner’s theorems. We then use the results in [3] to
generate familiar curves as roulettes in unusual ways. But first, we review some basic
facts, interesting in their own right, about pedals and roulettes.
2. ROULETTE CURVES. As one curve, C , rolls on another without slipping, the
locus of a point P that maintains a fixed position relative to C is called a roulette. We
focus on roulettes generated by curves rolling on straight lines. For example, a cycloid
is the roulette traced by a point P on the circumference of a circle as the circle rolls on
a line. If the point P starts at the origin and moves with a unit circle rolling along the
x-axis, then a parameterization of the cycloid is given by
(x, y) = (θ − sin θ, 1 − cos θ ),
(1)
where θ is the angle through which the wheel has turned. That y = d x/dθ in this parameterization is not a coincidence, as the following lemma shows. But before stating
the lemma, we wish to be sure that we can roll C smoothly on a line with a constant angular velocity. To this end, we require C to be a continuously differentiable plane curve
such that the angle of rotation of its tangent lines, as measured relative to some initial
position, is a strictly monotonic function of arc length. We call such curves rollable.
The monotonic condition implies that rollable curves have no inflection points, while
the strictness of the monotonicity precludes rollable curves from containing line segments. We invite the reader to consider generalizing the results in this paper to include
curves with inflection points as well as piecewise differentiable curves.
We assume throughout the paper that C is rollable. In the following lemma, and
throughout the paper, we let θ denote the angle of rotation, relative to some starting
position, of the curve C rolling on the x-axis. We take θ > 0 to correspond to a clockwise rotation.
Roulette Lemma. Let C be a rollable curve and let R be the roulette traced by a
point P moving with C as C rolls on the x-axis. If (x, y) = ( f (θ ), g(θ )) is a parameterization of R in terms of the angle of rotation of C , then f and g are continuously
differentiable and g(θ) = f 0 (θ).
doi:10.4169/amer.math.monthly.118.06.479
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Proof. The assumption that C is rollable ensures that we can parameterize R in terms
of θ , for as C rolls on the x-axis the sense of rotation of C will not change and θ will
increase. Let A be the point of contact between C and the x-axis, as shown in Figure 1.
Since A is the instantaneous center of rotation of C , the differential displacement of P
−→
is perpendicular to A P and has magnitude r dθ , where r = | A P|. So if ρ denotes the
−→
angle that A P makes with the negative x-axis and P has coordinates (x, y),
(d x, dy) = r dθ (sin ρ, cos ρ).
(2)
Since C contains no line segments, the instantaneous center of rotation moves continuously in the direction of the positive x-axis as C rolls on the axis, and the x-coordinate
of point A is a continuous function of θ . Also, since C is continuously differentiable, r
and ρ are continuous functions of θ , and hence f and g are continuously differentiable.
Finally, f 0 (θ ) = r sin ρ = g(θ).
y
R
P(x, y)
C
r
A0
A
x
Figure 1. The roulette lemma.
One immediate consequence of the roulette lemma is that the tracing point moves
to the right, that is d x/dθ > 0, precisely when it is above the x-axis. Or put another
way, the horizontal motion of the tracing point changes direction when the tracing
point crosses the x-axis. Thus, the vertical tangents to the roulette curve occur at the
x-intercepts. The case of a trochoid, traced by a point exterior to a circle as the circle
rolls on a line, nicely illustrates these ideas and is shown in Figure 2.
Figure 2. A trochoid.
We can use the lemma to parameterize roulette curves by expressing the ycoordinate of P in terms of θ and then integrating to determine the x-coordinate.
We assume the initial position of P, which we denote as P0 , is on the y-axis, and we
let y0 denote the y-coordinate of P0 . The y-coordinate of P after C has turned through
the angle θ is the signed distance, g(θ), from P0 to the tangent line to C at A0 , the
point of C that will come in contact with the x-axis after C has turned through the
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angle θ . This is shown in Figure 3, where A00 is the foot of the perpendicular from P0
to the tangent line. Figure 4 shows the curve after it has turned through the angle θ ,
where the points A and A0 are the rotated images of A0 and A00 , respectively. To ensure
that the y-coordinate of P has the correct sign, we define the signed distance to be
−−→
positive if the angle between the vector A00 P0 and the positive x-axis is θ + π/2, and
to be negative if the angle is θ − π/2. Then a parameterization of the roulette R is
given by
(x, y) =
Z
θ
(3)
g(t) dt, g(θ ) .
0
y
y
P0
g( )
A00
P(x, y)
C
g( )
C
A0
x
A0
A
Figure 3. Parameterizing a roulette, part I.
x
Figure 4. Parameterizing a roulette, part II.
We should point out that the standard parameterization of a roulette is usually of the
form
(x, y) = (s − r cos ρ, r sin ρ),
where r and ρ are as in Figure 1 and s is the arc length of C between A and the
point of C that was in contact with the origin. But by using (3) we avoid the arc length
computation. As an example, we give a short proof that the roulette traced by the focus
of a parabola rolling on a line is a catenary. In Figure 5 the curve C in its initial position
is the parabola y = x 2 /4 and P0 is the focus, (0, 1). A simple computation shows that
A00 is the point of intersection of the tangent line and the x-axis. Thus, g(θ ) = sec θ ,
y
A0
P0
A00
x
Figure 5. Parameterizing the roulette generated by the focus of a parabola.
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and a parameterization of the roulette is given by
(x, y) = (ln(sec θ + tan θ ), sec θ) .
It follows that the roulette generated by the focus of the parabola y = x 2 /4 rolling on
the x-axis is the catenary y = cosh x. See [1] for another proof.
3. PEDAL CURVES. Let P be a point in the plane of C . The pedal curve of C with
respect to P, which we denote by C P , is the set of the feet of all perpendiculars from
P to the tangent lines of C . The idea of a pedal curve is contained in Figure 3, where
the pedal, C P0 , is the set of points A00 as A0 varies along C . If we place the pole of
a polar coordinate system at P0 and let the polar axis point in the direction of the
negative y-axis in Figure 3, then a polar equation for C P0 is r = g(θ ). Thus, in (3), the
y-coordinate of the roulette is the polar radius of C P0 .
We use the notation of Figure 3 throughout, where capital letters denote points on
a curve and their primed counterparts denote the corresponding points on the pedal
curve. For now we give four examples.
1. Let C be the circle r = 2 cos θ and let P be the pole of the polar coordinate
system. Figure 6 illustrates that the cardioid r = 1 + cos θ is the pedal of C with
respect to P. To see why, note that since 1PQ0 D ∼ 1OQD and OD = sec θ ,
it follows that r = PQ0 = 1 + cos θ. Alternatively, we could have concluded
directly from the expression for the y-coordinate of the cycloid in (1) that the
pedal of the circle r = −2 cos θ with respect to the pole is the cardioid r =
1 − cos θ .
Q0
Q
1
P
1
O
D
C
CP
Figure 6. A cardioid as a pedal of a circle.
2. The pedal of a parabola with respect to its focus is the tangent line to the parabola
at the vertex (see Figure 5).
3. The pedal of an ellipse with respect to a focus is the circle having the major axis
of the ellipse as a diameter. See [5, p. 13] for a proof.
4. In a degenerate case, where the curve C is a point Q, the family of tangents to C
is the pencil of lines through Q and C P is the circle with diameter PQ.
A curve geometrically similar to C P , but twice as large, may be obtained by reflecting P across each of the tangent lines of C . The set of reflected points P 0 is called
the orthotomic of C with respect to P (see [5, p. 153]). The orthotomic may also be
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generated as the roulette traced by P 0 as it moves with a mirror image of C rolling on
C in such a way that the mirror image is the reflection of C about the tangent line at the
point of contact. This is illustrated in Figure 7, where C 0 is the reflection of C across
the tangent at A. We mention this to prove what we call the angle property of pedal
curves: the segments PA and PA0 from P to a point A on C and the corresponding point
A0 on C P make congruent angles with the tangents to these curves at A and A0 . That is,
the angles marked ρ at A and A0 in Figure 7 are congruent. To see why, note that since
the dilation centered at P with factor 2 maps A0 to P 0 and the pedal to the orthotomic,
the angles marked ρ at A0 and P 0 are congruent. But since A is the instantaneous center
of rotation of C 0 as it rolls on C , AP0 is perpendicular to the orthotomic at P 0 , and the
angles marked ρ at A and P 0 are also congruent. See [5, p. 14] for another proof.
P0
A0
CP
A
C0
C
P
Figure 7. The angle property of pedal curves.
4. STEINER’S THEOREMS. Given the close connection between pedals and
roulettes as expressed in Figures 3 and 4, it is not surprising that there are some
relationships between pedals and roulettes. Two of these are given by theorems discovered by Jakob Steiner (see [2] for generalizations). Let A and B be points of C ,
and let A0 and B 0 be the corresponding points on the pedal curve C P . Let R be the
portion of the roulette curve traced by P as C rolls on a line m between contact points
A and B.
Steiner Theorem 1. The arc length of R is equal to the arc length of C P between A0
and B 0 .
Steiner Theorem 2. The area between R and m is twice the area bounded by PA0 ,
PB0 and C P .
In particular, if C is a closed curve rolling on a line, the area between the line and
roulette after one revolution is twice the area of C P , and the arc length of the roulette
is equal to the arc length of C P . To illustrate, let C be a circle of radius r that rolls
through an angle of θ radians, and let P be the center of the circle. The roulette curve
is a line segment l of length r θ and the area of the rectangular region between l and
m is r 2 θ . Since Q 0 = Q for all points Q of C , the corresponding region of the pedal
curve is the sector of a circle with central angle θ , area 12 r 2 θ, and arc length r θ . For a
second example of Steiner’s theorems, take C to be a circle, let the tracing point P be
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a point on C , and let A and B be any two points on C . As illustrated in Figure 8, the
roulette, R, is an arc of a cycloid and the pedal, C P , is an arc of a cardioid. The two arc
lengths are equal and the area between the arc of the cycloid and the x-axis is twice
the area bounded by the cardioid and segments PA0 and PB0 .
C
Q
R
~
P
B
Q0
A
P
Q
CP
P
~
Q0
B0
~
Q
A0
Figure 8. An illustration of the Steiner theorems.
The Steiner theorems follow as corollaries to the roulette lemma. To see this, let C
roll along the x-axis with P initially on the y-axis, and let θ denote the angle through
which C has turned from its starting position. Using (3), a parameterization of R is
given by (x, y) = ( f (θ), g(θ)), where f 0 (θ) = g(θ ). If we place the pole of a polar
coordinate system at P with the polar axis pointing in the direction of the negative
y-axis, a polar equation of C P is r = g(θ). Then as C turns through the differential
angle dθ , the differential arc length of the roulette is
p
(d x)2 + (dy)2 =
p
(g(θ))2 + (g 0 (θ ))2 dθ,
(4)
and the differential arc length of the pedal is
p
p
(r dθ)2 + (dr )2 = (g(θ))2 + (g 0 (θ ))2 dθ,
(5)
proving the first Steiner theorem. The area element of the roulette is
y d x = (g(θ))2 dθ,
and the area element of the roulette is
1 2
1
r dθ = (g(θ))2 dθ,
2
2
proving the second Steiner theorem.
To get a dynamic picture of the Steiner theorems, let’s return for a moment to Figure 8, where we have shown the rolling circle in an intermediate position (dashed)
along with points P̃ and Q̃, the rotated images of the points P and Q, respectively.
Point Q̃ 0 is the foot of the perpendicular from P̃ to the x-axis and corresponds to
Q 0 on the pedal curve. Now let’s put this figure in motion. Imagine the dashed circle
rolling clockwise on the x-axis as P̃ traces out the arc of the cycloid. At the same time
point Q 0 moves counterclockwise around P and sweeps out the corresponding arc of
the cardioid. Steiner’s first theorem says that P̃ and Q 0 move with the same speed,
as shown by the equality of the differential arc lengths in (4) and (5). Given this, it’s
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easy to see why Steiner’s second theorem is true. As C turns through the angle dθ , the
segment PQ0 sweeps out a differential triangle, while the segment P̃ Q̃ 0 sweeps out a
differential rectangle. But since the segments are congruent and Q 0 and P̃ move at the
same speed, the area of the rectangle is twice that of the triangle. In Section 6 we give
a visual representation of Steiner’s first theorem. But first we must introduce the idea
of road-wheel pairs.
5. ROADS AND WHEELS. Many readers are probably familiar with a squarewheeled bicycle. If the wheels roll along a road consisting of portions of appropriately
chosen inverted catenaries, the axles of the wheels move horizontally and the ride is
smooth. Figure 9 shows another example of a road-wheel pair. An elliptical wheel rolls
without slipping on a sinusoidal road, while one focus, A, of the ellipse moves along
the x-axis. Hall and Wagon [3] generate many other examples of road-wheel pairs.
We give some of their examples below, but for now we present a modified version
of their approach which is more suited to our geometric point of view. We place the
road below the x-axis in a rectangular coordinate system and require that the wheel
roll on the road without slipping and that the axle of the wheel, which we initially
place at the origin, move along the x-axis. An equivalent way to describe the rolling
condition is that the point of contact of the road and wheel (Q in Figure 9) is the
instantaneous center of rotation of the wheel. This implies that the velocity of the axle
is perpendicular to the spoke AQ. Adding the requirement that the axle of the wheel
move along the x-axis forces Q to be directly below A at all times.
y
Q0
A
A0
x
Q
Figure 9. An ellipse rolling on a sine curve.
In describing a road-wheel pair, we adopt the convention of describing the wheel
in its initial position, with its axle at the origin. We do so by giving a polar equation
of the wheel, r = g(θ) > 0, where the pole coincides with the origin and the polar
axis points in the direction of the negative y-axis. Given the function r = g(θ ), we
wish to parameterize the corresponding road in terms of θ . Suppose the wheel rotates clockwise through an angle of θ while rolling on the road, so that the spoke of
the wheel, A0 Q 0 , which originally makes an angle of θ with the negative y-axis, has
moved to AQ and is now pointing downward. If Q(x, y) is the point of contact, then
y = −r = −g(θ). To finish parameterizing the road we must express x as a function
of θ , in a sense unwrapping the spokes of the wheel and placing them perpendicular
to the x-axis to form the road. Since Q is the instantaneous center of rotation of the
wheel, the differential displacement of the axle, A(x, 0), of the wheel is
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ROADS AND WHEELS, ROULETTES AND PEDALS
485
so that x = f (θ ) is the solution of the initial value problem
d x/dθ = g(θ),
x(0) = 0.
Thus, the wheel with polar equation r = g(θ ) has as its corresponding road the
curve
Z θ
(x, y) =
g(t) dt, −g(θ ) .
(7)
0
Note that (6) and (7) hold even if g(θ) < 0. In this case, the part of the road corresponding to g(θ) < 0 lies above the x-axis and the axle moves to the left while
the wheel is in contact with this part of the road. This is illustrated in Figure 10 (see
example 1 below). Here are some examples, generated using (7).
Figure 10. A limaçon rolling on a trochoid.
1. If the wheel is the limaçon r = 1 − d cos θ , the road is the trochoid (x, y) =
(θ − d sin θ, d cos θ − 1). If d = 1 the wheel is a cardioid and the road is a
cycloid. The case d = 1.5 is shown in Figure 10.
2. If the wheel is the horizontal line r = sec θ (or y = −1), the road is the catenary
(x, y) = (ln(sec θ + tan θ), − sec θ), as shown in Figure 11.
3. If the wheel is the circle r = cos θ , the road is the circle (x, y) = (sin θ, − cos θ ).
In this example a circle with radius 1/2, in its initial position with center at
(0, −1/2), rolls on the inside of the unit circle x 2 + y 2 = 1. The point on the
rolling circle initially at the origin oscillates on the x-axis between (−1, 0) and
(1, 0).
4. If the wheel is the parabola r = 1/(2 + 2 cos θ ), or y = x 2 − 1/4, the road is the
parabola (x, y) = (sin θ/(2 + 2 cos θ), −1/(2 + 2 cos θ )), or y = −x 2 − 1/4.
x
Figure 11. A line rolling on a catenary.
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More generally, as the mirror image of a fixed parabola, P , rolls on P in such
a way that the rolling parabola is the reflection of P across the tangent line at
the point of contact, the focus of the rolling parabola moves along the directrix
of P .
5. If the wheel is the rose r = cos nθ , the road is the ellipse (x, y) = ( n1 sin nθ,
−cos nθ).
6. ROADS, WHEELS, AND STEINER’S THEOREMS. In this section we describe how to generate road-wheel pairs geometrically and give a new geometric interpretation of the Steiner theorems. Let (R, W ) denote a road-wheel pair, and let capital
letters and their primed counterparts denote corresponding points on the road R and
wheel W , respectively. That is, if A is a point on the road, the point A0 is the point
on the wheel that comes into contact with the road at A. Then we have the following
two properties of road-wheel pairs, the first of which follows immediately from our
assumption that the wheel rolls without slipping:
Lemma 1. The length of the road between A and B is equal to the length of the wheel
between A0 and B 0 .
Lemma 2. The area between the road and the x-axis from A to B is twice the area
bounded by the wheel and the segments OA0 and OB0 , where O is the axle of the wheel.
To prove the second property, note from (6) that the area element between the road
and the x-axis is
d A = −y d x = −y r dθ = r 2 dθ,
which is twice the area element of the wheel. We can show this property geometrically
by using the spokes to unwrap the wheel as shown in Figure 12. The distance between
nearby unwrapped spokes of length r and R = r + 1r is taken to be r 1θ, where 1θ is
the angle between the spokes. The area of each trapezoid is then approximately twice
the area of the corresponding sector. This can be thought of as a generalization of the
grade school proof for the area of a circle. Alternatively, we can think of the latter
proof as constructing a road for a circular wheel.
r1
r
R
R
r
Figure 12. Unwrapping a wheel to form a road.
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The similarity of these lemmas with the Steiner theorems, as well as the similarity
between (3) and (7), suggests the following theorem. In the statement of the theorem
we follow our convention of describing the wheel in its initial position with its axle at
the origin.
Main Theorem. Let C be a rollable curve initially tangent to the x-axis at the origin,
O, and let P0 be a point on the y-axis. Let R0 be the reflection in the x-axis of the
roulette curve R generated by P0 as C rolls on the x-axis. Then translating the pedal,
−−→
C P0 , by the vector P0 O gives a wheel for the road R0 .
Proof. A comparison of (3) and (7) actually proves the theorem, but we give a more
geometric proof that illustrates why the theorem is true and describes the rolling motion of the wheel. As shown in Figure 13, let P(x, y) denote the position of P0 after C
has rolled some distance along the x-axis. Let A be the point of tangency between C
and the x-axis and let A0 be the corresponding point of C P . Also, let Q be the reflec−
→ −−→
tion of P in the x-axis. We show that translating C P by the vector PA0 = A0 Q gives the
wheel in its position when it is in contact with the road at Q; in particular, the initial
−−→
position of the wheel is the translation of C P0 by the vector P0 O. Let W denote the
−
→
translation of C P by PA0 . The translation ensures that W and R0 are in contact at Q.
It also sends the pedal point P to A0 , so that the translated copy of P moves along the
x-axis and serves as the wheel’s axle. It remains to show that W rolls on the road. We
first show that W and R0 are tangent at Q. By the angle property of pedals, the angles
at A and A0 labeled ρ are congruent. But the angle between R and PA0 at P also has
measure ρ. It follows that W and R0 are tangent at Q. That W is rolling, and not sliding, on R0 follows immediately from the first Steiner theorem, since the corresponding
arcs lengths of R0 and C P0 are equal. But since we would like to claim Steiner’s first
theorem as a corollary to this theorem, we give an alternative argument to show that
W is rolling on R0 . The roulette lemma implies that as C turns through the angle dθ ,
the horizontal component of the displacement of P, and hence the displacement of
y
R
P
C
A0
A
x
CP
Q
R0
Figure 13. Generating road-wheel pairs geometrically.
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A0 , is y dθ . But since W turns through the same angle as C , it follows that Q is the
instantaneous center of rotation of W .
We now show how to use our theorem to generate the first three road-wheel pairs
from the previous section. We describe the curve C in its initial position, tangent to the
x-axis at the origin. We also describe the wheel in its initial position, W0 , with its axle
at the origin. As usual, the polar equation of W0 assumes that the polar axis coincides
with the negative y-axis.
1. Take C to be the circle x 2 + (y − 1)2 = 1 and P0 = (0, 0). Then C P0 is the cardioid r = 1 − cos θ and the roulette R is the cycloid (x, y) = (θ − sin θ, 1 −
cos θ ). Since the pedal point is initially at the origin, W0 is the same cardioid.
Thus, the wheel r = 1 − cos θ rolls on the road R0 given by (x, y) = (θ −
sin θ, cos θ − 1).
2. Take C to be the parabola y = x 2 /4 and P0 = (0, 1). Then C P0 is the x-axis (see
Figure 5) and R is the catenary y = cosh x. The wheel, W0 , is the translation of
−−→
the x-axis by the vector P0 O = h0, −1i, or the line y = −1. So the line y = −1
is the wheel for the catenary y = − cosh x.
3. In a degenerate case, take C to be the origin and P0 to be the point (0, 1). The
pedal curve, C P0 , is the circle with diameter O P0 . The roulette curve R (and also
−−→
R0 ) is the unit circle centered at the origin. Since P0 O = h0, −1i, a wheel for
0
R is the unit circle centered at (0, −1/2).
Our theorem suggests that to visualize Steiner’s first theorem we should replace
our image of P tracing out R as C rolls on a line (as in Figure 8) by the image of
C P rolling on R0 . We use this idea to recast our dynamic interpretation of the Steiner
theorems given at the end of Section 4. To illustrate, return to Figure 9, showing an
elliptical wheel rolling on a sinusoidal road, and put the wheel in motion. As the ellipse
rolls on the sine curve, the point Q 0 moves on the fixed ellipse at the left. Recall Q 0
corresponds to the point, Q, on the rolling ellipse that is in contact with the road at
any instant. Expressed another way, the motion of the spoke A0 Q 0 on the fixed ellipse
is just the motion of the spoke AQ on the rolling ellipse as seen by an observer in the
reference frame of the rolling ellipse. We may restate Steiner’s first theorem as saying
Q and Q 0 move at the same speed. Of course this follows from our condition that the
ellipse roll on the road without slipping. But now consider the additional condition that
the axle of the wheel move along the x-axis. This requirement forces the spoke, AQ,
of the rolling ellipse to be perpendicular to the x-axis and thus sweep out a differential
rectangle as the ellipse turns through the angle dθ , while the spoke, A0 Q 0 , of the fixed
ellipse sweeps out a differential triangle. Since the speeds of A0 and A are equal, the
area of the differential rectangle is twice that of the triangle.
7. GENERATING ADDITIONAL ROAD-WHEEL PAIRS. We may use our main
theorem to generate additional road-wheel pairs by arbitrarily choosing a rollable
curve C . For example, let C be the cardioid r = 1 − cos(θ − π/2) and P0 be the origin. We leave it to the reader to verify that g(θ ) = 2 sin3 (θ/3). Using (3) we may
parameterize R as
(x, y) = (−6 cos(θ/3) + 2 cos3 (θ/3) + 4, 2 sin3 (θ/3)),
0 ≤ θ ≤ 3π.
The rolling motion of the cardioid is hinted at in Figure 14. The pedal curve, C P0 , is
Cayley’s sextic with polar radius given by the y-coordinate of R, or r = 2 sin3 (θ/3).
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ROADS AND WHEELS, ROULETTES AND PEDALS
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y
x
Figure 14. A cardioid rolling on the x-axis.
This is a wheel for the road R0 , as shown in Figure 15, where Cayley’s sextic is shown
in its initial position and some later position. The cardioid (dashed) is also shown to
illustrate our theorem; the position of the wheel is found by translating the pedal of the
−
→
cardioid by the vector PA0 .
y
P
x
A0
Figure 15. Cayley’s sextic rolling on a road.
As a second example, we find the road corresponding to a circular wheel of radius
a and center O whose axle, P, is ae units from O, with 0 < e < 1. Let C be the ellipse
with center O, one focus at P, and eccentricity e. Then by example 3 of Section 3, the
wheel is the pedal of C with respect to P. It follows that the road for the circular wheel
with axle at P is the roulette curve generated by P as C rolls on a line. The roulette,
an elliptic catenary, is shown in Figure 16 and the road-wheel pair in Figure 17. The
road-wheel pair consisting of a line rolling on a catenary, shown in Figure 11, may
be regarded as the limiting case of a circular wheel rolling on an elliptic catenary. If
we set a = 2e/(1 − e2 ) and let e → 1− , in any bounded region the elliptic catenary in
Figure 17 approaches the catenary y = − cosh x and the circular wheel approaches a
tangent line to the catenary.
Figure 16. An elliptic catenary generated as a
roulette, e = 0.8.
490
Figure 17. A circular wheel rolling on an elliptic
catenary, e = 0.8.
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8. GENERATING FAMILIAR CURVES AS ROULETTES. Given a road-wheel
pair, (R0 , W ), we can use our main theorem to express the reflection of the road in
the x-axis as a roulette by reversing the process described in Figure 13 to recover the
rolling curve C and the tracing point P. We describe this process when the wheel is in
its initial position, with its axle at the origin. So in Figure 13 we take A0 to coincide
with the origin, O, and we replace P and Q with P0 and Q 0 , respectively, to emphasize
that the wheel is in its initial position. Determining P0 is easy. We translate the axle of
−−→
the wheel (at O) by the vector Q 0 O. To determine C we first find the negative pedal
of W with respect to O; that is, we find the curve whose pedal with respect to O is
W . We denote this curve by W O . Finally, we recover C by translating W O by the
−−→
vector Q 0 O. To determine the negative pedal, note that the tangents to W O are the
lines through points S of W perpendicular to the segments OS. If this one-parameter
family of tangents has the parameterization f (x, y, t) = 0, then the negative pedal as
a set of points is a solution to the system
f (x, y, t) = f t (x, y, t) = 0,
(8)
where the subscript denotes partial differentiation with respect to the parameter t.
For example, suppose the wheel is the parabola y = x 2 /4 − 1, with corresponding road y = −x 2 /4 − 1. Since the initial point of contact of the wheel and road
is Q 0 (0, −1), the initial position of the tracing point is P0 (0, 1). Parameterizing the
wheel as (x, y) = (2t, t 2 − 1) gives a parameterization of the set of tangent lines to
W 0 as
f (x, y, t) = 2t x + (t 2 − 1)y − (t 4 + 2t 2 + 1) = 0.
−−→
We then translate the solution of the system (8) by the vector Q 0 O = h0, 1i to get a
parameterization of C in its initial position as
(x, y) = (3t − t 3 , 3t 2 ).
Thus, the the parabola y = x 2 /4 + 1 is the roulette traced by the point P0 (0, 1) as the
above curve rolls along the x-axis. In this case the rolling curve, the negative pedal
of a parabola with respect to its focus, is called Tschirnhausen’s cubic. Figure 18
shows Tschirnhausen’s cubic rolling along the x-axis as the point P0 (0, 1) traces out
a parabola. Tschirnhausen’s cubic is shown twice; once in its initial position (dashed)
and again after it has rolled some distance along the x-axis.
Figure 18. A parabola generated as a roulette curve.
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ROADS AND WHEELS, ROULETTES AND PEDALS
491
For the remaining examples we describe the wheel in its initial position by its polar equation, r = g(θ). In this case, replacing the parameter t with θ and keeping in
mind that the polar axis coincides with the negative y-axis, the system of equations (8)
becomes
x sin θ − y cos θ − g(θ) = x cos θ + y sin θ − g 0 (θ ) = 0,
with solution
(x, y) = (g(θ) sin θ + g 0 (θ) cos θ, −g(θ ) cos θ + g 0 (θ ) sin θ ).
(9)
For example, take the wheel to be the rose r = cos(nθ ), with corresponding road the
ellipse (x, y) = ( n1 sin nθ, − cos nθ). Since the initial point of contact is Q 0 (0, −1),
the initial position of the tracing point is P0 (0, 1). Using (9) and translating by the
−−→
vector Q 0 O = h0, 1i gives a parameterization of the rolling curve in its initial position
as
(x, y) =
n−1
(− sin(n + 1) θ, cos(n + 1) θ)
2
n+1
−
(sin(n − 1) θ, cos(n − 1) θ) + (0, 1).
2
(10)
The curves (10) are all hypocycloids. That is, they are the roulettes generated by a
point on the circumference of a circle of radius r as it rolls with internal contact inside
a fixed circle of radius R. Choosing r = (n − 1)/2 and R = n gives the hypocycloid
in (10). Since a hypocycloid is not differentiable at its cusps, we may apply our main
theorem only to each differentiable piece. However, a hypocycloid has well-defined
tangents at its cusps and the tangents to the hypocycloid vary continuously as the
curve is traversed. Thus, a hypocycloid can roll on a line and as it rolls it maintains its
sense of rotation. We can piece together the results from our main theorem to conclude
that when the curve (10) rolls on the x-axis, the point P0 (0, 1) traces out the ellipse
(x, y) = ( n1 sin nθ, cos nθ). The description of the rolling motion below makes this
transparent.
Figure 19 shows the case n = 2, where the rolling curve is an astroid. The dashed
astroid is the curve in its initial position and the solid astroid is the curve in its new
Figure 19. An ellipse as a roulette of an astroid.
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Figure 20. An ellipse as a roulette of a hypocycloid.
Figure 21. An ellipse as a roulette of a hypocycloid.
position after it has rotated θ = 0.5 radians. Figures 20 and 21 show the case n = 4,
where the initial and rotated (θ = 0.25) positions are shown separately for clarity.
Since the tangent lines to the hypocycloid at the cusps pass through the tracing point
P0 , the x-intercepts of the ellipse are traced when the cusps are tangent to the x-axis.
Thus, half of the ellipse is traced when one arc of the hypocycloid has completed
rolling along the x-axis. When the point of tangency of the hypocycloid and the x-axis
passes through a cusp, the tracing point crosses the x-axis and the point of tangency
switches between the “top” and “bottom” of the x-axis (more precisely, the astroid is
alternately above and below the x-axis near the point of tangency). As mentioned in
the remarks following the roulette lemma, it is at the moments when the tracing point
crossing the x-axis, in this case when a cusp is the point of tangency, that the direction
of the tracing point changes between a rightward and a leftward motion.
In our last two examples, the curve C also has cusps and we may apply our theorem
only to its differentiable pieces. But, as in the last example, the cusps have well-defined
tangent lines and the tangents to C vary continuously, and we may piece together the
roulettes generated by each differentiable part of C . We now consider the case of an
elliptical wheel rolling on a sinusoidal road, as one focus of the ellipse moves along the
x-axis, as shown in Figure 9. More specifically, suppose that the wheel is the ellipse
r = ed/(1 + e cos θ), 0 < e < 1. Using (7), the corresponding road is given by
f (θ) + e
1+e
ed
2ed
arctan
− arctan
,−
,
(x, y) =
a
a
a
1 + e cos θ
√
where a = 1 − e2 , f (θ) = tan(θ/2 + π/4), and the value of arctan(( f (θ ) + e)/a)
is the appropriate angle closest to θ/2 + π/4. A little trigonometry shows that an equation of the road in rectangular coordinates is given by
y=−
ed
(1 − e cos(cx)) ,
a2
(11)
where c = a/(ed). For an alternative approach see [3]. The negative pedal of an ellipse
with respect to a focus, P, is shown in Figures 22 and 23 for two cases, e > 1/2
and e < 1/2, respectively, where the ellipses are dashed. We follow Lockwood [4]
and call the negative pedal Burleigh’s oval and we call P the eye of the oval. If e >
1/2 the oval has two cusps. Otherwise, it is an egg-shaped curve. It follows that as
Burleigh’s oval rolls on a line, the eye traces out a sine curve. Using (9) and translating
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ROADS AND WHEELS, ROULETTES AND PEDALS
493
P
P
Figure 22. Negative pedal of an ellipse (e =
0.8).
Figure 23. Negative pedal of an ellipse (e =
0.45).
−−→
by the vector Q 0 O = h0, ed/(1 + e)i gives a parameterization of the rolling curve in
its initial position as
ed
ed
(x, y) =
.
(sin θ + e sin 2θ, − cos θ − e cos 2θ ) + 0,
(1 + e cos θ)2
1+e
If this oval is rolled on the x-axis, the eye, P0 (0, ed/(1 + e)), traces out the reflection
of the sine curve (11) in the x-axis. When e > 1/2 the point of tangency between the
oval and the x-axis changes between being above and below the x-axis at the cusps.
But in contrast to the previous example, the tracing point continues to move to the right
at all times since no tangent to the oval passes through its eye.
Figure 24. Roulette of focus of negative pedal of
an ellipse (e = 0.8).
Figure 25. Roulette of focus of negative pedal of
an ellipse (e = 0.45).
This example can be generalized by taking the wheel to be the curve
r = ed/(1 + e cos(λθ )),
(12)
where λ is a constant. Then the corresponding road is parameterized as
2ed
f (λθ) + e
1+e
ed
(x, y) =
arctan
− arctan
,−
,
aλ
a
a
1 + e cos(λθ )
with rectangular equation
y=−
494
ed
(1 − e cos(cx)),
a2
(13)
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√
−−→
where a = 1 − e2 and c = aλ/(ed). Using (9) and translating by the vector Q 0 O =
h0, ed/(1 + e)i gives a parameterization of the rolling curve in its initial position as
(x, y) =
ed
{(sin θ, − cos θ ) + e(sin θ cos λθ, − cos θ cos λθ )
(1 + e cos λθ)2
ed
+ λe(sin λθ cos θ, sin λθ sin θ )} + 0,
.
(14)
1+e
If λ is chosen to be a positive integer, the wheel is a simple closed curve with
λ “lobes”
and rolls over λ periods of the road as it rotates
√
√ once. If we choose e =
2
2
1/ 1 + λ and d = λ , then the road has equation y = − 1 + λ2 + cos x. The wheel
and the road for λ = 4√
are shown in Figure 26. As the curve√given by (14) rolls on the
x-axis, the eye, P0 (0, 17 − 1), traces out the curve y = 17 − cos x, as shown in
Figure 27.
√
Figure 26. A wheel for the road y = − 17 + cos x.
Figure 27. The curve y =
√
17 − cos x generated as a roulette.
While we have used our theorem to generate some familiar curves as roulettes, there
are many other possibilities. We invite the interested reader to explore more examples
of road-wheel pairs in [3] and [6] to generate additional roulettes.
ACKNOWLEDGMENTS. I am deeply grateful to several referees whose many suggestions greatly improved
this article. In particular, I would like to thank one referee for generously providing me with an example of
how to use Mathematica and another for pointing out the angle property of pedals and providing the geometric
proof of the main theorem. I would also like to thank my student, Mark Pembrooke, for suggesting the roulette
lemma, and my teacher, John P. Titterton, for introducing me to the beauty of curves thirty years ago.
REFERENCES
1. A. Agarwal and J. E. Marengo, The locus of the focus of a rolling parabola, College Math. J. 41 (2010)
129–133. doi:10.4169/074683410X480230
2. T. Apostol and M. Mnatsakanian, Area & arc length of trochogonal arches, Math Horizons 11(2) (2003)
24–30.
June–July 2011]
ROADS AND WHEELS, ROULETTES AND PEDALS
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3. L. Hall and S. Wagon, Roads and wheels, Math. Mag. 65 (1992) 283–301. doi:10.2307/2691240
4. E. H. Lockwood, Negative pedal of the ellipse with respect to a focus, Math. Gaz. 41 (1957) 254–257.
doi:10.2307/3610116
5.
, A Book of Curves, Cambridge University Press, Cambridge, 1961.
6. G. B. Robison, Rockers and rollers, Math. Mag. 33 (1960) 139–144. doi:10.2307/3029034
FRED KUCZMARSKI received his B.A. from the University of Pennsylvania in 1984 and his Ph.D. from
the University of Washington in 1995 under the guidance of Paul Goerss. His passion for geometry was later
sparked by James King. He currently teaches at Shoreline Community College in Seattle. His other interests
include baking bread and helping the Forest Service look for fires in Washington’s North Cascades.
Department of Mathematics, Shoreline Community College, Shoreline, WA 98133
[email protected]
Kovalevskaya on Learning Calculus
“When we moved permanently to the country, the whole house had to be redecorated and all the rooms had to be freshly wallpapered. But since there were many
rooms, there wasn’t enough wallpaper for one of the nursery rooms . . . [which]
just stood there for many years with one of its walls covered with ordinary paper. But by happy chance, the paper for this preparatory covering consisted of
the lithographed lectures of Professor Ostrogradsky on differential and integral
calculus, which my father had acquired as a young man.
These sheets, all speckled over with strange, unintelligible formulas, soon
attracted my attention. I remember as a child standing for hours on end in front
of this mysterious wall, trying to figure out at least some isolated sentences . . . .
Many years later, when I was already fifteen I took my first lesson in differential calculus from the eminent Petersburg professor Alexander Nikolayevich
Strannolyubsky. He was amazed at the speed with which I grasped and assimilated the concepts of limit and of derivatives, ‘exactly as if you knew them in
advance.’ . . . And, as a matter of fact, at the moment when he was explaining
these concepts I suddenly had a vivid memory of all this, written on the memorable sheets of Ostrogradsky; and the concept of limit appeared to me as an old
friend.”
Sofya Kovalevskaya, A Russian Childhood,
trans. B. Stillman, Springer-Verlag, New York, pp. 122–123
—Submitted by Robert Haas, Cleveland Heights, OH
496
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118