the geometry of rolling curves john bloom and lee whitt, texas a&m university

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The Geometry of Rolling Curves John Bloom and Lee Whitt, Texas A&M University, College Station, TX 77843

The American Mathematical Monthly, June–July, 1981, Volume 88, Number 6,

pp. 420–426.

Roll a closed convex curve along a line and follow the path of any chosen point onthe curve. In the simplest case, the well-known cycloid is traced by a point on a

rolling circle. In general, the set of (pointed) closed convex curves produces a

wide variety of traced curves. Which curves are produced this way? Given a curve, can

it be traced by rolling a (pointed closed convex) curve? If so, which one? In this paper,

we give the necessary and sufficient conditions for traceability in terms of the normals

to the curve and construct the curve to be rolled.

Suppose that the tracing point is allowed to be inside or outside of the rolling curve.

Suppose further that the “line of roll” is replaced by a curve and that nonconvex curves

are allowed to roll, i.e., by requiring that the point of contact move smoothly with no

sliding (arclengths must agree) and the tangent lines agree at the contact point. In all

cases, we solve the local inverse problem, as before, in terms of the normals to the

curve.

The geometry of rolling curves has been studied extensively by mechanical engineers

and others (see bibliography; Besant’s book is the earliest systematic study) but their

solution of the inverse problem is somewhat incomplete. We thank Dr. Rundell for

suggesting this problem. All curves are plane curves. For simplicity,

differentiability is assumed unless explicitly stated otherwise. This topic may be suitable

for an honors calculus class.

1. Necessary Condition.

We begin with a simple case. Let be a closed convex curve which can roll along a linei.e., the curvature of is positive except possibly on a nowhere dense set and so there

are no “straight sides.” This condition guarantees that the point of contact is well

defined and behaves as the arclength parameter on both and (cf. §3). The tracing

point can be placed inside, on, or outside These are illustrated in Fig. 1 with

a circle.

If is regarded as the origin, then can be described by polar coordinates If is

the angle between the tangent line and radial line of then traces out the curve

(Fig. 2) given by

(1) y ϭ r sin x ϭ s Ϫ r cos

C PC ,

r , .C P

C .P

LC

s

C L,C

C ϱ



We claim that the radial line is normal to Using the well-known equation

we calculate

and so

and

Figure 1

dy

d ϭ cos r ϩ tan

dr

d .

dx

d ϭ

ds

d Ϫ

dr

d cos ϩ r sin ϭ sin r ϩ tan

dr

d

ϭ sec dr ,

ϭ Ί 1 ϩ tan2 dr

ϭΊ 1 ϩ rd dr 2

dr

ds ϭ Ί dr 2 ϩ r 2d 2

tan ϭ rd dr ,

C .

2



Figure 2

Hence and the radial line is normal to In particular,

CONDITION 1. The normals to intersect in increasing order.

This condition is sufficient for the local construction of (Lemma 1).

An important observation is that the angle sum is the angle between the tangent

lines of and a fixed line (the polar axis). Since is the curvature of by

definition, the curvature assumption on is equivalent to with

equality only on a nowhere dense set.

CONDITION 2. The function is positive except possibly on a nowhere

dense subset of

It suffices to show

(2)

By construction,

and

So,

ϭ

s Ϫ x 2

yϪ ysec

dr

dxϩ y ϩ r tan

dr

dx

r 2.

ϭ

dy

dx s Ϫ x Ϫ ydsdx Ϫ 1

s Ϫ x 2 ϩ y2ϩ

r tan

r 2 dr

dx

d

dxϩ

d

dx ϭ

d

dxtanϪ1 y

s Ϫ x ϩtan

r dr

dx

dy

dxϭ

s Ϫ x

y.

y

s Ϫ xϭ tan

1

y dx

dsϭ

d

dsϩ

d

ds.

C .

1 y dx ds

d ϩ ds ≥ 0,C

C d ϩ dsC

ϩ

C

LC

C .dy dx ϭ cot

3



Since we obtain

which is equivalent to equation (2).

Equation (2) has several amusing interpretations. From the curvature assumption on it

follows that moves forward above and backward below This is equivalent to the

popular brain-teaser: What part of a train is moving backward? The answer is: the part

of the inner wheel flange that drops below the track. Also, if crosses , then it crosses

orthogonally, i.e., is defined everywhere and so whenever

If nonconvex curves are rolled, then the position of (above or below ) and the

curvature of at the contact point determine the direction travels. It follows that the

nonconvexity of can be an obstruction to the smoothness of . For example, if is

nonconvex and if lies above then changes sign and moves forward and

backward above If is smooth, then it has a vertical tangent line. But the normal

lines to must intersect and hence cannot be smooth.

2. Sufficient Conditions

Let be a plane curve which satisfies condition 1 and which is periodic with respect to

a line In this way, the arclength parameter on also parametrizes The

differentiability assumption on is that the length of the normal vector from tois smooth and (or equivalently has a smooth

extension over all its singularities. These smoothness conditions do not imply that

is smooth.

LEMMA 1. If and are as above, then there is a smooth closed, not necessarily

convex or simple, curve C and a distinguished point P which traces any finite piece

of Proof. Let be the length of the normal vector from to and let satisfy the

differential equation

The curve given by is smooth by assumption. To see that this curve is it

suffices to check that the arclength parameter of is

Since equations (1) still hold, it follows that

and

dy

dxϭ

cot .

dy

d ϭ

cos r ϩ

tan

dr

d

s. r s , sC , r s , s

d

dsϭ

dr

rdstan .

L,C r C .

C L

C

1 y dx ds Ϫ d ds dr rds tan L

C r sC

C . Ls L.

C

C L,C

C L.

P 1 y dx L,C

C C C

PC

LP

y ϭ 0.dx|C ϭ0 ϩ

LC

L. LP

C ,

d

dxϩ

d

dxϭ

1

y΄ s Ϫ x 2 ϩ y2

r 2 ΅ ϭ 1

y,

ysec ϭ r tan ,

4



Hence

and, on differentiating equations (1),

Hence and is the arclength of If our construction of does

not result in a smooth closed curve, then we easily complete to obtain the desired

curve (Figs. 4 and 5). Q.E.D.

Figure 3

The local construction of contains several peculiarities. For example, if all of is

used, and if is the graph of then is a spiral. Another example isbest illustrated by a curve (solid curve) lying slightly above a cycloid (dotted curve)

(Fig. 3). Our main theorem requires a period of and it is possible for different periods

to produce different curves Over a period like our construction of produces a

corner at the origin (solid curve) which is then completed (dotted curve) to a smooth

closed curve (Fig. 4). Over a period like our construction of produces a nonclosed

curve which is then completed (Fig. 5). This dependence on the period can be resolved

by introducing the following integral condition (3) into our main theorem.

Figure 4 Figure 5

C B,

C A,C .

C ,

C C y ϭ log x, x ≥ 1,C

C C

C

C C .sds ϭ Ί dr 2 ϩ (rd 2

ϭΊ 1 ϩ rd dr 2dr

d .

ϭ Ί 1 ϩ tan2 dr

d

ds

d ϭ sec

dr

d

dx

d ϭ sin r ϩ tan

dr

d

5



THEOREM 1. Let L and be as in Lemma 1. Assume further that satisfies

Condition 2 and

(3)

where is a period of relative to L. Then there is a unique smooth closed convex

curve C and a distinguished point P which traces all of

Proof. At the endpoints of the normals agree. Also, the change in on the

integral condition (3), and equation (2) determine the change in Namely, if changes

by 0, or then changes by or 0, respectively, where these correspond to,

for example, the prolate, ordinary, and curtate cycloids in Fig. 1. If changes by or

0, then applying Lemma 1 to produces a closed curve If changes by then, by

the periodicity of must be zero at the ends and also is closed. The smoothness of

follows from the continuity (equivalently, closure) of and the smoothness of the

tangent vector to The convexity follows from Condition 2 and

equation (2). Finally, is unique up to a rotation of the -plane because is

uniquely determined by and is defined up to a constant, or equivalently, a rotation.

Q.E.D.

3. Relaxation of Some Conditions

As mentioned earlier, a nonconvex curve can roll under a suitable definition of “roll,”

and the traced curve will not be smooth. The nonsmoothness of can also be caused

by flat sides on . Such sides force a discontinuity in the normal vector field to If a

square is rolled along a line and the tracing point is chosen to be a corner, then we

obtain Fig. 6. Here we clearly see the effects of corners and flat sides. When rolls at a

corner, the normals meet the line in a stationary fashion. The flat sides cause the

singularities. Notice also that is not convex, even though is. If is rounded slightly

to produce a smooth convex curve, the traced curve will still be nonconvex, althoughit will now be smooth.

Figure 6

C

C C C

C

P

C .C

C C

C

C

r r , C

C . dr ds, d ds

C C

C r C ,

C .C 1

2

,2 , 2 , ,

.

C 1, C

1,

C .

C C 1

C͵ 1

1

y dx ϭ 2

C C

6



4. Rolling Curves Along Curves.

Let a curve with tracing point (not necessarily on ) be rolled along a curve to

produce a traced curve (Fig. 7). We parametrize these curves by and

respectively. Let be the angle between the normal to and the axis.

Figure 7

We claim that, as before, the line between and the contact point is perpendicular to

Consider

(4)

where is the angle between the normal to and the tangent to at the contact point.

Since the arclength parameter on agrees with that on we obtain

From and it follows that

sec dr ϭ Ϫsec␣ du ϭ csc␣ dz.

dz du ϭ Ϫtan␣,du dz ϭ tan 90Њ ϩ ␣

Ί dz2 ϩ du2 ϭ ds ϭ sec dr .

C ,C

C C

y ϭ r sin90Њ ϩ ␣ Ϫ

ϩ u ϭ r cos

␣ Ϫ

ϩ u

x ϭ r cos 90Њ ϩ ␣ Ϫ ϩ z ϭ Ϫr sin ␣ Ϫ ϩ z

C .P

C ␣ x, y ,

z, u , r , ,C

C C PC

7



Differentiating equation (4),

Similarly,

and so

establishing perpendicularity.

Analogous to §2, the local construction of proceeds by solving the differential

equation

where and are obtained from and To see that is the desired curve, we work

backward through the equations above to show that the arclength parameter on agrees

with that on Also, we need to calculate as before, for closure and convexity

considerations.

ϭ1

y Ϫ uϩ

d ␣

dx.

ϭ1

y Ϫ uϩ

1

r dr

dx ΄r d ␣dr

ϭ1

y Ϫ uϩ

1

r dr

dx΄sin ␣ Ϫ cos␣

cos Ϫ

cos ␣ Ϫ sin␣

cos ϩ tan ϩ r

d ␣

dr

ϭ1

r 2 dr

dx΄ z Ϫ xcos␣

cos Ϫ y Ϫ u

sin␣

cos ϩ r tan ϩ r 2

d ␣

dr ϩ

1

r 2΄ z Ϫ x 2

y Ϫ uϩ y Ϫ u΅

ϭ 1

r 2΄ z Ϫ x tan ␣ Ϫ ϩcos ␣

cos dr

dx Ϫ y Ϫ u sin ␣cos dr

dxϪ 1 ϩ r 2

d ␣

dxϩ r tan

dr

dx΅

ϭ

d y Ϫ u

dx z Ϫ x Ϫ y Ϫ u dzdx Ϫ 1

r 2ϩ

d ␣

dxϩ

tan

r dr

dx

d dxϩ d

dx ϭ d

dx΄tanϪ1

y Ϫ u z Ϫ x ϩ ␣ ϩ 90Њ΅ ϩ tan

r dr dx

d ϩ d ,C .

C

C C .C r

d

dr ϭ

tan x

r ,

C

dy

dxϭ tan ␣ Ϫ ,

dy

d ␣ϭ sin ␣ Ϫ ΄Ϫr 1 Ϫ d

ds ϩ tan dr

d ␣΅,

ϭ cos ␣ Ϫ ΄Ϫr 1 Ϫ d

ds ϩ tan dr

d ␣΅.

ϭ Ϫdr

d ␣sin ␣ Ϫ Ϫ r cos ␣ Ϫ 1 Ϫ d

d ϩ sin ␣ Ϫ ϩ sec dr

d ␣

dx

d ␣ ϭ Ϫ

dr

d ␣sin ␣ Ϫ Ϫ r cos ␣ Ϫ 1 Ϫ d

d ␣ ϩdz

d ␣

8



Hence, is convex if and only if

The integral condition for closure is

where the integral and the change in are taken over one period. Note that and are

the “heights” of the curves and but not necessarily above the same point. A point

lies above only when or, equivalently, the normal to is

vertical.

Previous considerations carry through with the obvious changes. Convexity corresponds

to the curvature of being greater than the curvature of . Flat sides correspond to

congruent pieces of and which roll against each other.

Finally, one can ask the general question: Given , is there a pair of curves so

that rolls on to produce If no smoothness is required, the answer is yes, but somecurves admit no pairs with the curvature of greater than the curvature of

smooth. Details are left to the reader.

References

1. I. I. Artobolevskii, Mechanisms for the Generation of Plane Curves, Pergamon Press,

New York, 1964.

2. W. H. Besant, Roulettes and Glissetes, Deighton, Bell, London, 1870.

3. R.C. Yates, A Handbook on Curves and Their Properties, 2nd ed., Edwards, Ann

Arbor, Mich., 1942.

C s , C , C

C sC , C C C .C C

C , C C

C C

C sC s

x, y␣ Ϫ ϭ 90Њ z, u x, yC ,C

u y␣͵1

y Ϫ u dx ϭ 2 Ϫ ⌬␣

1

y Ϫ uϩ

d ␣

ds ≥ 0.

C

9

the geometry of rolling curves john bloom and lee whitt, texas a&m university

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