# Tumbling Downhill Along a Given Curve

Jean-Pierre Eckmann
Yaroslav I. Sobolev
Tsvi Tlusty

Communicated by Notices Associate Editor Chikako Mese

## 1. The Problem

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? See Fig. 1 for an example.

This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve—most often two copies will suffice, but some curves require an arbitrarily large number of copies.

## 2. Rolling Stones

There is an ample mathematical literature on rolling objects, but for our purpose, it may suffice to mention the study of rolling acrobatic apparatus Seg21 and the rolling of balls on balls or Riemann surfaces Lev93. “Rolling” here means pure rolling motion without slipping or pivoting (that is, rotating around an axis perpendicular to the surface). With these rules of the game, a cylindrical stone will roll down along a straight line, indefinitely, in the projected direction of gravity.

But besides straight lines traced by cylinders, can one chisel objects that will follow more interesting planar paths—curved, convoluted, and self-crossing? This problem was posed and amply discussed in SDT23, and the aim of the current contribution is to go into more mathematical detail and add some new results about the general set of paths for which solutions exist.

A solution means that we have a way to sculpt such a wobbly stone, which we call “trajectoid” because it has the following property: Once placed at the starting point of the trajectory, appropriately oriented, and released, the trajectoid stone should roll along a pre-described infinite trajectory, and only along it. This periodic path is made by concatenating translated copies of an original finite path , as shown in Fig. 1.

The principle of trajectoid design (whose realization in practice we will discuss later) is the following: We take a heavy ball and coat it with some lightweight material, which we assume is weightless. Thus, the rolling object is inhomogeneous, and we will see that this allows the existence of trajectoids. The featherlight solid envelope is precisely chiseled (or “shaved”), such that it will lift the heavy ball whenever it tries to depart from the prescribed path. Gravity will therefore ensure that the object’s center of mass (CM) always stays at a constant height above the plane, and as a result, this trajectoid obediently follows the desired path. The concentration of all the mass in the heavy ball allows us define “following” simply as

We typically design and build such gadgets for rolling downhill on an inclined plane (as in Fig. 1). But from a mathematical point of view, it makes more sense to think about a slightly more general scenario where one puts the trajectoid on a horizontal plane into a certain starting position and orientation. One then starts rolling it carefully by hand, without ever sliding, or rotating the trajectoid around the axis perpendicular to the plane (“pivoting”), or ever lifting its CM, as in Fig. 2. In this scenario, “uphill” or self-crossing paths are allowed and feasible simply by changing the direction of the rolling hand.

We will discuss the mathematics of this problem, and note that the question is somewhat different from what is seen in oloids Sch13, balls rolling on balls or Riemann surfaces Lev93, or a general discussion of acrobatic rolling apparatus Seg21. In these cases, the objects are relatively simple, while for our inverse problem, each given path requires its own adapted wobbly stone, and complicated paths require extremely elaborate chiseling of the stone.

## 3. The Shaving Solution

To see how one can design a rolling stone as was done in SDT23, we consider first a simple polygonal curve . The reader then might think of many other curves that are well-approximated by polygons with infinitesimal segments. We now repeat this curve periodically by adding successively identical copies to its end while maintaining the overall orientation in . More precisely, if and are beginning and end of the path then we consider . We want to construct an object that will roll indefinitely along this infinite curve, either on an inclined plane, using gravity, or when we roll it by hand with the constraints described above. In our opinion, the beauty of this problem lies in it being a quite simple question with a relatively deep answer.

Our starting point is the very simple observation that a cylinder rolls along any piece of a straight line, with the axis of the cylinder perpendicular to that segment. Hence, by intersecting two cylinders, we can construct an object that rolls along two consecutive straight segments, as illustrated in Fig. 2.

We now want to generalize this idea so we can follow a polygonal path with many segments. This path is parameterized by its arc-length , where is the overall length of . For our purpose, it is essential to note that:

1.

The planar path will always touch the inner, heavy ball, while the specifically-shaped weightless shell serves to stabilize the motion. We take the radius of the heavy ball to be , and the maximum radius of the weightless shell to be .

2.

Therefore, by the definition of “following” (Def. 1), the rolling motion can be decomposed into the pure translation of the heavy ball’s CM, at a height above the prescribed path , and a pure rotation of the heavy ball around its CM.

Throughout, the radius of the heavy ball will be , but for some figures and calculations, it will be convenient to use instead where is the length of one (primitive) period of the path.

This allows us to formalize the shaving procedure by following the rolling motion in the frame of reference of the moving CM (now located at the origin ). In these coordinates, rolling is a pure rotation around the CM, and the current contact point is always at . As the object keeps rolling, the contact point leaves a trace on the surface of the inner ball. Let us examine the shaving procedure for just one straight segment in the plane. This corresponds to an arc on the ball, as shown in Fig. 2. At each point along the arc, we shave away any portion of the envelope shell lying beyond the plane tangent to the inner ball at , which serves as the “razor.” For each point of the curve, let be the cut at . This is the set of points between the tangent plane at and the outer shell:

This is shaved away. Thus, after traversing the whole arc, the shaved away part is

The remaining piece is the ball of radius minus the shaved away piece . We could have shaved away the relevant piece all at once at the end of the procedure, rather than at each step. But the first description corresponds better to the mechanical view of Fig. 2.

Neglecting the mass of the outer shell and the effects of inertia (or equivalently, inclining the plane so gently that the gadget rolls very slowly), this construction ensures that the object will roll stably and precisely over the arc. The object can follow a general polygon by similarly shaving along each segment, and this holds for any curve which can be approximated by finer and finer polygons. The computed object will follow , but only once, and will not necessarily follow its repetitions.

We note that shaved surfaces of the trajectoid are locally cylindrical (having zero Gaussian curvature), and therefore can be isometrically flattened (or “developed”) onto a plane without any stretching, compressing, shearing, or tearing. It is easy to see from Eq. 1 that when the outer shell is thick enough (large ratio), the whole outer boundary () is shaven off, (except if the path is a straight line, in which case there always remain spherical cups on the cylinder) such that the whole trajectoid surface is “developable.” Hence, if a trajectoid solution exists for a given path, then by enlarging the outer shell, we can always obtain a developable trajectoid for this path. Such trajectoids belong to the class of developable rollers, which includes sphericons, polycons, and platonicons HS20.

## 4. The Rotation Group

Up to this point, no deep mathematics is needed for the construction. But we would like to achieve more: Namely, if we repeat the original drawn path indefinitely, is there still a solution? Clearly, we need that—after having run over the first repetition of the path—the gadget should be in exactly the same orientation as at the starting point. In more technical jargon, one then says that the holonomy is a pure translation.

In Fig. 1, we show a path and the trajectoid which was fabricated to follow this particular path “indefinitely.” The figure shows just four repetitions of the path, with the prescribed path shown in black and the actually followed path in blue, demonstrating the reachable quality of making the gadget by 3D-printing, as explained in Section 7.

The question we now ask is when such a construction is possible, and here, some deeper mathematics comes in. Before we go into details, one should note the scaling involved in the problem. Namely, if one has a solution, then by scaling the curve and the object by the same factor, one again has a solution. In the discussion below, we will therefore fix the curve and only adapt the radius of the ball until a certain mathematical condition is satisfied. If there exists such a radius , we say that a trajectoid exists for the given curve, and if there is no such , the curve has no solution.

Running over a polygon connecting to , the cumulative effect of rolling is just the product of a sequence of rotations of the shaved ball:

when there are segments in the polygon, with each rotation . To require that the orientation of the piece at the endpoint of the trajectory returns to its initial orientation at the starting point is the same as requiring that the rotation product is , the identity.

In Fig. 3, we illustrate how a given path is actually mapped from the plane onto the ball. We denote the mapped objects by an overarc . The condition means that, when mapped onto the ball, the initial point meets the final point , namely, the curve on the ball (as in Fig. 3) must be closed.

But we also need a condition to guarantee that the orientation of the ball is the same at and at . For this, we need to understand the rotation on the surface of the ball, which involves the use of an index: Consider a path , parameterized by its arc length , where is the length of . Then we can describe by the normal to the path (blue in Fig. 3) in the plane on which the ball rolls, and is the angle this normal forms (in coordinates on the plane). Equivalently, we can specify the path using its in-plane (geodesic) curvature , when the initial angle is given.

We consider for the path the integral over the curvature,

where we integrate over one period of .

When we map onto the ball (Fig. 3), as , then, by isometry, the integral is also defined on the ball, and is equal to the integral of the planar curve: . This follows directly from the conservation of the geodesic curvature by the mapping.

Now, if the path on the ball is closed, then this integral is an index. For example, a path that does not self-intersect is of 0-index, , while each self-intersection adds a phase of (with the sign reflecting left/right handedness), or to the index. Therefore, for the sake of simplicity, in the remainder of the paper, we consider enclosed areas modulo and indices modulo .

By the Gauss-Bonnet theorem, (see, e.g., the book Kob21) the surface enclosed by the path on the ball has area

(The Gauss-Bonnet theorem on the sphere relates the curvature of a closed curve to the enclosed area. This is in contrast to, say, plane triangles, where, giving the angles does not determine the size of the triangle. Harriot discovered in 1603 that for a triangle on a sphere with interior angles , , the area of a triangle is equal to the excess of the sum of the interior angles over (on a sphere of radius 1):

This generalizes to polygons on the sphere, and, by continuous limits to curves on the sphere.) Therefore if we require , then is of 0 index, , and therefore also , implying that the orientation of the ball at the end of is the same as at the beginning .

We see that a trajectoid exists if the curve drawn on the ball encloses exactly half of its surface, namely . Note that for a given curve, there can be many radii for which this condition is fulfilled, as shown in Fig. 4 (B).

Before we discuss which paths have a trajectoid that fulfills the area condition, we generalize the problem somewhat.

## 5. $n$-Paths

The “experimental” situation seems to be as follows: In general, it seems that most path are not -paths. In other words, in this case there is no object that can recover its original orientation after tracing just 1 copy of the (irreducible) path. However, in stark contrast, it seems that “most” paths seem to be 2-paths, although we have no proof nor a good formulation for that (see the discussion in the next section). On the other hand, the following two theorems hold:

In other words, for any path one can construct a trajectoid which will recover its orientation after passing trough copies of , for some finite .

Let denote the space of differentiable functions with the derivative . By any path, we think of a finite path in with . The path may also contain corners and polygons, which may require some trivial reparameterization.

On the other hand, for arbitrary finite one can easily construct paths which are -paths with as we show next:

This means that for any there are indeed paths for which any trajectoid must pass over more than copies to recover its original orientation.

While Thm. 2 is quite general, it is perhaps a little unsatisfactory, as it does not guarantee that for any there is a path which is an -path. We have provided such an example for an explicit choice, as shown in Fig. 6. In this case, one can show, by an explicit multiplication of the 4 rotation matrices of the 4 segments of the path, that for every , there is a for which that path is an -path. The calculation was done for K and . It is important that K is irrational, as otherwise, the left part of the figure is a 1-path, which we also had to exclude in the proof of Thm. 2.

## 6. Prevalence of $2$-Paths

We come back to the Gauss-Bonnet theorem, and will explain, but not be able to prove, why almost any planar curve is a -path (what “almost” means is discussed later). Recall that any trajectoid is related to a closed curve on the ball, enclosing half of its surface, i.e., . This is difficult to obtain with a primitive path, (a path which is not a multiple of some shorter one) but now consider 2-paths (the following discussion is adapted from SDT23), and let us examine what happens after having crossed just one copy of the path , the green curve in Fig. 7.

This curve is not closed, but we close it by the red geodesic arc between M and A. (This shortest arc is unique unless and are antipodal, in which case a slightly modified argument applies.) Consider then the quantity corresponding to the green area of Fig. 7. In other words, we call the area spanned by the (green) curve, together with the red geodesic connecting with .

It turns out experimentally, that, for almost all polygonal paths , we can find a radius for which the enclosed area is either zero, or exactly . Assuming that we found such an then by the Gauss-Bonnet theorem the index of the closed green + red curve equals one half, namely . However, the integral of the curvature vanishes along the great arc (because it is a geodesic), and along (due to the periodicity of the planar curve , ). It follows, that the only contribution to the index comes from the corners and whose angles add up to .

This facilitates the following construction, using Fig. 7: we rotate the green curve by 180 degree about the midpoint of the red arc to form the blue curve. Because the two angles add up to , the blue and green curves are connected smoothly, that is, without corners. The connected curve is therefore the mapping to the ball of a two-period repeat of . Now, since the green area is and is equal to the blue area , the two areas add up to . Thus, by doubling the nonclosed path, we achieved the condition that the enclosed area is that of a half-ball, and hence a trajectoid exists for . It is easy to see that the same argument applies also to paths that begin at a sharp corner, simply by shifting the starting point to a smooth point on the curve.

But, we still have to find an for which the green area equals . We therefore consider the normalized green (or blue) area as a function of given path and a ball of radius . Unfortunately, the function does not have nice monotonicity properties. On the other hand, it proves relatively difficult to construct paths for which for all , as we have seen in Thm. 2. (In Fig. 4 we use the more natural variable for the dependence on the radius. This does not resolve the monotonicity problem.)

So at this point, we are left with the empirical result that for “most” paths we indeed find an for which a trajectoid exists when one passes two copies of the path . Numerical experimentation shows that for randomly chosen paths , one always seems to find an for which . This motivates us to formulate this as a conjecture, but the reader should be aware that the space of finite curves on the plane is infinite-dimensional. In such contexts, notions like measure or genericity are delicate, and it can happen that sets of full measure only contain “uninteresting” examples.

A possible way out could be the piecewise affine interpolation of the curve, which is perhaps closest to the way trajectoids are constructed. Another possibility is to work in Fourier space, imposing conditions on the Fourier coefficients, see, e.g., Kah68.

Let denote the space of differentiable functions with the derivative .

The “piecewise” condition can be omitted if one allows for reparameterizations of the .

Still we can add the following density result: Assume that some path is a -path, i.e., . In general, all paths in a small open neighborhood of are again -paths. Indeed, if the function defined as

traverses transversely—which is the generic case—then, by continuity of the area, for curves near that transversality is maintained and we have a -path. One sees that the 2-path property is mostly an open condition in the space of paths. On the other hand, we conjecture above that close to any non-2-path, there is one which leads to a 2-path.

We illustrate the typical generic and nongeneric cases in Fig. 8. The generic situation appears when crosses the level transversally. Two notable nongeneric cases appear if is tangent to the level , or if jumps because for some the points and are antipodal.

Note that, for fixed , the function oscillates (irregularly) in as a function of , and so, we can indeed see (cf. Fig. 4) that in general, there can be many trajectoids for a given curve (usually with decreasing radii ). It seems that the trajectoid with largest will roll most accurately along its path.

## 7. Fabricating a Trajectoid and Further Remarks

1.

2.

Because of the half-surface condition mentioned above, it turns out that the trajectoid for a -path is actually made of two identical pieces as shown in Fig. 1 and Fig. 9.

3.

One can approximate any smooth curve by finer and finer polygons, shaving off each time a tiny piece for each infinitesimal segment.

4.

We do not know precisely which primitive paths are 1-paths (“primitive” excludes paths that are just concatenations of two (or more) identical pieces). But any path which is a “V” (e.g., the lines connecting with and ) is a -path. The problem probably needs a careful analysis of symmetries of the paths.

## Acknowledgment

We thank the referees for their careful reading of the manuscript and pointing out a number of inconsistencies in the original text. JPE is partially supported by SwissMap. YS and TT are supported by the Institute for Basic Science, Republic of Korea, project code IBS-R020-D1.

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## Credits

Figure 1, Figure 3, and Figure 7 are courtesy of the authors. Previously appeared in SDT23.

Figure 2, Figures 4–6, Figure 8, and Figure 9 are courtesy of Jean-Pierre Eckmann, Yaroslav I. Sobolev, and Tsvi Tlusty.

Photo of Jean-Pierre Eckmann is courtesy of Mathematisches Forschungsinstitut Oberwolfach gGmbH (MFO). CC-BY-SA 2.0.

Photo of Yaroslav I. Sobolev is courtesy of Yaroslav I. Sobolev.

Photo of Tsvi Tlusty is courtesy of Tsvi Tlusty.