Tumbling Downhill along a Given Curve

Abstract

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 $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? 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.

Figures (9)

• Figure 1: The performance of an actual trajectoid rolling downhill (left-to-right). To demonstrate that trajectoids can be fabricated for rather general paths, we take as a prescribed path (black curve) the shape of "a boa constrictor digesting an elephant", often mistaken to be simply "a hat", from The Little Princeprince, which is repeated periodically. The blue curve is followed by the actual fabricated trajectoid shown on the left, which has outer diameter of 4.128cm. Black in-plane 1-cm scale bar is shown on the right. Deviations from the ideal black curve are attributed to the precision of the 3D printing, some inertia, and the errors of determining the location of the center of mass (CM) by the projection centroid method xxx. Note that this trajectoid is made of two identical pieces (green and pink), and returns to its original orientation after rolling along two periods (this double periodicity is denoted by alternating solid and empty black circles along the flat path). Adapted with permission from Fig. 4j of xxx.
• Figure 2: Top: The shaving. Left: A green ball of radius 1 surrounded by a light blue ball of radius 1.4. A cylindrical surface is shaved from the blue envelope, across the violet line. The resulting object can roll along the red curve, which is a great arc on the inner green ball, while keeping its center of mass (CM) at fixed height (1, the inner ball's radius). Like a cylinder, the object will roll along a straight line, but its CM will be elevated as soon as it would try to roll beyond one of the ends of the red curve. Right: After a second shaving, the object will turn and roll along the blue line, again keeping the CM at the same constant height of 1. Bottom: The effect of rolling beyond the cut. Left: The position when the object reaches the end of the red arc. Right: The CM of the object is lifted as soon as it rolls on beyond this point, and such a motion is therefore prevented by gravity. Similarly, tilting sideways also lifts the CM.
• Figure 3: When the ball rolls downhill, the path ${\mathrm{P}} = {\mathrm{A}\Omega}$ is mapped isometrically onto the ball as $\overset{\frown}{{\mathrm{P}}}$. In particular, the geodesic curvature of the planar path, $\kappa(t)$, is conserved. The sufficient condition for a trajectoid to exist is that this curve encloses half of the ball's surface (the rightmost stage). Adapted with permission from Fig. 2a of xxx.
• Figure 4: An illustration of the existence of multiple solutions i.e., radii $r$, for the path shown in (A). The area enclosed by the trace of a single period (green in Fig. 7) and the geodesic arc connecting its ends (red in Fig. 7) is plotted against $\sigma={L}/(2\pi r)$ on the horizontal axis. Here, $L$ is the length of single period of the path, $r$ is the radius of the ball. Five two-period trajectoid solutions are marked by red dots in (B). Two-period spherical traces for the first two solutions (at $\sigma \approx 0.707$ and at $\sigma \approx 1.398$) are shown in (C) and (D), respectively. Color coding in (C) and (D) is the same as in (A).
• Figure 5: A family of paths ${\mathrm{P}}$ that are made of two identical pieces $\mathrm{W}$, where the second $W$ is rotated by an angle $\beta$ with respect to the first. For small enough $\beta$, ${\mathrm{P}}$ is an $n$-path with large $n > \pi/\beta$, unless $\mathrm{W}$ is a 1-path. Two balls below show spherical traces for trajectoid solutions for ${\mathrm{P}}$ with $n=5$ and $n=8$. In both cases, a fixed $\mathrm{W}$ and $\beta=\pi/4$ were used. The enclosed shaded green areas are $2\pi r^2/5$ and $2\pi r^2/8$, respectively.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Theorem \oldthetheorem
• Theorem \oldthetheorem
• proof : Proof of Thm. \ref{['thm:1']}
• proof : Proof of Thm. \ref{['thm:3']}
• Conjecture 1