Smooth-Rolling Knots

Year: 2025 Authors: Max Brodeur; Michele Vidulis; Liliane-Joy Dandy; Mark Pauly

Core claim

A two-stage optimization can transform Morton’s knots and related torus knots into perfect smooth-rolling curves with zero center-of-mass oscillation.

Topics

knot geometry, rolling motion, convex hull optimization, curve approximation

Domains

differential geometry, optimization, torus knots, convex geometry, kinetic sculpture, jewelry design, 3D printing, metal casting

Methods

parametric curve modeling, gradient-based optimization, L-BFGS-B, spline sampling

Media

3D-printed model, metal cast, earring, video

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2025 Conference Proceedings

Smooth-Rolling Knots

Max Brodeur ${}^1$ , Michele Vidulis ${}^2$ , Liliane-Joy Dandy ${}^3$ , and Mark Pauly ${}^4$

Geometric Computing Laboratory, EPFL, Switzerland

1max.brodeur@epfl.ch 2michele.vidulis@epfl.ch

3liliane-joy.dandy@epfl.ch 4mark.pauly@epfl.ch

Abstract

Morton’s knots are a family of space curves to which there is no plane simultaneously tangent in three distinct points. This property enables a physical instance to roll on a plane while having at most two contact points with it. These knots are not smooth-rolling, i.e. they require a force to provoke a rolling motion, during which their center of mass oscillates up and down. By drawing a connection between Morton’s knots and smooth-rolling Two-Disk Rollers, we design new smooth-rolling knots. These curves preserve the mesmerizing rolling behavior of Morton’s knots while requiring only an infinitesimal force to be set in motion. We apply our method to a variety of knots beyond Morton’s family, obtaining tightly-winding, smooth-rolling knots with different topologies.

Figure 1: Left: The center of mass heights $z_{\mathrm{cm}}$ along the rolling motion of knots with $a=0.9$ . Right: Multijet 3D-print (blue) and metal cast (gold) of a smooth-rolling knot with $a=0.7$ .

Morton’s Knots and Smooth-Rolling Structures

Morton’s knots [5] are a geometric family of trefoil knots defined by the following parametrization:

where $a\in (0,1)$ is a shape parameter, $a^2+b^2=1$ , $c=\frac{a}{1+b}$ is an isotropic scaling factor, and $z$ encodes an anisotropic scaling factor along the $z$ -axis. Eget and coauthors [3] have studied these knots for their rolling behavior. Knots in this class are tritangentless according to Morton’s definition [5], i.e. there exists no plane tangent to the curve in three distinct points. Tritangentless knots can roll continuously on a plane without ever having more than two contact points with it.

To quantify rolling smoothness, Eget et al. measure the range-to-average ratio, denoted by $\rho$:

$\rho =\frac{\max z_{\mathrm{cm}}-\min z_{\mathrm{cm}}}{z_{\mathrm{cm}}},$ (2)

where $z_{\mathrm{cm}}$ is the center of mass height and ${\overline{z}}_{\mathrm{cm}}$ its average over the rolling motion. The smaller $\rho$ is, the smoother the motion. The smoothest-rolling objects have $\rho =0$ and are said to be smooth-rolling, such as a right cylinder with a circular base. With an elliptical base, we get $\rho >0$ due to vertical oscillations of the center of mass with periodic exchanges between kinetic and potential energy. For smooth-rolling objects, the onset of rolling motion can be achieved with an arbitrarily small force, or equivalently with an arbitrarily small tilt of the support plane. The variation of the center of mass height is visualized in Figure 1 for Morton’s knot with $a=0.9$.

Eget and colleagues look for the smoothest-rolling of Morton’s knots by minimizing $\rho$ with respect to $a$. To further reduce $\rho$ for all values of $a$, they propose to optimize over the $z$ scaling factor. Building upon these results, Dzojic and colleagues [2] apply additional scaling transformations to obtain yet smaller values of $\rho$. However, none of these methods yield smooth-rolling knots.

We introduce a new method to obtain smooth-rolling knots by transforming Morton’s knots. The method stems from the observation that the optimally stretched knots by Eget and colleagues have external lobes that visually resemble a Two-Disk Roller (TDR) [4].

TDRs are made of two identical orthogonal elliptical disks characterized by $\alpha$ and $\beta$, the lengths of the ellipse half-axes, and by the distance $\gamma$ between the two ellipse centers (see Figure 2). The oloid, an iconic example of a rolling structure [6], can be seen as the convex hull of such disks, with $\alpha =\beta =\gamma$ and both circles passing through each other’s centers. As is the case for any generic TDR, the center of mass of the oloid oscillates up and down while rolling. Engelhardt and coauthors [4] identify a two-parameter class of smooth-rolling TDRs characterized by the relation ${\gamma }^2=4{\alpha }^2-2{\beta }^2$.

We frame the search for smooth-rolling curves that approximate a given knot as an optimization problem, and we solve it in two stages. First, a smooth-rolling TDR is fit to the curve allowing vertical stretching of the knot. Then the curve is morphed within the convex hull of the TDR, controlling the trade-off between curve smoothness and knot approximation, while preserving the smooth-rolling behavior. We also apply our method to knots outside of Morton’s family, leading to the design of new sets of smooth-rolling curves with various topologies.

3 Method

We want to find an optimal knot $K^{\ast }$ that minimizes a distance measure $d$ from a family of curves $\mathcal{C}$ subject to a smooth-rolling constraint. We are interested in the case $\mathcal{C}=K_a(z)=\{(x_M(t),y_M(t),z_M(t))\in {\mathbb{R}}^3,t\in [0,2\pi ]\}$, i.e. knots from family (1), where the shape parameter $a$ is fixed and the scaling parameter $z$ is free.

Assuming a homogeneous mass density along the curve, by symmetry, the center of mass of $K_a$ is always located at the origin. In such symmetric knots, only the points lying on the convex hull of the curve affect the rolling behavior. We can therefore restrict the search for $K^{\ast }$ to curves whose convex hull guarantees $\rho (K^{\ast })=0$, such as the ellipses of a smooth-rolling TDR.

Figure 2 illustrates the computation of a smooth-rolling curve in two stages. We first look for the TDR that best fits the exterior lobes of the target knot, i.e. the portions of the curve defining the convex hull. The vertical stretching proposed by Eget et al. aligns the exterior lobes to orthogonal ellipses (left vs. middle knots in Figure 2). We then fix the exterior to this smooth-rolling TDR, and morph the interior segments by minimizing a metric that strikes a balance between global curve smoothness and approximation of $K_a$.

In the first stage, we determine the optimal parameters ${\alpha }^{\ast },{\beta }^{\ast },z^{\ast }$ of the smooth-rolling TDR $T(\alpha ,\beta )$ that best fits $K_a(z)$, with $a$ given, by minimizing the distance $d(T(\alpha ,\beta ),\mathcal{H}[K_a(z)])$, where $\mathcal{H}[\mathcal{C}]$ denotes

Smooth-Rolling Knots

Figure 2: Given Morton’s knot $K_a$ (left), we compute the $z$-stretch and the best-fitting TDR (center), then morph the knot to match the convex hull of the TDR while preserving curve smoothness (right).

the portions of the curve $C$ belonging to its convex hull. Since affine transformations preserve convexity, a scaling of $C$ results in a corresponding scaling of $\mathcal{H}[C]$. Therefore, by pre-computing the convex hull, we circumvent the issue of differentiation through a potentially discontinuous operation and efficiently solve the optimization problem via gradient-based optimization. We discretize $K_a$ and $T$, compute the average minimum distance $d$ from the knot’s hull to the TDR ellipses, and solve the distance minimization problem in Python via automatic differentiation and L-BFGS-B [8]. A result of this step is shown in Figure 2, center.

In the second optimization stage, we project $K_a(z^{\ast })$ onto the convex hull of $T({\alpha }^{\ast },{\beta }^{\ast })$. Using naive closest-point projection would create discontinuities and sharp kinks at the junction between the interior points of the knot and the TDR convex hull, as shown in Figure 2, top right. We formulate a second optimization problem over the curve by introducing an interpolating cubic spline with approximately equispaced control points sampled from the result of the previous stage. The exterior control points are pinned in place while the interior ones, collectively named $\mathbf{q}$, are the variables of the following problem:

Here, $E_{\mathrm{knot}}$ penalizes the distance between the optimized knot and $K_a(z^{\ast })$, while $E_{\mathrm{curvature}}$ and $E_{\mathrm{TDR}}$ ensure curve smoothness by penalizing the discrete curvature at the interior-exterior junction and attracting interior points to the TDR. All quantities are computed by sampling the spline. Assuming $w_{\mathrm{knot}}=1$, the weights $w_{\mathrm{TDR}}$ and $w_{\mathrm{curvature}}$ are free parameters to be set by the user that allow exploration of the trade-off between curve smoothness and approximation quality (see Figure 2). The union of the TDR convex hull with the interior points of the optimized curve gives us a candidate smooth-rolling knot $K^{\ast }$. Since some points might be pushed outside of $\mathcal{H}[T]$ during the optimization process, we computationally check that $\rho (K^{\ast })=0$. If needed, we increase $w_{\mathrm{TDR}}$ until this condition is met.

Results

The first two columns of Figure 3 show the results of our algorithm for Morton’s knots with $a=0.3$ and 0.5. While the center of mass of Morton’s knot and the stretched knot exhibit vertical oscillations (\rho > 0), its height is constant for all optimized knots. Besides Morton’s knots, which are (3,2)-torus knots, we also consider other $(p,2)$-torus knots with odd integer $p$, as suggested by Eget et al. Even though they are not tritangentless anymore, these knots still roll. They feature two exterior lobes, analogous to TDRs, and have intriguing, tightly-winding curves in their interior. Figure 3 showcases our results for values of $p=5,7$ and 9. Our code, as well as videos and 3D models, can be found online [1].

Brodeur et al.

Figure 3: Morton’s knots, vertically stretched knots, and optimized knots for various $a$ and $p$ values.

Conclusion

In this work, we achieve perfect rolling smoothness by transforming Morton’s knots to match the convex hull of smooth-rolling TDRs. It would be worthwhile to explore the general problem of finding such curves without relying on TDRs. A potential application of our results is their combination with trajectory optimization [7] for mechanical systems or robotics design, where rolling components with smooth and predictable movements are crucial. We envision artistic applications with smooth-rolling knots used for designing jewelry pieces (see Figure 4) or serving as the foundation for elegant kinetic sculptures.

Figure 4: Cast earring

References

[1] https://go.epfl.ch/smooth_rolling_knots. [2] K. Dzojic, Sciortino. Rolling Knots. 2023. [3] A. Eget, S. Lucas, and L. Taalman. “Optimizing Morton’s Tritangentless Knots for Rolling.” Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture. 2020. [4] C. Engelhardt and C. Ucke. “The Two-Disc-Roller - a Combination of Physics, Art and Mathematics.” 2017. [5] H. R. Morton. “Trefoil Knots without Tritangent Planes.” Bulletin of the London Mathematical Society, vol. 23, no. 1, 01 1991, pp. 78–80. [6] P. Schatz. Rhythmusforschung und technik. Verlag Freies Geistesleben, 1975. [7] Y. I. Sobolev, R. Dong, T. Tlusty, J.-P. Eckmann, S. Granick, and B. A. Grzybowski. “Solid-body trajectoids shaped to roll along desired pathways.” Nature, vol. 620, no. 7973, Aug. 2023, pp. 310-315. [8] C. Zhu, R. Byrd, P. Lu, and J. Nocedal. “Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization.” ACM Trans. Math. Soft., vol. 23, no. 4, Dec. 1997, pp. 550-560.