Smooth-Rolling Knots
Left: simulation of a smooth-rolling knot (top), the colored curves represent the height of the centre of mass. Right: two physical prototypes, a multijet 3D-print (top, blue) and metal cast (bottom, gold).
- Project Overview
- Mortonâs 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 Mortonâs knots and smooth-rolling Two Disk Rollers, we design new smooth-rolling knots. These curves preserve the mesmerizing rolling behavior of Mortonâs knots while requiring only an infinitesimal force to be set in motion. We apply our method to a variety of knots beyond Mortonâs family, obtaining tightly-winding, smooth-rolling knots with different topologies.
Our optimized knots (bottom) roll smoothly, i.e., their centre of mass is always at the same distance from the rolling plane. We generate smooth rolling torus knots with increasing topological complexity.
Given a Mortonâs knot (left), we compute the vertical stretch and the best-fitting Two-Disk Roller (TDR, center), then morph the knot to match the convex hull of the TDR while preserving curve smoothness (right).
The optimization process smooths out sharp kinks while preserving smooth-rolling behaviour.
Out method can be applied to (p, 2)-torus knots with p â„ 3, making them smooth-rolling.Â
Our optimized knots can be 3D printed: they indeed roll smoothly!
