Extended Data Fig. 2: Deforming the input path non-uniformly to make it have a one-period trajectoid.
From: Solid-body trajectoids shaped to roll along desired pathways
a, Angle \(\theta \) between starting and final orientations of a sphere rolled along a target path (same as in Fig. 2a and Fig. 4d) that was deformed non-uniformly in two directions by scaling factors \({k}_{x}\) and \({k}_{y}\) (see Methods for details). Examples of deformed path on the left correspond to \({k}_{y}=1,2,3\) and fixed \({k}_{x}=1\); examples below correspond to \({k}_{x}=0.5,2\), and \({k}_{y}=1\). Fulfilling the \({R}_{{\rm{A}}\Omega }=1\) condition requires that \(\theta \) be zero (blue regions). b,c, Curves traced over the sphere by sphere-plane contact point when using paths corresponding to the local minima marked as b and c in the \(\theta ({k}_{z},{k}_{y})\) map in panel (a). d–f, Deformations additional to non-uniform scaling affect the minimal mismatch on the \(\theta ({k}_{z},{k}_{y})\) map: a path (blue, same as in Fig. 2a and Fig. 4d) is partially modified by adjusting its Bezier curve nodes; the modified section is shown in orange in e. Maps (d,f) of mismatch angle \(\theta \) for both the original path (d) and its modified version (f) are plotted for the vicinity of the useful minimum of the original path (see full map in panel a). Note the increase of mismatch in f compared to d.
