Solid-body trajectoids shaped to roll along desired pathways.

Area of Science:

• Mathematics
• Physics
• Robotics
• Optics

Background:

• Rolling motion typically involves simple shapes like cylinders and spheres moving linearly.
• Exotic solids like oloids and sphericons exhibit curvilinear rolling paths but are limited to sinusoid-like trajectories.
• Existing research on rolling bodies has focused on specific shapes and their limited path diversity.

Purpose of the Study:

• To determine if a general solution exists for designing a rolling body to trace a given infinite periodic trajectory.
• To develop an algorithm for creating such bodies, termed 'trajectoids'.
• To experimentally validate the designed trajectoids by 3D printing and tracking their rolling paths.

Main Methods:

• Development of a computational algorithm to design trajectoids based on desired infinite periodic trajectories.
• 3D printing of the designed trajectoid shapes.
• Experimental tracking and analysis of the rolling paths of the printed trajectoids.

Main Results:

• Successful design and validation of trajectoids capable of following arbitrary infinite periodic paths.
• Demonstration of trajectoids tracing complex paths, including those that close onto themselves.
• Observed intermittent uphill motion of the center of mass for certain trajectoid designs.

Conclusions:

• A general method for designing rolling bodies (trajectoids) for any infinite periodic trajectory has been established.
• Trajectoids offer a significant expansion in the diversity of rolling motion beyond simple curvilinear paths.
• The study has potential implications for quantum and classical optics due to exact mappings with trajectoid dynamics.