Putting Rigid Bodies to Rest
ACM Transactions on Graphics (SIGGRAPH 2025)
Abstract
This paper explores the analysis and design of the resting configurations of a rigid body, without the use of physical simulation. In particular, given a rigid body in R^3, we identify all possible stationary points, as well as the probability that the body will stop at these points, assuming a random initial orientation and negligible momentum. The forward version of our method can hence be used to automatically orient models, to provide feedback about object stability during the design process, and to furnish plausible distributions of shape orientation for natural scene modeling. Moreover, a differentiable inverse version of our method lets us design shapes with target resting behavior, such as dice with target, nonuniform probabilities. Here we find solutions that would be nearly difficult to find using classical techniques, such as dice with additional unstable faces that provide more natural overall geometry. From a technical point of view, our key observation is that rolling equilibria can be extracted from the Morse-Smale complex of the support function over the Gauss map. Our method is hence purely geometric, and does not make use of random sampling, or numerical time integration. Yet surprisingly, this purely geometric model makes extremely accurate predictions of rest behavior, which we validate both numerically, and via physical experiments. Moreover, for computing rest statistics, it is orders of magnitude faster than state of the art rigid body simulation, opening the door to inverse design---rather than just forward analysis.
Dice Rolling Experiments Video
Acknowledgments
This work was generously supported by NSF awards 2212290, 1943123, NSERC Discovery (RGPIN–2022–04680), the Ontario Early Research Award program, the Canada Research Chairs Program, a Sloan Research Fellowship, the DSI Catalyst Grant program and gifts from Adobe Systems. The authors also thank Joseph Sharp for resin printing the dice, Zoë Marschner and Olga Guțan for helping with conducting and recording the dice experiments, and Carnegie Mellon University TechSpark 3D printing facilities for enabling and helping with dice fabrication.