When it comes to toys, it doesn't get much simpler than a spinning top: an aerodynamic shape with a point that will spin in place once you set it in motion. It's elegant, graceful and fascinating. The toys themselves are also perfectly symmetrical; if they're not, they're going to topple.
This is due to something called moment of inertia, which is the rotational inertia of an object -- that is, the resistance of that object to being rotated. This is affected by the mass of the object and the rotation axis; objects with evenly distributed mass resist rotation less than objects with uneven mass, particularly if the axis is in the dead centre of that mass.
So a perfectly symmetrical object, where the mass is evenly distributed -- such as a spinning top -- will spin beautifully on its point. If you were to try to spin a teapot, however, even if you did have a smooth, rounded foot for it to spin on, it would topple in very short order, since the handle and spout would unbalance the rotation.
Disney Research Zurich's algorithm is designed to counter this lack of balance in asymmetrical objects without, where possible, altering the external shape. Instead, it redistributes the mass inside.
"Since the moment of inertia depends on the entire volume, rather than just on the surface geometry, we preserve the appearance of the input design by prioritising changes to the internal mass distribution," the team wrote in its paper, presented at SIGGRAPH 2014.
The team started by adjusting solid models with hollow voids in the interior. The centre of mass is also placed as low on the rotation axis as possible. This balances the asymmetrical form factor of the object, allowing it to spin, while 3D printing allowed the team to rapidly prototype their objects.
For objects where hollowing wasn't effective, the team used "cage-based deformation", deforming both the interior and exterior voids to smooth the shape into one that better allowed for hollowing. Finally, they used a technique they called "dual-density optimisation"; that is, using materials of multiple densities as fill, allowing the external shape to remain unmodified.
"Our approach is effective on a wide range of models, from characters such as an elephant balancing on its toe, or an armadillo break-dancing on its shell, to abstract shapes," said Moritz Bächer, a post-doctoral researcher at Disney Research Zurich. "It's well-suited to objects that can be produced with a 3D printer, which we used to make tops and yo-yos with unusual shapes but remarkably stable spins."