Spin-It: Optimizing Moment of
Inertia for Spinnable Objects
By Moritz Bächer, Bernd Bickel, Emily Whiting, and Olga Sorkine-Hornung
Spinning tops and yo-yos have long fascinated cultures
around the world with their unexpected, graceful motions
that seemingly elude gravity. Yet, due to the exceeding difficulty of creating stably spinning objects of asymmetric shape
in a manual trial-and-error process, there has been little
departure from rotationally symmetric designs. With modern 3D printing technologies, however, we can manufacture
shapes of almost unbounded complexity at the press of a button, shifting this design complexity toward computation.
In this article, we describe an algorithm to generate
designs for spinning objects by optimizing their mass distribution: as input, the user provides a solid 3D model and
a desired axis of rotation. Our approach then modifies the
interior mass distribution such that the principal directions
of the moment of inertia align with the target rotation frame.
To create voids inside the model, we represent its volume
with an adaptive multiresolution voxelization and optimize
the discrete voxel fill values using a continuous, nonlinear
formulation. We further optimize for rotational stability by
maximizing the dominant principal moment. Our method
is well-suited for a variety of 3D printed models, ranging
from characters to abstract shapes. We demonstrate tops
and yo-yos that spin surprisingly stably despite their asymmetric appearance.
Spinning toys have existed since antiquity as playful objects
that capture the imagination. Invented independently all
over the world, spinning tops are referenced in ancient
Greek literature, 12 and evidence of clay tops has been found
in ancient cities dating as early as 3500 B.C. Similarly, while
yo-yos are believed to have been invented in China, there
are many historical references, including in Mozart’s The
Marriage of Figaro where a yo-yo is spun to relieve stress. 17
Despite the long tradition of these toys, until today creating
new designs has been a trial-and-error process, calling on
the intuition and meticulous patience of artists and hobbyists. Moreover, there has been little departure from rotationally symmetric designs.
Much attention has been devoted in the field of classi-
cal mechanics to explaining the motion of spinning objects;
however, the focus has been primarily on analysis8, 9, 19, 21
rather than design. In this article, we investigate the unique
geometric properties of shapes that spin, with an eye on
digital modeling and free-form design. A stable spin has
requirements on rotational inertia, including precise posi-
tioning of the center of mass and correct alignment of the
primary axes of the body. We propose an algorithm to opti-
mize for these inertial properties, for example, to design a
spinning top that rotates smoothly and stably and can be
fabricated using 3D printing.
In our approach, users provide an initial design for a
spinning model, specified as a 3D surface mesh. Along with
the input geometry, the user may specify the desired axis of
spinning and the contact point with the support. The mass
distribution is then optimized to ensure that the primary
axis for the moment of inertia aligns with the desired axis
of rotation. 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 changing the
internal mass distribution as we illustrated in Figure 1 on an
We first formulate a nonlinear functional that measures
the spinnability of a solid shape about a user-defined axis.
Using this measure, we then devise constrained optimization
problems that align the principal axes for moment of inertia
with user-specified rotation axes. To this end, we maximize
the ratio of principal moments in the dominant and lateral
directions and place the center of mass on the rotation axis.
For our tops, we further improve stability by lowering the
center of mass, simultaneously reducing the mass.
The original version of this paper was published in
Proceedings of SIGGRAPH’ 14, August 2014, ACM.
Figure 1. We describe an algorithm for the design of spinning tops
and yo-yos. Our method optimizes the inertia tensor of an input
model by changing its mass distribution, allowing long and stable
spins even for complex, asymmetric shapes.
Our spinning top design
Hollowed, optimized model
Elephant in motion