The Oloid’s Inertia

The oloid is the convex hull of two unit circles in perpendicular planes, each passing through the other’s center. It rolls on its entire surface — every point touches the ground during a full revolution — and it develops a ruled surface between two curved edges. Despite this elegant geometry, its moment of inertia tensor has resisted analytical expression.

The difficulty is the integration domain. The oloid is defined implicitly as a convex hull, not by an explicit surface equation. Computing the inertia tensor requires integrating mass times squared distance over a volume whose boundary is the ruled surface connecting two non-coplanar circles. The boundary conditions change character depending on angle — sometimes bounded by one circle’s projection, sometimes by the other’s, sometimes by both.

The analytical result now exists and is confirmed numerically. The derivation requires carefully decomposing the volume into regions where the boundary constraints take different functional forms, computing the integral in each region, and summing. The tensor inherits the oloid’s symmetry: two principal moments are equal (the object has a twofold rotational symmetry axis) and the third differs.

The oloid is one of very few convex bodies whose surface is entirely developable — it can be unrolled flat without stretching. This makes it useful in industrial mixing (every point contacts the vessel wall) and mathematically interesting (developable surfaces are rare among convex bodies). Having the inertia tensor in closed form completes the mechanical characterization: you can now predict how an oloid responds to torques, vibrations, and rotational perturbations without numerical integration.