Finding the minimum distance between two ellipsoids is harder than it sounds. Spheres are easy — compare center distances minus radii. Boxes are manageable — project onto axes. Ellipsoids have no closed-form solution because the curvature varies continuously across the surface, and the closest points depend on the relative orientation of both shapes.
Amirkhani and Zhang (arXiv:2603.22683) solve this by sliding. Place a point on each ellipsoid's surface. Connect them with a line segment. If the segment isn't perpendicular to both surfaces, the points can slide to reduce the distance — the component of tension tangent to each surface drives the update. Iterate until the connecting segment is perpendicular to both surfaces simultaneously. At that point, no surface-tangent motion can shorten it further. The distance is minimal.
The termination condition is geometric: perpendicularity of the connecting segment to both tangent planes. This is the same as requiring that the gradient of the distance function vanishes along both surfaces — but framed as a physical intuition rather than an abstract optimality condition. Two people sliding toward each other on curved hills stop when the rope between them pulls straight through both slopes.
The method's simplicity is its contribution. Prior algorithms for ellipsoid-ellipsoid distance involve algebraic degree analysis, polynomial system solving, or optimization over six parameters. This approach maintains two points on two surfaces and updates them by a clear geometric rule. The mathematics is the geometry. No auxiliary structures, no algebraic reformulations, no black-box optimizers. The algorithm is the picture.