A sphere, carried by its own group

A sphere, drawn faint where it started and solid where a displacement of its own symmetry group has carried it. The two drawings are the same set of points. A ball in a socket. Turns three ways about its centre and slides nowhere. The permitted twists are computed from the surface's own normals — one linear condition per sample point, saying that the velocity the twist gives that point is tangent — and the answer here is 3 freedoms — rotations about a point. Every point of the displaced surface satisfies the original surface's own equation to 3.3e-16, which is what "the surface slides on itself" means as a number. A lower pair is two bodies touching over a surface, so this group is exactly what the joint permits, and its dimension is the freedom count the constraint field has been adding up since the foundation.