Rotations About Body Points: Sharp Bounds for Rigid Motion

Let B ⊂ R^n be a compact rigid body with nonempty interior and diameter L, with n ≥ 2. We study the problem of realizing an orientation-preserving isometry of R^n as a composition of rotations whose centers lie on the current configuration of B. An earlier version of this work claimed that at most n+1 such rotations always suffice. That claim is false: a pure translation v requires at least ⌈|v|/(2L)⌉ rotations, so no bound depending only on n can hold. In this paper we establish three results. (1) General bound: if B is uniformly fat (contains a ball of radius cL), then for an arbitrary orientation-preserving isometry, O(n + d/L) rotations suffice, where d = |T(O) − O| is the displacement of the inscribed ball center O. The matching lower bound Ω(d/L) follows from the displacement obstruction. (2) Bounded case: if there exist distinct points P_1, P_2 ∈ B such that |T(P_1) − P_1| ≤ 2|P_1 − P_2|, then at most n+1 rotations suffice. (3) Translation-augmented variant: if a single translation is permitted as an additional operation, then every orientation-preserving isometry can be realized in at most 2 operations.