FIGURE 1.

After Caudai et al. (2019b). (A) An illustrative example of the bead chain at some scale represented as a single, modified bead at the immediately coarser scale. The modified bead is characterized by the two endpoints of the reconstructed chain, its centroid and its physical size, represented by the semi-transparent large sphere. This allows us to adequately place and rotate each bead when reconstructing the chain at the coarser scale. (B) A chain at the immediately coarser scale made of four beads, properly placed and rotated. (C) Geometric relationships between two consecutive beads. The two beads determine, respectively, planes P ₁ and P ₂ , and the two centroids C ₁ and C ₂ with the common endpoint E ₁₂ ≡E ₂₁ determine plane P ₁₂ . Angles $\zeta =\widehat{{\mathbf{P}}_{\mathbf{1}}{\mathbf{P}}_{\mathbf{12}}}$ and $\xi =\widehat{{\mathbf{P}}_{\mathbf{1}}{\mathbf{P}}_{\mathbf{2}}}$ are used to establish the 3D rotations of the two beads. Angles α ₁ and α ₂ are fixed when modeling the beads at the immediately finer scale.