Figure 1

Steps to obtain a transformation. (1) put a _i at one of the $O\left({\left(\frac{1}{\epsilon }\right)}^3\right)$ grid points d-ball of b _i′ . (2) put a _j at a grid point on the intersection of the sphere centered at a _i with radius |a _i a _j | and d-ball of b _j′ . There are at most $O\left({\left(\frac{1}{\epsilon }\right)}^2\right)$ grid points on the intersection. (3) use a _i and a _j as the rotation axis.