Symmetrical solution to Continuous transmission pulley problem

We study a geometric closure problem arising in belt–pulley mechanisms, where a prescribed belt length induces an implicit relation between the radii of two pulleys. For a given discrete radius profile r(t), the corresponding profile R(t) is defined pointwise on a uniform grid by solving an implicit belt-length equation, which yields an explicit mapping R = g(r) via a one-dimensional root-finding step. To enforce symmetry, we propose a simple discrete symmetrization iteration that couples only the symmetric grid pairs (t _i , 1−t _i ) and preserves the closure relation at every iteration. For each fixed pair, we prove geometric decay of the symmetry error under a Lipschitz condition on g in the open-belt regime. Based on the observed regularity of the computed profiles, we additionally derive a low-order quadratic approximation for the symmetric discrete radius samples and provide an a posteriori bound on the maximum deviation using standard interpolation error estimates. Numerical experiments illustrate the convergence of the iteration and the accuracy of the quadratic approximation for representative parameter values.