Rigid-body chain following on piecewise analytic paths is a fundamental subroutine in motion planning and multibody simulation. The problem is nontrivial when only the leader trajectory of the first node is available: enforcing fixed inter-node distances reduces to circle–curve intersection, which is generally multi-valued and becomes particularly challenging near non-smooth junctions. We present a Dichotomy Geometric Path Search (DGPS) framework that converts each constraint into a one-dimensional root-finding task and resolves the branch selection through no-backtracking ordering: at every time step, the admissible solution for the current node is the nearest feasible root in the past relative to its immediately preceding node. DGPS combines backward bracketing with bisection, achieving robust convergence. Compared with the inverse Jacobian method, which maps end-effector velocities to joint velocities via explicit inversion, the proposed approach avoids Jacobian inversion and globally coupled nonlinear solves. We further characterize the local structure of the zero set and establish monotonicity/uniqueness conditions that justify stable root selection across piecewise junctions. Extensive tests on representative piecewise trajectories (line–arc–line, polylines with corners, piecewise sinusoids, and time reparameterization) show that DGPS enforces distance constraints to near machine precision, produces interpretable speed/acceleration transients around non-smooth events, and exhibits computational costs consistent with iteration difficulty. The results support DGPS as a general, efficient solver requiring only the prescribed leader trajectory.