The present paper focuses on the problem of the connected road intersection of multiply connected Lie group covering learning which was recently shown to possess a cover learning based on the connectivity of Lie group on the author's previous studies. It discusses a geodesic curve for optimal mapping of roads to minimize the correlation of roads from different connected spaces and maximize the correlation of roads within the same connected space. A review on some relevant notions from Lie-group connectivity theory is provided, followed by a brief introduction of multiply connected covering learning algorithm. New path optimization algorithms are then proposed. Some numerical experiments compared with classical covering learning methods, Lie group means learning algorithms and the author's previous algorithm serve to illustrate the better classification performance of the presented optimization algorithms.