Search algorithms for optimizing a complex problem are mainly categorized as gradient-driven and stochastic search, each with its advantages and shortcomings. A newly developed algorithm, GO-POLARS, is proposed with a hyperspherical coordinate framework, which could perturb a given direction with well-controlled variation. It designs a steerable stochastic search algorithm that explores toward a promising direction, such as the gradient, at any desired levels. In this note, we provide an analytical study on the hyperspherical coordinates and the corresponding random distributions and, thus, prove the local convergence property of the GO-POLARS. Extensive numerical experiments are illustrated to show its advantages compared to conventional search algorithms.