This paper reviews a dynamics discretization method called spatially quantized dynamics (SQD)and shows that time scale transformation yields the SQD. Time scale transformation also derives con-tinuous spatial dynamics representations, which are equivalent to the original dynamics. By introducinga path parameter and path frames, the SQD are expanded into the 3-dimensional Euclidean space. Anefficient method to calculate actual time steps evolving system state is proposed. As a result, trajectoryoptimization using the proposed SQD enables a kinematic walking pattern to satisfy dynamic constraints,and how to generate a biped gait with stretched knees along an arc on a horizontal plane is presented.