## 1 Introduction
Therefore, on the premise of not affecting the degree of convergence, after every corrective polishing iteration in the pre-polishing stage, the residual error distribution that is more uniform and almost free from the influence of the parent surface form error distribution should be obtained as much as possible, which is of great theoretical significance and engineering value. The mid-spatial-frequency (MSF) error, i.e., the ripple error of optical components, has an important impact on the performance of laser systems and high quality imaging systems. At present, some scholars mainly have been focusing on the suppression of ripple error. However, there is not much research on the genetic phenomenon of the error distribution and its relationship with ripple error. When using jet polishing to correct the surface with a large difference in surface form error, [PERSON] et al. proposed a selective region path planning strategy to optimize the residual error distribution and successfully avoided the introduction of path error in the region where the surface form accuracy meets the requirements [11]. The influencing factors of ripple error mainly include initial surface error, TIF and polishing path. In terms of the polishing path, the ripple error can be suppressed by changing the randomness of the path to optimize the period of the ripple error and choosing a reasonable path spacing to optimize the height of the ripple error [12]. [PERSON] et al. [13], [PERSON] et al. [14] and [PERSON] et al. [15] proposed the six-direction, four-direction and circular pseudo-random paths, respectively. Their experimental results showed that the random variability of the path direction can effectively suppress the MSF error. In the deterministic surface correction process, a small path spacing should generally be preferred. Too large a path spacing is not conducive to correcting the surface form error and suppressing the ripple error. However, due to changes in material removal and the speed limitation of the axes of the polishing machine, polishing along an overly-dense path will introduce additional dwell time and cause excessive material removal, especially in regions where material removal is low, which is unfavorable for the convergence of surface accuracy. [PERSON] et al. proposed to use the inflection point of the approximately reversed \"L\"-shaped curve between path spacing and ripple error to determine the optimal path spacing, which provides a fast and efficient method for optimal selection of path spacing [16]. [PERSON] et al. proposed to divide the surface into several regions from the perspective of suppressing the MSF error and plan polishing paths with different pitches in different form error regions [17]. In their implementation process, the transition of polishing paths in different regions and the avoidance of the path edge effect are complicated. [PERSON] et al. also proposed a polishing path with random spacing based on the distribution of surface form error [18]. Their choice of path spacing is based on empirical formulas, and the intrinsic relationship between form error and path spacing has not been studied in depth. In this paper, the influence law of path spacing and bulk material removal depth on residual error is deeply studied. It is not difficult to understand that the influence law of the distribution of surface form error on residual error can be revealed by studying the effects of different bulk material removal depths on residual error. On this basis, mathematical modeling is used to demonstrate the feasibility of the novel idea consisting of the normalizing residual error distribution by optimizing the path spacing under the condition that other polishing conditions including TIF are unchanged. This idea considers the fine adjustment of the path spacing to compensate for the effect of the surface form error on the residual error. A complete residual error optimization strategy based on the form error distribution is proposed.