A new category of circular pseudo-random paths is proposed in order to suppress repetitive patterns and improve surface waviness on ultra-precision polished surfaces. Random paths in prior research had many corners, therefore deceleration of the polishing tool affected the surface waviness. The new random path can suppress velocity changes of the polishing tool and thus restrict degradation of the surface waviness, making it suitable for applications with stringent mid-spatial-frequency requirements such as photomask blanks for EUV lithography.