A method for polishing at least one of a magnetic, optical and semiconductor substrate in the presence of a polishing medium with a polishing pad [100, 400, 500, 600, 700], the substrate being fixed within a carrier fixture, the carrier fixture having a channel-free surface, the method comprising: a) securing the substrate in the carrier fixture with the channel-free surface adjacent and parallel to a polishing surface of the polishing pad [100, 400, 500, 600, 700], the polishing pad [100, 400, 500, 600, 700] having multiple grooves [116, 616], the multiple grooves [116, 616], having a high-rate path, at least fifty percent of the high-rate path being within twenty percent of a groove trajectory φ ( r) in polar coordinates referenced to a concentric center of the polishing pad [100, 400, 500, 600, 700] and defined in terms of distance R between the concentric center of the polishing pad [100, 400, 500, 600, 700] and the rotational center [ O '] of the substrate being polished, radius R c of the carrier fixture, and local angle θ c0 of imaginary grooves in the carrier fixture, as follows: φ r = ∫ R − R C r R R C sin φ c + tan θ c 0 R R C cos φ c + 1 R R C cos φ c + 1 − tan θ c 0 R R C sin φ c dr ′ r ′ where φ c = cos − 1 R 2 + R C 2 − r 2 2 RR C − π for values of r from ( R - R C) to ( R + R C) b) applying polishing medium to the polishing pad [100, 400, 500, 600, 700] adjacent the carrier fixture; and c) rotating the polishing pad [100, 400, 500, 600, 700] and carrier fixture to polish the substrate with the polishing pad [100, 400, 500, 600, 700] and the polishing medium wherein the channel-free surface of the carrier fixture presses against the polishing pad [100, 400, 500, 600, 700] to impede flow of the polishing medium into the substrate and the high-rate groove paths traverse the carrier fixture to promote flow of the polishing medium to the substrate. a) securing the substrate in the carrier fixture with the channel-free surface adjacent and parallel to a polishing surface of the polishing pad [100, 400, 500, 600, 700], the polishing pad [100, 400, 500, 600, 700] having multiple grooves [116, 616], the multiple grooves [116, 616], having a high-rate path, at least fifty percent of the high-rate path being within twenty percent of a groove trajectory φ ( r) in polar coordinates referenced to a concentric center of the polishing pad [100, 400, 500, 600, 700] and defined in terms of distance R between the concentric center of the polishing pad [100, 400, 500, 600, 700] and the rotational center [ O '] of the substrate being polished, radius R c of the carrier fixture, and local angle θ c0 of imaginary grooves in the carrier fixture, as follows: φ r = ∫ R − R C r R R C sin φ c + tan θ c 0 R R C cos φ c + 1 R R C cos φ c + 1 − tan θ c 0 R R C sin φ c dr ′ r ′ where φ c = cos − 1 R 2 + R C 2 − r 2 2 RR C − π for values of r from ( R - R C) to ( R + R C) b) applying polishing medium to the polishing pad [100, 400, 500, 600, 700] adjacent the carrier fixture; and c) rotating the polishing pad [100, 400, 500, 600, 700] and carrier fixture to polish the substrate with the polishing pad [100, 400, 500, 600, 700] and the polishing medium wherein the channel-free surface of the carrier fixture presses against the polishing pad [100, 400, 500, 600, 700] to impede flow of the polishing medium into the substrate and the high-rate groove paths traverse the carrier fixture to promote flow of the polishing medium to the substrate.