Chemical mechanical polishing can be used for “touch-up polishing” in which polishing is performed on a limited area of the front surface of the substrate. The contact area between the polishing pad and the substrate can be substantially smaller than the radius surface of the substrate. During polishing, the polishing pad can undergo an orbital motion. The polishing pad can be maintained in a fixed angular orientation during the orbital motion. The contact area can be arc-shaped. The contact area can be provided by one or more lower portions projecting downward from an upper portion of the polishing pad. A perimeter portion of the polishing pad can be vertically fixed to an annular member and a remainder of the polishing pad within the perimeter portion can be vertically free.