PURPOSE: To enable a curved surface in the angular corner part of a work to be automatically polished, by causing the work of T-shape to perform a combined shuttle- turning motion mounting it to the second arm rotatably supported by an axis to and turned around the axis at a right angle with that of the first arm which turns around an axis, serving as a center, parallel to a polishing buff spindle.

CONSTITUTION: When an angular corner part 10 in a work 9 is polished by a buff 17 by respectively driving arms 4, 7, the work 9, when it turns around a turning axis (a) displacing its both axial centers j1, j2 in a vertical plane containing the axis (a), is synchronously turned around a turning axis (b). Then a tilt angle A of the axial center j1, as it increases with respect to a horizontal axis, successively spreads a polishing area (k) of the work in a circular arc direction along a top surface 10a of the angular corner part 10 to be transferred from the side of a trunk pipe 9a to the side of a branch pipe 9b. While simultaneously, the work 9 in each of its turning attitude around the axis (b) is turned around the axis (a) each 90° to both sides. In this way, a curved surface in the angular corner part 10 can be automatically polished over the full area by polishing an upper side half peripheral surface in each angular corner part 10 in each vertical section, passing the axis (b), of the angular corner part 10.