A curve interpolation method capable of obtaining a curve approximating an original curve based on a sequence of command points within a tolerance set for the original curve, and performing interpolation on the obtained curve. Points Q1, ..., Q2n are interpolated between respective two adjacent command points (P0, P1), (P1, P2), ..., (Pn-1, Pn) as shape-defining points. The shape-defining points are positioned within a tolerance width 2w set to the original curve. One shape-defining point and shape-defining points surrounding the one shape-defining point are successively selected and an approximate curve for the selected shape-defining points is successively created. The one shape-defining point is moved towards the approximate curve to determine a modified shape-defining point for the one shape-defining point. A smooth curve passing a sequence of the modified shape-defining points is created and interpolation for machining is performed on the created curve. Since the shape-defining points are positioned closer to an original target curve within a tolerance width set to the original curve than the command points, the created smooth curve passing the sequence of modified shape-defining points approximates well to the original target curve within the tolerance width.