Automated inspection of manufactured parts is increasingly getting attention as it helps to make a rapid decision on product conformity. In this context, the aim of this paper is to present a new scanner path planning method for part inspection using an industrial 6-axis robot The novelty of the approach is to generate a scan path with the control of the overlap between 2 adjacent scanning paths based on the use of the least-squares conformal maps, which stretches a 3D mesh surface on a 2D plane. Equidistant paths calculated in the 2D space are then transformed into equidistant paths in the 3D space. The effective performance of controlling the overlap can improve digitizing quality and save digitizing time by managing the coverage of the laser beam. Furthermore, the digitizing quality is also ensured by keeping a constant scanning distance and executing a continuous control of the scanner orientation relatively to the part surface for all the driven points of the scan path. An experimental application of this new approach is proposed for a laser-plane scanner mounted on an industrial robot with 6 degrees of freedom, which demonstrates the interest of such an approach.