The manufacture of solids using material deposition has boomed in recent years, and its use has grown steadily, generating challenges to improve printing. In addition, new mechanisms of six degrees of freedom have been designed focused on manufacturing parts, creating problems such as their trajectory planning. Therefore, this paper presents a new path-planning methodology based on the contour-parallel pattern and, through propagation techniques and geodesic flattening, transforms curved spaces into flat contours and vice versa. Furthermore, this methodology generates continuous trajectories for manufacturing parts for mechanisms of six degrees of freedom, focused on improving the surface quality and the quantity of deposited material, eliminating unwanted effects such as ladder and stringing, which usually occur in 3D printing.