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Mapping a desired 2D pattern onto a curved surface has many applications. This includes motion planning for mobile robots to perform coverage path planing, robot end effector trajectory design for tasks such as printing, depositing, wielding on a 3D surface. This problem becomes more difficult if we want the mapped pattern to keep the properties of the original pattern (i.e, least possible mapping distortion), and pass over some desired points and/or remain bounded in a specific region on the surface. In this paper, we apply surface parameterization and mapping distortion analysis, which is rarely used in robot motion planning works, to map a pattern onto 3D surface. To meet additional goals such as passing over certain points, a planar mapping determined by constrained optimization is employed on the original pattern. Our focus is on printing/depositing materials on curved surfaces, and simulations and experiments are provided to confirm the performance of the approach.