## 2 Literature Review
### Non-Planar Path Planning for 3D Printing
Conformal printing has emerged as a process to deposit silver inks on curvilinear surfaces to create conductive paths [42]; however, some recent studies were published about path planning for conformal 3D printing using FFF. [PERSON] et al. [43] proposed an algorithm for conformal printing using non-planar layers and evaluated the differences in roughness between a surface finish when printed using planar layer slicing and the proposed algorithm. The algorithm aims at collision-free trajectory planning using a projection method: (1) a grid is created on the XY plane (0.5 mm spacing); (2) vertices of each triangle are projected to the XY plane; (3) specific points of the grid belong to a particular triangle; (4) the equation of the plane of the triangle is calculated from three vertices; and then (5) the z value for these points inside the triangle is calculated and mapped back to the non-planar surface. A zigzag pattern at two different angles is used to improve the finishing of the surface. [PERSON] et al. [44] proposed an algorithm to locate conformally one tessellated structure onto a second tesselllated surface (substrate). The algorithm achieves the following: (1) it generates a curved slicing surface by offsetting the top of the substrate; (2) it obtains the boundaries of the pattern to be printed by the intersection of the structure and the slicing surface; and (3) 2D printing patterns are projected to create 3D patterns. To achieve conformal trajectories, this algorithm has the restriction that the bottom of the 3D structure must fit the freeform substrate, and in the case of a mismatch, the free spaces are filled to connect both structures. The algorithm outputs the G-code for 3D printing. A different approach for printing quality improvement was proposed by [PERSON] et al. [45], who developed an algorithm for planar and non-planar slicing. Their main contribution is the detection of the parts suitable to be printed using non-planar slicing assuring collision-free toolpaths, using a simplified printhead model defined by the maximum nonplanar angle and the maximum nonplanar height. The printing trajectories presented are focused to achieve smooth surfaces (zigzag pattern). [PERSON] et al. [46] implemented a five-axis machine (a delta printer plus a platform rotating) and proposed an algorithm for curved layer material extrusion. Their main contribution is the reduction of the material used for the mandril to achieve conformal curved printing; hence, the printing time is also reduced. They generated a conformal surface offset and a toolpath using the geodesic distance as the shortest zigzag along the facet edges of the STL file. The path planning consisted in equidistantly offsetting the starting curves. Most of the work published, to the best of our knowledge, about conformal-curved trajectory printing is focused on the improvement of the surface finish (reducing the stairstepping effect) with improvement in the mechanical performance as a consequence, and a reduction in the number of layers needed for the part to be printed. Due to this, the most-used paths for curved printing are limited to zigzag or circular patterns [43; 44; 46]. In these works, printing paths consisted of equidistantly offsetting starting curves, which restrict the points to define the printing trajectory. A limited number of works were focused on the generation of more complex trajectories, such as those needed for the manufacturing of lattices or specific paths for the deposition of fibers or conductive materials. Complex trajectories or patterns, such as sinusoidal lattices, were reported, and these were printed on analytical surfaces (mathematically parametrized) but not using tessellated surfaces [40; 41]. Complex freeforms, such as those encountered in biomedical applications (normally obtained from scanned data), may not be mathematically parametrized. Hence, approaches such as the one presented in this paper, gain relevance; any surface, parametrizable or not, could be modeled using a tessellated surface.