## 2 Literature Review
### Non-Planar Path Planning for 3D Printing  
The term curved layer printing and its variants mainly refer to slicing a model using curved layers instead of the traditional planar layer slicing or a combination of planar and curved. A very complete review of planar and non-planar slicing methods and path planning for AM is presented by [PERSON] and [PERSON] [30]. Different approaches are also found in the literature related to conformal printing on non-planar surfaces that are not commonly associated with the curved slicing model, but which are included here due to their importance in this study. The term CLFDM was first introduced by [PERSON] et al. [31] who formulated a theoretical method, mainly based on CNC traditional concepts, for the manufacturing of thin curved shells to improve the mechanical properties and reduce the stair-step effect. Their work was based on employing longer length tool paths, focused on the proper orientation of the filament and appropriate bonding between adjacent filaments. Their formulation used a parametric surface, calculated the partial derivative of it to obtain the normal vector and then generated an offset surface. The concept of CLFDM was experimentally reported by a number of researchers, such as [PERSON], [PERSON], and [PERSON] [32; 33; 34]. They generated the path planning (flat layers) to produce a mandril as a support structure where the curved layers were later deposited, following the contour of the part. Later, [PERSON] et al. [35] improved their algorithm of the cross product of four vector by considering a vertical plane passing through three consecutive surface points. Even though all their contributions laid the foundations of experimental CLFDM, some important details about the implemented algorithms were skipped. They studied the generation/selection of data points to produce the offset curved layer directly from the G-code or M-code generated by a CAM software, and as a result, the printing trajectory was limited to those points. Some authors have developed curved layer slicing by modeling and fitting the surface using B-spline, such as [PERSON] et al. [36], who fitted an STL mesh surface with a B-spline surface with two independent parameters (\(u\) and \(v\)). They modified the original tessellated surface to reduce the number of triangles of the STL file and then fitted the surface. The first printing path (the author recommends it to be along one of the edges of the design) defines the next paths, which are generated by a certain offset (equidistant). They reported some limitations in the processing of the part surface due to the CAD and CAM software used. [PERSON] et al. [37] optimized the number of curved layers needed for printing by preserving the critical features. They modeled a B-spline surface using selected critical points and generated curved offset layers optimized by the application of genetic algorithms and surface-surface intersection. Their results included simulations but a nonphysical implementation. [PERSON] and [PERSON] [38] used a delta configuration system and generated the printing path of a surface or skin defined mathematically and a core component (infill pattern) as a matter of contrasting distinct structural or physical functions, as [PERSON] et al. [39] later also demonstrated by producing models with aesthetic and structural properties. The algorithm consisted of converting the analytical surface in a grid XY following the points in order and calculating z for dynamic movements. [PERSON] and [PERSON] [6] presented the procedure to fabricate non-planar lattice-shells on non-planar equation-defined surfaces (parametric Bezier surfaces of arbitrary order), whereas [PERSON] et al. [40] generated and fabricated a lattice using rectangular equations and studied the mechanical behavior when force is applied. [PERSON] and [PERSON] [41] presented a mathematical approach to parametrize lattices onto Bezier surfaces to fabricate non-planar chirality lattices and studied them under cyclic loading.