## 1 Introduction  
One of the most used additive manufacturing (AM) technologies is fused filament fabrication (FFF), which emerged as a prototyping process. During FFF, extruded material is deposited on a flat platform, and the 3D component is built by stacking layers that cause functional weakness. A new trend in research aims to deposit extruded material using out-of-plane trajectories, following the shape of curved surfaces rather than planar layer-by-layer printing. Non-planar printing has facilitated the fabrication of complex geometries, improving their mechanical properties. Several terms are used in the literature to refer to non-planar printing, such as curved layer fused deposition modeling (CLFDM), curved layer fused filament fabrication (CLFF), and curved layer fused deposition (CLFD). Path planning concepts relevant to these terms overlap in terms of following a non-planar trajectory and depositing fused material onto curved surfaces. Conformal additive manufacturing usually features direct write (DW) technologies, causing a new field related to conformal fabrication using FFF to emerge. The literature also describes the use of conformal fabrication to add fused material to uneven or non-planar surfaces via one or more different materials to refer to non-planar printing. According to [PERSON] and [PERSON] [1], the current path planning strategies for improving 3D printing are divided into three groups: (1) those that improve the printing surface's qualities, shape accuracy, or infill patterns; (2) those that save material and/or time; and (3) those that achieve specific printed properties (e.g., mechanical, topological, or functional). Constant-depth CLFD has made a very significant contribution to increasing stiffness and strength [2]. Further, variable-depth CLFD [3,4], has improved surface finishing by reducing/eliminating the stair-stepping effect. To improve structural and mechanical properties while reducing material use, [PERSON] and [PERSON] [5] and [PERSON] and [PERSON] [6] proposed manufacturing non-planar lattices or cellular structures to serve as potential mechanical metamaterials. Depositing conductive materials on a non-planar substrate, reducing material and assembly complexity, has made functional printed electronics a reality [7]. Therefore, curved layer printing has confirmed its capability to achieve and improve manufacturing properties through non-planar slicing processes, eliminating or reducing supports, creating efficient infill pattern layouts, and depositing multi-material for heterogeneous composites. The path planning problem for conformal printing requires that material be deposited on any target surface of a body digitally represented by a model. These digital models, created with a variety of computer-aided design (CAD) file formats, are highly important, as they contain information about raster geometry representation, density, and material. Some representations are the exclusive file formats of a company, whereas others are open-source and can be transferred between different CAD systems developers to achieve a flexible product creation value chain. CAD data may contain either of the two types of model object representation: analytic geometry or faceted geometry. Analytic geometry is produced for CAD design systems and is the preferred method for design and manufacturing processes, due to its unlimited resolution and flexibility. The most common analytic representation of a continuous surface is the non-uniform rational basis spline (NURBS), which describes curves and surfaces using mathematical functions. In contrast, faceted geometry (for a detailed explanation, see [8]) is a discrete form of geometry and comprises non-parametric representation consisting of points that connect to form a tessellation or groups of polygons (often triangles) as the boundaries of a body. As [PERSON] et al. [9] explained, faceted geometry has limited resolution but a flexible and straightforward definition.