# Quasi-Uniform Density Non-Solid Infill Strategy for Axisymmetric Non-Planar Additive Manufacturing
## 1 Introduction  
Additive manufacturing (AM) has emerged as a transformative technology for producing complex geometries with high precision [1, 2]. Nevertheless, traditional planar AM processes introduce inherent anisotropies in mechanical properties and limit geometric versatility. These planar layers limit the range of manufacturable parts, as they introduce significant anisotropy between the mechanical strength along the layer direction and the strength in the perpendicular direction. This effect is particularly pronounced in material extrusion (MEX) and directed energy deposition (DED) processes [3, 4, 5]. In response, Non-Planar Additive Manufacturing (NPAM) has gained increasing attention as a methodologyfor forming structures from additively deposited curved layers [6; 7]. Once stacked, non-planar layers enhance structural integrity, surface quality, and material efficiency. NPAM allows for local variations in layer deposition direction, optimizing mechanical properties, increasing fatigue life, and reducing the stair-stepping effect studied for planar slicing techniques [8; 9; 10]. The development of NPAM led to the rise of robotic systems that, by adding degrees of freedom to the end effector, enable material deposition in multiple directions. This is a technique known as Robot-based Non-Planar Additive Manufacturing (RbNPAM) [11]. This approach addresses the limitations of traditional additive manufacturing machines, and opens new possibilities for complex geometries, enhanced surface finishes, and optimized part performance in multiple applications [12; 13]. Despite these advantages, systematic methodologies for non-planar slicing and infill generation remain an open research problem [14; 15]. The first challenge in the field of non-planar layer path generation is the definition of non-planar layers. Various research studies aim to define non-planar layers by varying the \(z\)-coordinate in traditional additive manufacturing machines [16; 17]. Additionally, building on planes with a small angular offset is used to produce geometries with planar layers in different orientations [18]. Alternatively, curved \(x\)-\(y\) grids have been proved to work for some specific cases [19]. Other authors have explored the use of isothermal curves and maximum local stress-oriented surfaces [20; 21]. However, the use of non-Cartesian coordinate systems like polar or spherical coordinates [22; 23; 24], as well as helicoidal [25], can be observed to facilitate the generation of such surfaces. These methods, which may be called embedded map methods, become more complex when attempting to generalize the problem to parametrically defined layers while becoming more versatile. Curving the build surfaces introduces a distortion of the Cartesian space in the embedded map, making it challenging to handle using traditional methods found in commercial slicers for Planar Additive Manufacturing [26]. Within an embedded map, the surface curvature distorts the deposited material's line thickness and alters the distance between target points along the toolpath, significantly complicating the development of infill patterns. Additionally, it exacerbates the issues already analyzed in the literature regarding the occurrence of voids in infill patterns [27]. Non-solid infill strategies have been extensively studied in the context of planar AM, owing to their benefits in terms of material, build cycle time, cost, and energy savings [28]. Common approaches to non-solid infill generation include the use of contouring combined with spiral or Hilbert curve techniques [29; 30; 31; 32]. Other studies propose maze infill strategies [33] or adaptive filling based on local density requirements [34]. Additionally, there have been pixel-based approaches where pixel size is defined by the mesh point distortion [35; 36]. Even so, no algorithm has yet been developed to adapt way-points density to non-planar surfaces.