# Quasi-Uniform Density Non-Solid Infill Strategy for Axisymmetric Non-Planar Additive Manufacturing
## 3 Results and Discussion
### Generatrix Definition, Continuity, and Geodesics  
The continuity requirements proposed in Section 2.1 are satisfied by all build platform geometries through the use of degree-six Bezier spline compositions for their generatrices. Both generatrices, together with their first three derivatives \(\mathcal{S}\), \(\mathcal{\bar{S}}\), \(\mathcal{\bar{S}}\) and curvature \(\kappa\), present sufficient continuity at the spline knots. This fact confirms that the curvature is differentiable and demonstrates that the build platforms are Riemannian manifolds. In turn, this enables the differentiability required for feasible and efficient geodesic path computations in NPAM trajectory planning. This is further substantiated through the analysis of the curvature \(\kappa\), the metric tensor components \(g_{ij}\), and the Christoffel symbols \(\Gamma^{i}_{jk}\). Figure 4 presents these three magnitudes for the hourglass build platform, the more complex of the two. The C\({}^{3}\) nature of the spline composition is verified by the smoothness of the curvature and all non-zero symbols. Of note, the curvature is non-constant, with both positive and negative regions. This implicitly validates the methodology for a wide range of curvature values of any given build platform. Additionally, this smoothness allows for a stable resolution of the geodesic equation throughout the domain except near the poles, where \(g_{\theta\theta}\to 0\) and \(\Gamma^{\theta}_{u\theta}\to\infty\). This is caused by the loss of invertibility of the metric tensor around the poles. Therefore, these polar regions should be avoided for stable geodesic path calculations, and result in a suitable area for the mechanical interface of the end effector with the robotic manipulator. Figure 4: Curvature \(\kappa\), non-zero metric tensor components, and Christoffel symbols of the hourglass-shaped build platform. Color relates to the variable magnitude. Results of the Repulsion Energy Minimization Algorithm and Validation of the Uniform Density Hypothesis  
The proposed geodesic repulsion energy algorithm demonstrated its capacity to distribute points uniformly on both barrel-shaped and hourglass-shaped surfaces. The case \(\Omega_{1}\), with \(u\in[0.8,\ 1.2]\), \(\theta\in[0,\ 1]\), and \(\rho_{c}=50\%\) (\(n=1\), \(N=538\)), was initialized via EBNS owing to the large number of points \(N\). Figure 5 represents the GRE-optimized distribution of points on the left-hand side of the figure, whereas the right-hand side graphs indicate the distribution in the embedded map, the KDF evaluation for each point, and the values of KDF sorted in ascending order for both the EBNS initial seed and the GRE optimized distribution. The embedded map distribution shows a noticeable improvement in point homogeneity, which is further supported by the KDF evaluation for each point. This implies that a path generated through these points will retain a mean infill density close to command. Additionally, this methodology enables the creation of quasi-uniform infill densities across any given layer, introducing or removing points according to the slice total area \(A_{\Sigma}\). The sorted KDF function also validates the quasi-uniform hypothesis, as it is possible to differentiate three regions: the vertices, the domain boundary, and the interior. After applying GRE, each of these regions is separated from each other by means of a noticeable increase in kernel density function, a behavior not observed in the EBNS initial seed distribution. The second case study \(\Omega_{2}\) in the same barrel geometry featured \(\rho_{c}=25\%\) (\(n=1\), \(N=173\)) and a non-rectangular slice boundary, with concave and convex areas. To solve this, the ERE initial seed algorithm was used, combined with a non-linear constraint to force all points towards the interior of the slice polygon.