# Quasi-Uniform Density Non-Solid Infill Strategy for Axisymmetric Non-Planar Additive Manufacturing
## 2 Methodology
### Uniformly Meshing a Riemannian Manifold  
To generate uniform density non-solid infills, it is necessary to create a uniform distribution of points that account for build platform curvature. Additionally, the number of points \(N\) to distribute must be determined based on process parameters selected by the user, namely, command infill density, infill pattern, and line width. Both challenges may be addressed utilizing Riemannian geometry concepts.  
First, uniformly distributing points on a Riemannian manifold requires a metric that accommodates local curvature. In this work, each slice is uniformly populated by means of a geodesic repulsion energy algorithm (GRE). Every pair of points \((i,j)\) is associated with a potential \(E_{ij}=\left(d_{ij}^{geo}\right)^{-m}\), where \(d_{ij}^{geo}\) is the geodesic distance between points \(i\) and \(j\), and \(m>1\) is a penalizing coefficient (in this work, \(m=5\) to highly penalize close pairs of points). Summing over all non-repeating pairs yields a total mesh energy \(E\). The algorithm iteratively adjusts point positions to minimize \(E\) by displacing each point along the direction defined by the repulsion force \(\mathbf{F}_{i}=-\Sigma\
abla E_{ij}\).  
Since the pair-wise geodesic distance must be determined, the manifold metric tensor \(\mathbf{g}\) and Christoffel symbols of the second kind \(\Gamma_{jk}^{i}\) are required. Points \(\{u_{i},\theta_{i}\}\) and \(\{u_{j},\theta_{j}\}\) in the embedded map are connected by the shortest path governed by a second-order ordinary differential Equation (1), with boundary conditions (2). Solving this equation and integrating along a parameter, \(\tau\) in (3), yields the geodesic distance, which replaces Euclidean measures in the repulsion potential. Although this approach is computationally more complex, it adapts the distribution to the manifold intrinsic curvature and avoids clustering in regions of negative curvature.  
\[\begin{cases}\ddot{u}=-\Gamma_{uu}^{u}(\dot{u})^{2}-\Gamma_{\theta\theta}^{u}( \dot{\theta})^{2}\\ \ddot{\theta}=-2\Gamma_{\theta\theta}^{u}\dot{\theta}\end{cases} \tag{1}\]  
\[\begin{cases}\mathcal{G}_{ij}(\tau=0)=\{u_{0},\theta_{0}\}=\{u_{i},\theta_{i}\} \\ \mathcal{G}_{ij}(\tau=1)=\{u_{1},\theta_{1}\}=\{u_{j},\theta_{j}\}\end{cases} \tag{2}\]  
\[d_{ij}^{geo}=\int_{0}^{1}\sqrt{g_{uu}\mathcal{G}_{ij}^{u}(\tau)+g_{\theta \theta}\mathcal{G}_{ij}^{\theta}(\tau)}d\tau \tag{3}\]  
Second, the number of points \(N\) for each slice is determined by the area \(A_{\Sigma}\) of the domain in the embedded map, the command infill density \(\rho_{c}\), the Cartesian nominal line width \(e\), and the number of equi-angled lines \(n\) that pass through each point. This \(n\) parameter depends on the infill pattern. This number of points may be simplified to (4), considering the worst-case infilling scenario, where each of the equi-angled \(n\) rasters traversing a point covers the least area around it. This relationship provides a direct means to estimate the mesh number of points in the infill by adjusting \(\rho_{c}\), while also considering the desired line width and complexity of the path pattern.  
\[N=\frac{A_{\Sigma}\rho_{c}^{2}}{n^{2}e^{2}} \tag{4}\]The meshing procedure begins by scattering \(N\) points within the parameter domain \(\Omega\), which corresponds to the slice polygon. These initial placements can come from random sampling or more refined strategies. A subsequent repulsion-minimization stage updates each point's location by applying the forces \(\mathbf{F}_{i}\) derived from \(-\
abla E\). When changes in point positions or overall energy become negligible, the algorithm terminates, leaving a near-uniform distribution that reflects the manifold curvature.