## 2 Surface Construction Methods  
The lattice structure is to be constructed as a free-form surface [18], meaning that it does not belong to a more easily recognised class of surfaces like cylinders or spheres due to the desired smoothness and irregular topology. According to [18], free-form surfaces are here constructed using a segmentation into local parametrisations of the surface, also denoted as _surface patches_[19]. To ensure smoothness, \(G^{1}\)-continuity is imposed between neighbouring patches. Two surfaces having a common boundary curve are called _geometric continuous_, written \(G^{1}\)-_continuous_, if they have a continuously varying tangent plane along that boundary curve [20]. If two surfaces have a common boundary curve but different tangent planes along that curve, the transition is called \(G^{0}\)-_continuous_. Within the proposed algorithm, surface patch construction methods for the following three base cases are required:  
1. Given two curves \(b,c:[0,1]\to\mathbb{R}^{3}\), the _ruled surface_\(r:[0,1]\times[0,1]\to\mathbb{R}^{3}\) defined by \(r(s,t):=s\cdot b(t)+(1-s)\cdot c(t)\) linearly interpolates \(b\) and \(c\)[20]. Figure 1a shows an example for the latter. In the developed algorithm, they are used for the definition of cross-boundary tangent conditions to ensure \(G^{1}\)-continuity of further constructed surface patches.
2. If the ruled surface matches additional cross-boundary tangent conditions along \(b\) and \(c\), it is denoted as \(G^{1}\)-_continuous ruled surface_. An example is illustrated in Figure 1b. A method to impose geometric continuity on ruled surfaces can be found in [21]. This type of surface patch is used for the construction of struts and for smooth transitions between strut ends that converge to the same node.
3. Given a set of \(N\) boundary curves \(c_{1}(t),\dots,c_{N}(t)\) forming a loop, a surface patch filling the region inside the loop and meeting cross-boundary tangent conditions is denoted as \(G^{1}\)-_continuous \(N\)-patch_. In Figure 1c, a \(G^{1}\)-continuous 3-patch is sketched. Algorithms to construct this type of surface patch can be found in [22; 23; 24]. They are used in the lattice construction algorithm to fill regions between strut ends and their smooth transitions.  
Based on the wireframe topology and local shape parameters of the lattice structure, a set of ruled surfaces is constructed to define cross-boundary tangent conditions for the subsequently defined \(G^{1}\)-continuous ruled surface patches that form the struts and that define the node transition. Lastly, \(G^{1}\)-continuous 6-patches are used to fill the remaining holes in the lattice surface at the transitions. The construction steps are explained in more detail in Section 3.3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Automatisation** & **Irregular Topology** & **Exact Parametrisability** \\ \hline \hline PMM & ✓ & ✓ & ✗ \\ ISM & ✓ & ✗ & ✓ \\ SPM & ✗ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of polygon mesh methods (PMMs), iso-surface methods (ISMs), and surface patch methods (SPMs) regarding the features’ automation, irregular topology, and exact parametrisability.  
Figure 1: Sketches of the three types of surface patches that are required for the lattice construction algorithm developed in this work. (**a**) Ruled surface. (**b**) \(G^{1}\)-continuous ruled surface. (**c**) \(G^{1}\)-continuous 3-patch.