@@ -22,18 +22,18 @@ $\mathcal{M}=\mathcal{B}$. Thus, the initialization immediately terminates by co
In our implementation we decide for an initialization with input parameters ''$\mathcal{B}, V_\mathcal{M}\subset V_\mathcal{B}$'', since this is the easiest for the user. We also implement a way to randomly select $V_\mathcal{M}$ by giving either a ratio or a total number of vertices in $V_\mathcal{B}$ to be selected. An initialization with inputs ''$\mathcal{B}, \Phi, \mathcal{M}$'' is implicitly supported as it already defines the whole embedding and needs no further work. And an initialization of the type ''$\mathcal{B}, \mathcal{M}$'' can be simulated by inputting $\mathcal{B}, V_\mathcal{M}\in\mathcal{M}$ and then flipping edges of $\mathcal{M}$ until it reaches the desired connectivity.
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\caption{Delaunay triangulation pipeline}
\label{fig:RockerarmTriangulation}
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Now the task is creating a meta mesh $\mathcal{M}$ and an embedding $\Phi$ on the base mesh $\mathcal{B}$ given an inpute of $\mathcal{B}$ and $V_\mathcal{M}\subset V_\mathcal{B}$. This can be done using Delaunay Triangulation \cite{delaunay1934sphere} on $\mathcal{B}$ with $V_\mathcal{M}$ as feature points to grow Voronoi regions from.
Figure \ref{fig:RockerarmTriangulation} illustrates the entire process. (a) shows the input, a base mesh $\mathcal{B}$ with selected feature vertices $V_\mathcal{M}$ marked in red. In the second step (b) Voronoi regions are grown around each feature point. This is done by growing regions outwardly from each feature vertex, and assigning each base vertex to the feature vertex it is closest to (by navigating over surface edges). Lastly, these regions can be used to derive connectivity for$\mathcal{M}$ as seen in Figure \ref{fig:RockerarmTriangulation}.(c).
The Delaunay Triangulation process is illustrated on the right. (a) shows the input, a base mesh $\mathcal{B}$ with selected feature vertices $V_\mathcal{M}$ marked in red. In the second step (b) Voronoi regions are grown around each feature point. This is done by growing regions outwardly from each feature vertex, and assigning each base vertex to the feature vertex it is closest to (by navigating over surface edges). In the next step, the edges $E_\mashcal{M}$ and thus the whole connectivity of $\mathcal{M}$ are derived; (c) shows the meta mesh $\mathcal{M}$. Lastly, $\mathcal{M}$ is embedded onto $\mathcal{B}$ creating the meta mesh embedding $\Phi(\mathcal{M})$, shown as the colored edges in (d).
In practice, we derive the connectivity from the Voronoi regions by iterating over the faces of $\mathcal{B}$. Since $\mathcal{B}$ is a triangle mesh, there has to be one unique face in $\mathcal{B}$ for which the Voronoi regions of each of its three vertices differ, per face in $V_\mathcal{M}$. Thus we derive the edges $E_\mathcal{M}$ of $\mathcal{M}$.
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