@@ -12,7 +12,7 @@ It is clear that one of the input parameters has to be $\mathcal{B}$, but afterw
\begin{enumerate}
\item$\mathcal{B}, \Phi, \mathcal{M}$: With all parameters given, the embedding is ready and no further initialization needs to be done. On the other hand this leaves a difficult task to the user in having to input a pre-initialized embedding.
\item$\mathcal{B}, \mathcal{M}$: Giving just the two meshes $\mathcal{B}$ and $\mathcal{M}$ and then embedding $\mathcal{M}$ onto $\mathcal{B}$ creates a similar initialization as that of \cite{praun2001consistent}, and could be handled with a similar method for finding the embedding $\Phi$. However, designing a fitting mesh $\mathcal{M}$ in such a fashion is no trivial task, and should still be simplified.
\item$\mathcal{B}, V_\mathcal{M}\subset V_\mathcal{B}$: Giving a subset $V_\mathcal{M}\subset V_\mathcal{B}$ of the vertices of $\mathcal{B}$ as an input implicitly defines the vertex embedding which is part of $\Phi$. The connectivity and actual edges of $\mathcal{M}$ can then be triangulated. This type of input is the easiest for a user, since all that needs to be given is a set of feature points V_\mathcal{M}\subset V_\mathcal{B}$.
\item$\mathcal{B}, V_\mathcal{M}\subset V_\mathcal{B}$: Giving a subset $V_\mathcal{M}\subset V_\mathcal{B}$ of the vertices of $\mathcal{B}$ as an input implicitly defines the vertex embedding which is part of $\Phi$. The connectivity and actual edges of $\mathcal{M}$ can then be triangulated. This type of input is the easiest for a user, since all that needs to be given is a set of feature points $V_\mathcal{M}\subset V_\mathcal{B}$.
\begin{enumerate}[label=(\alph*)]
\item A special case of this input type is $V_\mathcal{M}=V_\mathcal{B}$, in which case $E_\mathcal{M}=E_\mathcal{B}$ is a trivial triangulation, and with that
$\mathcal{M}=\mathcal{B}$. Thus, the initialization immediately terminates by copying $\mathcal{B}$ and adding the necessary connecting pointers.
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@@ -35,7 +35,7 @@ Now the task is creating a meta mesh $\mathcal{M}$ and an embedding $\Phi$ on th
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).
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}.
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 $\mathcal{M}$.
Lastly, these edges $E_\mathcal{M}$ still need to be traced. Given a very fine mesh $\mathcal{M}$, or thin features on $\mathcal{M}$, it can happen that a Voronoi region borders with the same Voronoi region twice. In such a case it is necessary to ensure that edges are traced through the correct border (eg. from two sides around a handle rather than twice around the same side). This is easily done by restricting edge traces during the initial triangulation by disallowing them from crossing any Voronoi borders \textit{except} the one corresponding to its edge.
