I have a cloth in 3 dimensions represented as a triangle mesh, which is a kind of spatial graph. The nodes of each triangle are specified in a clockwise manner, such that I can consistently determine the surface normals of each triangle.
Now, I want the user to be able to start from the boundary of the cloth and make a cut from a vertex $v_1$ to another vertex $v_2$. I don't want the user to be able to cut right in the "middle" of the cloth, so I'm limiting myself to these boundary cuts.
TL;DR: In this graphic, I am not sure how to compute whether to associate the yellow-highlighted edges with $v_1'$ or to keep them associated with $v_1$. (The figure shows the desired outcome, but I'm not sure how to compute it).
I think I have a proper working definition of a "boundary edge," which is the following: the user can direct a cut from vertex $v_1$ to vertex $v_2$ if and only if there exists some edge $e$ incident to $v_1$ such that $e$ is only an edge of one triangle (as opposed to two triangles).
Hopefully, everything I have said so far makes sense. Now, here is my main issue. I am trying to figure out how to actually modify the graph to reflect the cut. So far, I have figured out the following for a cut directed from $v_1$ to $v_2$:
- Create a node $v_1'$, and an edge $\{v_1', v_2'\}$.
- Replace all edges $\{v_1, w\}$ with $\{v_1', w\}$ if $w \neq v_2$ and ... some condition $C$ holds.
As you can see, the trouble is with the condition $C$; I'm not sure how to mathematically decide whether an edge incident to $v_1$ should be associated with $v_1$ or $v_1'$.
I suspect it could have something to do with cross products, i.e. if I take the cross product of $(v_1 \rightarrow v_2)$ with $(v_1 \rightarrow w)$, that could give me some useful information. But I'm not quite sure where to go from there.
I think I've figured it out.
Consider the edge $\{v_1, v_2\}$ and the two triangles $t_1$ and $t_2$ associated with that edge. I "sweep" through the edges incident to $v_1$ by iterating through adjacent triangles in a clockwise fashion (WLOG), starting with $t_1$. I reassign only those edges to $v_1'$. The rest stay assigned to $v_1$.