Question

I’m implementing an algorithm in which I need manipulate a mesh, adding and deleting edges quickly and iterating quickly over the edges adjacent to a vertex in CCW or CW order.

The winged-edge structure is used in the description of the algorithm I’m working from, but I can’t find any concise descriptions of how to perform those operations on this data structure.

Answer 1

I’ve learned about it in University but that was a while ago.

In response to this question i’ve searched the web too for any good documentation, found none that is good, but we can go through a quick example for CCW and CW order and insertion/deletion here.
Have a look at this table and graphic:

from this page:

http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/winged-e.html

The table gives only the entry for one edge a, in a real table you have this row for every edge. You can see you get the:

• left predecessor,
• left successor,
• right predecessor,
• right successor

but here comes the critical point: it gives them relative to the direction of the edge which is X->Y in this case, and when it is right-traversed (e->a->c).

So for the CW-order of going through the graph this is very easy to read: edge a left has right-successor c and then you look into the row for edge c.

Ok, this table is easy to read for CW-order traversal; for CCW you have to think “from which edge did i come from when i walked this edge backwards”. Effectively you get the next edge in CCW-order by taking the left-traverse-predecessor in this case b and continue with the row-entry for edge b in the same manner.

Now insertion and deletion: It is clear that you cant just remove the edge and think that the graph would still consist of only triangles; during deletion you have to join two vertices, for example X and Y in the graphic. To do this you first have to make sure that everywhere the edge a is referred-to we have to fix that reference.

So where can a be referred-to? only in the edges b,c,d and e (all other edges are too far away to know a) plus in the vertex->edge-table if you have that (but let’s only consider the edges-table in this example).

As an example of how we have to fix edges lets take a look at c. Like a, c has a left and right pre- and successor (so 4 edges), which one of those is a? We cannot know that without checking because the table-entry for c can have the node Y in either its Start- or End-Node. So we have to check which one it is, let’s assume we find that c has Y in its Start-Node, we then have to check whether a is c's right predecessor (which it is and which we find out by looking at c's entry and comparing it to a) OR whether it is c's right successor. “Successor??” you might ask? Yes because remember the two “left-traverse”-columns are relative to going the edge backward. So, now we have found that a is c's right predecessor and we can fix that reference by inserting a's right predecessor. Continue with the other 3 edges and you are done with the edges-table. Fixing an additional Node->Vertices is trivial of course, just look into the entries for X and Y and delete a there.

Adding edges is basically the reverse of this fix-up of 4 other edges BUT with a little twist. Lets call the node which we want to split Z (it will be split into X and Y). You have to take care that you split it in the right direction because you can have either d and e combined in a node or e and c (like if the new edge is horizontal instead of the vertical a in the graphic)! You first have to find out between which 2 edges of the soon-to-be X and between which 2 edges of Y the new edge is added: You just choose which edges shall be on one node and which on the other node: In this example graphic: choose that you want b, c and the 2 edges to the north in between them on one node, and it follows that the other edges are on the other node which will become X. You then find by vector-subtraction that the new edge a has to be between b and c, not between say c and one of the 2 edges in the north. The vector-subtraction is the desired position of the new X minus the desired position of Y.