# Straight Skeleton “Opposite Edge” of V

I'm trying to implement the straight skeleton algorithm as described by this paper: http://www.dma.fi.upm.es/personal/mabellanas/tfcs/skeleton/html/documentacion/Straight%20Skeletons%20Implementation.pdf

I am currently stuck trying to understand the author's description of the intersection point B as created from bisector of V, on page 4-5: I am very confused by this description.

Point B can be characterized as having the same perpendicular distance to the straight line carrying the "opposite" line segment to the vertex V and from both straight lines containing the line segments starting at the vertex V . We have to find such an "opposite" line segment.

• What exactly does opposite mean in this case?
• Does "straight lines containing the line segments starting at the vertex V" refer to its adjacent edges of V?
• Why does Figure 4 show a pre-constructed straight skeleton (the dashed line)? Or.. what is that even? I'm not sure what this diagram is supposed to show.

In this case, the edge opposite a vertex $$V$$ is the edge found by bisecting the interior angle at $$V$$ and travelling along that bisector through the polygon until an edge is reached. There seems to be an implicit assumption that you won't end up in the position of a square, where each vertex has no opposite edge because each vertex is, in fact, opposite another vertex.
You are right with your second statement; the straight lines containing the line segments starting at the vertex $$V$$ are indeed the adjacent edges of $$V$$. Note that in the figure $$4a$$) and $$4b$$) these lines are extended (as dotted lines) to show where the intersect the 'opposite' edge.
Figure $$4$$ is, I think, showing you how the skeleton is constructed, which is why the dotted line is there. Only the bits inside the shaded area are constructed using the point $$B$$; outside the shaded area they may change direction in order to meet up with the next constructed point (see the first Figure and compare it with the 'magnifications' of Figure $$4$$).