To represent them, we'll use a duality transform. A line y=ax+b is associated with the point (a,b).
The set of lines going through a point is a line in the dual space.
The visibility complex is a partition of the maximal free segments according to the edge of a polygon they touch.
A face is a connected component of segments which see the same objects.
scene  dual space 
It is bounded by edges which are segments tangent to one object.


scene  dual space 
The vertices correspond to the bitangents of the scene.


scene  dual space 
Many segments can be colinear. Thus we have to add a third dimension which is not continuous.
For example, the segments on the right and on the left of the triangle share the same (a, b) coordinates but have different z values.


scene  dual space 
The visibility complex is a 2D cellular complex immersed in 3D.


scene  dual space 
Here we can see the vertices.


scene  dual space 
And a dual line corresponding to a point in the scene.


scene  dual space 
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