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.
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| scene | dual space |
It is bounded by edges which are segments tangent to one object.
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| scene | dual space |
The vertices correspond to the bitangents of the scene.
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| 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.
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| scene | dual space |
The visibility complex is a 2D cellular complex immersed in 3D.
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| scene | dual space |
Here we can see the vertices.
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| scene | dual space |
And a dual line corresponding to a point in the scene.
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| scene | dual space |
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