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Dual Contouring octree enumeration.

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I have been following http://www.frankpetterson.com/publications/dualcontour/dualcontour.pdf to impelement duel contouring. However up until this point it was over a uniform grid; Now I am attempting to implement it over an octree and am very confused on the paper's enumeration method. (page 6 of the paper)

 

I understand up until how you are supposed to use the minimal edge, if it was two adjacent cubes you could test if the surface intersected between them and generate that 'quad', covering that area of the surface, however their use of four cubes around an edge loses me. The main issue I am having is, under this implementation, are they attempting to find the surface perpendicular or parallel to the four nodes; and to that end how do they plan to hook up the connecting 'quads' when the endpoints of that line would be one of the vertices of the 'quads'. Please help.

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