Construction principle for the dual of the intersection of cylinders

Intersection of cylinders typically have

• no flat faces (and are therefore not polyhedra)
• normal vertices where the cylinder generatrices meet
• "flat" vertices where edges meet that have vanishing profiles and where the cylinder generatrices form concentric polygones.

These flat vertices play a special role in the construction process for the dual of the cylinder intersection.

Let this dual be the limit of the duals of polyhedral approximations of the intersection of cylinders.

Take only these flat vertices and form the bissectors to replace the old edges at this vertex and span a generalized cross vault. When all flat vertices are transformed in this way there are left flat faces. The dual volume of the intersection of cylinders is delimited by flat faces and by curved (cylindrical) faces.

We show now as an example the intersection of 3 cylinders o3

and the dual of it:

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