Primal-dual circle representation

Simultaneous circle-packing representations of a map M and its dual M are called primal-dual circle representation of M if it holds ... 

See Figure 1.2 for an illustration of this feature and an example of a primal-dual circle representation. 

Figure 1.2 Illustration of primal-dual circle representations...

A map M is called reduced (see [Moh97, Section 3]) if its universal cover is 3-connected and is a cell-complex. It is shown in [Moh97, Corollary 5.4] that reduced maps admit unique primal-dual circle packing representations on a Riemann surface of the same genus moreover, a polynomial time algorithm allows one to find the coordinates of those points relatively easily. This means that the combinatorics of the map determines the structure of the Riemann surface. 

The edges, vertex circles and face circles of a primal-dual representation... 

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