Remember in polygonalization of the surface, following rules must be followed. Any two polygons 

(i)  share a common edge,  

(ii)  Share a common vertext, 

(iii)  Arc disjoint. 

Figure (a) on the next page, gives an example of an admissible approximation of a surface using triangles, called triangulation. Further, Figure (b) and (c) give examples of some cases that are not admissible as per the rules given above.  This means an edge of a polygon cannot be a part of an edge of another polygon.

Similarly, no two polygons can overlap leading to a common non-empty interior.