Finding and defining the generators of genus-n objects for constructing topological and cut graphs

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The topology of an object is commonly represented through a topological graph or a cut graph (polygonal scheme). Over the past few years, many studies have focused on extracting the topological and cut graphs of complex freeform objects that are represented by meshes. For an object with genus-n, the topological graph has n cycles, while the cut graph contains 2n cycles. These loops, however, do not always explicitly represent the holes in the objects. That is, a cycle in the graph can be a cycle around a solid (meridian), a cycle around a hole (longitude), or almost any combination of the two. The task of classifying the cycles (generators) as cycles around holes (longitude) and cycles around solids (meridians) on the mesh is not straightforward. Every closed orientable 2-manifold with genus-n can be seen as a collection of n toruses stitched together, so that each hole in the object can be referred to as a torus with two generators. This paper proposes a method that extracts the generators from which the longitudes and the meridians are found. The topological graph is defined by the longitudes and by a spanning tree constructed between them. The cut graph is constructed using the same concept. The advantage of the proposed method over other methods is that each loop in the topological graph explicitly represents a hole in the object.