Parameterization of 3D meshes is important for many graphic and CAD applications, in particular for texture mapping, remeshing, and morphing. Current parameterization methods for closed manifold genus-$g$ meshes usually involve cutting the mesh according to the object generators, adjusting the resulting boundary and then determining the 2D parameterization coordinates of the mesh vertices, such that the flattened triangles are not too distorted and do not overlap. Unfortunately, adjusting the boundary distorts the resulting parameterization, especially near the boundary. To overcome this problem for genus-$g$ meshes we first address the special case of closed manifold genus-1 meshes by presenting cyclic boundary constraints. Then, we expand the idea of cyclic boundary constraints by presenting a new generalized method developed for planar parameterization of closed manifold genus-$g$ meshes. A planar parameterization is constructed by exploiting the topological structure of the mesh. This planar parameterization can be represented by a surface which is defined over parallel $g$-planes that represents $g$-holes. The proposed parameterization method satisfies the nonoverlapping requirement for any type of positive barycentric weights, including asymmetric weights. Moreover, convergence is guaranteed according to the Gauss-Seidel method.

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