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In this article, we present an original unified framework for analyzing, describing, and encoding 3D facial surfaces. This framework allows the derivation of topologically ordered structures from triangular mesh surfaces, addressing thus the lack of ordered structure in such a modality. After describing the foundations of the framework and highlighting its advantages with respect to close representations, we show its adaptability to a variety of facial mesh surface processing tasks which includes mesh regularity assessment, facial surface cropping, facial surface compression, and facial surface alignment. In addition, it can be used for the extraction of a rich variety of local and global face descriptors. We validate this framework by testing it with raw 3D facial mesh surfaces.