A triangle mesh is a partition of a bounded domain with simplices (triangles in 2D, tetrahedra in 3D) so that any two of these simplices are either disjoint or sharing a face. The resulting triangulation (also called tetrahedralizations in 3D) provides a discretization of space through both its primal (simplicial) elements and 148its dual (cell) elements. Both types of element are crucial to a variety of numerical techniques. A growing trend in numerical simulation is the simultaneous use of primal and dual meshes: Petrov–Galerkin finite element/finite volume methods [24,271,298] and methods based on exterior calculus [65,119,176] use the ability to store quantities on both primal and dual elements to enforce (co)homological relationships, which for instance appear in Hodge theory. The choice of the dual, defined by the location of the dual vertices, is however not specified a priori. The barycentric dual, for which barycenters are used instead of circumcenters, is used for certain finite volume computations, but it fails to satisfy both the orthogonality and the convexity conditions on general triangulations. A very common dual to a triangulation in R d $ \mathbb R ^d $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315153452/15c305c2-37ac-42a2-a986-bfbc1a242481/content/inline-math9_2.tif"/> is the cell complex that uses the circumcenters of each d-simplex as dual vertices. It corresponds to the Voronoi diagram in the case of Delaunay triangulations [137,313]. The circumcentric Delaunay–Voronoi duality has been extensively used in diverse fields. Building on a number of results in algebraic and computational geometry, a more general primal-dual pair of complexes can be defined , as we briefly describe in the first sections of this chapter. A barycentric representation of primal-dual triangulations is then presented as a parameterization of this space of generalized triangulations.