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I am green in manifold and I have some geometry background. I have some question about "manifoldness" of a triangular mesh. To my understanding, charts {$\phi_{\alpha},U_{\alpha}$} and transition maps $\tau_{\alpha\beta} $ define every point p on manifold M a unique location. A mesh T is a set of space points $P\in R^3$ with index set {$V,E,F$} specifying the topology. It usually use simple bilinear Barycentric map to represent every triangular face in both charts {$\phi_{\alpha},U_{\alpha}$} and the realization $\psi_{\alpha}(\phi_{\alpha}(U_{\alpha}))\in R^3$ (Am I correct?). The transition function $\tau_{\alpha\beta} $ then is well defined both in manifold space and $R^3$ because automorphism(right?).

Then I have a question about the possibility of not using bilinear Barycentric map for $\psi_{\alpha}$ to all triangles that keeps the surface the same manifold representation. For example a cubic polynomial map for $\psi_{\alpha}$ while keeping $\phi_{\alpha}$ unchanged for some triangular faces. E.g. $\psi_{\alpha}(\phi_{\alpha}(U_{\alpha}))$ is polynomial and $\psi_{\beta}(\phi_{\beta}(U_{\beta}))$ is bilinear. Then the transition map $\tau_{\alpha\beta}$ should not be affected because it works in the domain of $\psi_{\alpha}$ and $\psi_{\beta}$.(right?) However, $R^3\ni\psi_{\alpha}\ne\psi_{\beta}\in R^3 $ because they have different degree of continuity. So, the mesh now becomes topologically manifold but geometrically "half-edge" mismatched. Would it be the situation?

Thanks for any reply and I am glad to learn from.

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Triangular meshes and baricentric coordinates are useful things, but they don't give you a manifold structure. It is a mistake to think that a manifold is "sewn together" from charts. The charts in a manifold atlas overlap, and this is an important and useful feature of an atlas. For example, it allows us to find partition of unity subordinate to charts, and use it to localize some computations. Think of each chart as the coverage zone of a cell network tower. We need overlaps for uninterrupted coverage as we move around.

The Wikipedia article on manifolds explains them as a "patchwork" of charts, which unfortunately brings to mind the wrong picture of sewing edges together.

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Thank you. I get your point. The concept of the topological structure is clearly explained by Eular-Poincare Characteristics. However, because the manifold-structure of a triangulation of space $X$ itself is topologically proven, then the problem I am think of is the incompatibility of the realization maps $\psi_\alpha$. – l l Aug 13 '12 at 2:33

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