Intrinsic Delaunay triangulation (IDT) naturally generalizes Delaunay triangulation from R^2 to curved surfaces. Due to many favorable properties, the IDT whose vertex set includes all mesh vertices is of particular interest in polygonal mesh processing. To date, the only way for constructing such IDT is the edge-flipping algorithm, which iteratively flips non-Delaunay edges to become locally Delaunay. Although this algorithm is conceptually simple and guarantees to terminate in finite steps, it has no known time complexity and may also produce triangulations containing faces with only two edges. This article develops a new method to obtain proper IDTs on manifold triangle meshes.We first compute a geodesic Voronoi diagram (GVD) by taking all mesh vertices as generators and then find its dual graph. The sufficient condition for the dual graph to be a proper triangulation is that all Voronoi cells satisfy the so-called closed ball property. To guarantee the closed ball property everywhere, a certain sampling criterion is required. For Voronoi cells that violate the closed ball property, we fix them by computing topologically safe regions, inwhich auxiliary sites can be addedwithout changing the topology of theVoronoi diagram beyond them.Given a meshwith n vertices, we prove that by adding at most O(n) auxiliary sites, the computed GVD satisfies the closed ball property, and hence its dual graph is a proper IDT. Our method has a theoretical worst-case time complexity O(n^2 + tn log n), where t is the number of obtuse angles in the mesh. Computational results show that it empirically runs in linear time on real-world models.

We verify a long-standing conjecture of Nazarov, Treil, and Volberg.

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For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, the decomposition of $\mathcal{C}$ is determined by the decomposition of $\mathcal{T}$ satisfying "vanishing condition" of negative extension groups, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $\mbox{Hom} _{\mathcal{C}}(\mathcal{T} _i[t],\mathcal{T} _j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that for any two cluster tilting objects $T, T'$ in a $2-$Calabi-Yau triangulated category $\mathcal{C}$, the Gabriel quivers of endomorphism algebra End$_{\mathcal{C}}T$ of $T$ is connected if and only if so is End$_{\mathcal{C}}T'$. As an application, we prove that indecomposable $2-$Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs.