Tetrahedral Mesh Construction for Unit Tangent Bundle over Genus-Zero Surfaces

Unit tangent bundle of a surface carries various information of tangent vector fields on that surface. For 2-spheres (i.e. genus-zero closed surfaces), the unit tangent bundle is a closed 3-manifold that has non-trivial topology and cannot be embedded in R. Therefore it cannot be constructed by existing mesh generation algorithms directly. This work aims at the first discrete construction of unit tangent bundles over 2-spheres using tetrahedral meshes. We propose a two-stage algorithm for the construction, which starts from constructing two local bundles and then combines them into a global bundle.