Degenerate 3D Tensors

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Topological analysis of 3D tensor fields starts with the identification of degeneracies in the tensor field. In this chapter, we present a new, intuitive and numerically stable method for finding degenerate tensors in symmetric second order 3D tensor fields. This method is based on a description of a tensor having an isotropic spherical component and a linear or planar component. As such, we refer to this formulation as the geometric approach. In this chapter, we also show that the stable degenerate features in 3D tensor fields form lines. On the other hand, degenerate features that form points, surfaces or volumes are not stable and either disappear or turn into lines when noise is introduced into the system. These topological feature lines provide a compact representation of the 3D tensor field and are useful in helping scientists and engineers understand their complex nature.