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This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points, as previously thought. Second, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Third, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields.