  The analysis of contours of scalar fields plays an important role in
visualization. For example the contour tree and contour statistics can be used
as a means for interaction and filtering or as signatures. In the context of
tensor field analysis, such methods are also interesting for the analysis of
derived scalar invariants. While there are standard algorithms to compute and
analyze contours, they are not directly applicable to tensor invariants when
using component-wise tensor interpolation. In this chapter we present an
accurate derivation of the contour spectrum for invariants with quadratic
behavior computed from two-dimensional piece-wise linear tensor fields. For
this work, we are mostly motivated by a consistent treatment of the anisotropy
field, which plays an important role as stability measure for tensor field
topology. We show that it is possible to derive an analytical expression for
the distribution of the invariant values in this setting, which is exemplary
given for the anisotropy in all details. Our derivation is based on a
topological sub-division of the mesh in triangles that exhibit a monotonic
behavior. This triangulation can also directly be used to compute the accurate
contour tree with standard algorithms. We compare the results to a na\"ive
approach based on linear interpolation on the original mesh or the subdivision.
