  Mode surfaces are the generalization of degenerate curves and neutral
surfaces, which constitute 3D symmetric tensor field topology. Efficient
analysis and visualization of mode surfaces can provide additional insight into
not only degenerate curves and neutral surfaces, but also how these features
transition into each other. Moreover, the geometry and topology of mode
surfaces can help domain scientists better understand the tensor fields in
their applications. Existing mode surface extraction methods can miss features
in the surfaces. Moreover, the mode surfaces extracted from neighboring cells
have gaps, which make their subsequent analysis difficult. In this paper, we
provide novel analysis on the topological structures of mode surfaces,
including a common parameterization of all mode surfaces of a tensor field
using 2D asymmetric tensors. This allows us to not only better understand the
structures in mode surfaces and their interactions with degenerate curves and
neutral surfaces, but also develop an efficient algorithm to seamlessly extract
mode surfaces, including neutral surfaces. The seamless mode surfaces enable
efficient analysis of their geometric structures, such as the principal
curvature directions. We apply our analysis and visualization to a number of
solid mechanics data sets.
