  Extracting level sets from scalar data is a fundamental operation in
visualization with many applications. Recently, the concept of level set
extraction has been extended to bivariate scalar fields. Prior work on vector
field equivalence, wherein an analyst marks a region in the domain and is shown
other regions in the domain with similar vector values, pointed out the need to
make this extraction operation fast, so that analysts can work interactively.
To date, the fast extraction of level sets from bivariate scalar fields has not
been researched as extensively as for the univariate case. In this paper, we
present a novel algorithm that extracts fiber lines, i.e., the preimages of so
called control polygons (FSCP), for bivariate 2D data by joint traversal of
bounding volume hierarchies for both grid and FSCP elements. We performed an
extensive evaluation, comparing our method to a two-dimensional adaptation of
the method proposed by Klacansky et al., as well as to the naive approach for
fiber line extraction. The evaluation incorporates a vast array of
configurations in several datasets. We found that our method provides a speedup
of several orders of magnitudes compared to the naive algorithm and requires
two thirds of the computation time compared to Klacansky et al. adapted for 2D.
