  This paper demonstrates that parallel vector curves are piecewise cubic
rational curves in 3D piecewise linear vector fields. Parallel vector curves --
loci of points where two vector fields are parallel -- have been widely used to
extract features including ridges, valleys, and vortex core lines in scientific
data. We define the term \emph{generalized and underdetermined eigensystem} in
the form of
$\mathbf{A}\mathbf{x}+\mathbf{a}=\lambda(\mathbf{B}\mathbf{x}+\mathbf{b})$ in
order to derive the piecewise rational representation of 3D parallel vector
curves. We discuss how singularities of the rationals lead to different types
of intersections with tetrahedral cells.
