Mathematical Foundations & Numerical Methods
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 generalized and underdetermined eigensystem in the form of Ax+a=Bx+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.