Exact Analytical Parallel Vectors

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 = $\lambda$(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.