Exact Analytical Parallel Vectors
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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=λ(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.
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