Exact Analytical Parallel Vectors

TL;DR

This paper proves that parallel vector curves in 3D vector fields can be represented as piecewise cubic rational curves, enabling better feature extraction in scientific data analysis.

Contribution

It introduces a novel analytical method to represent parallel vector curves as piecewise rational curves using a generalized eigensystem framework.

Findings

Parallel vector curves are piecewise cubic rational in 3D vector fields.

Singularities of rational curves cause different intersection types with tetrahedral cells.

The method facilitates feature extraction like ridges and vortex lines.

Abstract

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.