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.

中文翻译：

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精确解析平行向量

本文论证了平行向量曲线是 3D 分段线性向量场中的分段三次有理曲线。平行矢量曲线——两个矢量场平行的点的轨迹——已被广泛用于提取科学数据中的山脊、山谷和涡旋核心线等特征。我们以 $\mathbf{A}\mathbf{x}+\mathbf{a}=\lambda(\mathbf{B}\mathbf{x}+\mathbf {b})$ 以导出 3D 平行向量曲线的分段有理表示。我们讨论了有理数的奇异性如何导致与四面体单元格的不同类型的交集。