We introduce a representation of a 2D steady vector field ${{\mathbf v}}$ by two scalar fields $a$, $b$, such that the isolines of $a$ correspond to stream lines of ${{\mathbf v}}$, and $b$ increases with constant speed under integration of ${{\mathbf v}}$. This way, we get a direct encoding of stream lines, i.e., a numerical integration of ${{\mathbf v}}$ can be replaced by a local isoline extraction of $a$. To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields $a$ and $b$ can be computed. We show several evaluations on non-trivial vector fields.

翻译：本文引入一种用两个标量场 $a$、$b$ 表示二维稳态向量场 ${{\mathbf v}}$ 的方法，其中 $a$ 的等值线对应 ${{\mathbf v}}$ 的流线，而 $b$ 沿 ${{\mathbf v}}$ 积分时以恒定速度增加。由此，我们实现对流线的直接编码，即 ${{\mathbf v}}$ 的数值积分可被替换为 $a$ 的局部等值线提取。为确保解在任何情况下均存在，我们引入梯度保持割线，使得标量场在数值上可间断但梯度保持连续。通过结合分段线性离散化与割线的合理布置，可计算场 $a$ 和 $b$。我们基于非平凡向量场进行了多项验证。