Scalar Representation of 2D Steady Vector Fields

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.