As-Perpendicular-as-possible surfaces for flow visualization

We define APAP surfaces, surfaces that are as perpendicular as possible to steady 3D vector fields, and present a method to construct discrete representations of them. Since, in general, a perfectly perpendicular surface to a vector field does not exist, we propose and minimize an error metric to enforce perpendicularity as much as possible. Our algorithm constructs an APAP surface by deforming a seed surface anchored in a domain point. In the discrete setting this minimization results in iteratively solving linear least-squares problems and integrating a locally scaled version of the vector field. The definition of the error metric and its numerical minimization guarantee that the minimum zero is attained for the perfectly perpendicular surface if it exists. Otherwise, the minimization converges to the same local minimum independent of the seed configuration, and the resulting surface is - in a least-squares sense - as perpendicular as possible to the flow. We apply these APAP surfaces as an interactive flow visualization tool which we demonstrate on a number of synthetic and real flow data sets.