Vector field curvature and applications

The treatment of tangent curves is a powerful tool for analyzing and visualizing the behavior of vector fields. Unfortunately, for sufficiently complicated vector fields, the tangent curves can only be implicitly described as the solution of a system of differential equations. In this work we show how to compute the curvature of tangent curves and discuss its usefulness for analyzing and visualizing vector fields. In particular, we investigate the curvature behavior around critical points. We show that the curvature of the tangent curves of a 2D vector field and its perpendicular vector field uniquely describe the vector directions in the vector field. We will also describe special curvature properties of linear vector fields in 2D. Applying the ideas of vector field curvature to vector fields over general parametrized surfaces, we are able to compute the curvature of particular tangent curves on a surface, such as contour lines, lines of curvature, asymptotic lines, isophotes and reflection lines. For special tangent curves, we introduce ”thickness” as another characteristic measure. We discuss the application of the curvature of tangent curves on surfaces as a surface interrogation tool. Finally, using the concepts of curvature of tangent curves, we deduce geometric conditions (necessary and sufficient) for G continuity of surfaces.