Using Geometric Algebra for Visualizing Integral Curves

The Differential Geometry of curves is described by means of the Frenet-Serret formulas, which cast first, second and third order derivatives into curvature and torsion. While in usual vector calculus these quantities are usually considered to be scalar values, formulating the Frenet-Serret equations in the framework of Geometric Algebra exhibits that they are best described by a bivector for the curvature and a trivector for the torsion. The bivector curvature field is directly suitable for visualization of integral curves for vector fields, providing “Frenet Ribbons” which are much richer in their visual expressiveness than lines. The set of quantities in the Frenet-Serret formalism allows to study numerical pitfalls for computing Frenet Ribbons. We show how to address them and demonstrate the applicability of the technique upon a complex numerical data set from computational fluid dynamics.