# Rotation Invariant Vortices for Flow Visualization

# Extracting, Tracking, and Visualizing Magnetic Flux Vortices in 3D Complex-Valued Superconductor

Both of these papers involve the extraction of a particular kind of feature, a vortex,
which is defined as a set of points satisfying a certain mathematical condition. The
definition of flow vortices (Günther et al.) and magnetic flux vortices (Guo et al.)
are different, but in both cases, the feature definition is by design
invariant with respect to certain changes in the given data. In the case of flow
vortices, past vortex measures have been invariant with respect to
affine transforms of space-time (these are called *Galilean-invariance*). In the
case of magnetic flux vortices in scalar (but complex-valued) fields, vortices
are by definition invariant to uniform phase changes (this is called *gauge-invariance*).

As discussed in Section 6.1 of the Algebraic Vis paper, the algebraic vis terminology
can become counter-intuitive when applied to feature-based Sci Vis like this.
One could consider a vortex feature itself as the “visualization”, in which
case the symmetries of the data (which leave the feature unchanged) are **confusers**;
but those confusers are not bad, despite the judgement implied by the term:
they are in fact an essential defining property of the feature.

Alternatively, the vortex feature could be considered the real “data”, in which case
the symmetrices of the actual given dataset (the vector or complex scalar field) are
really changes in **representation**. The feature definition, and a correct implementation
of its extraction, would then avoid **hallucinators** by ensuring that the same features result
from any possible representation. The importance of this is explicitly mentioned in the
magnetic flux vortex paper by Guo et al. The innovation in the paper
by Günther et al., on the other hand, is to formulate a vortex feature with a new of symmetrices
(rotation around a fixed axis) rather than standard Galilean invariance.