  The efficiency of modern computer graphics allows us to explore collections
of space curves simultaneously with "drag-to-rotate" interfaces. This inspires
us to replace "scatterplots of points" with "scatterplots of curves" to
simultaneously visualize relationships across an entire dataset. Since spaces
of curves are infinite dimensional, scatterplots of curves avoid the "lossy"
nature of scatterplots of points. In particular, if two points are close in a
scatterplot of points derived from high-dimensional data, it does not generally
follow that the two associated data points are close in the data space.
Standard Andrews plots provide scatterplots of curves that perfectly preserve
Euclidean distances, but simultaneous visualization of these graphs over an
entire dataset produces visual clutter because graphs of functions generally
overlap in 2D. We mitigate this visual clutter issue by constructing
computationally inexpensive 3D extensions of Andrews plots. First, we construct
optimally smooth 3D Andrews plots by considering linear isometries from
Euclidean data spaces to spaces of planar parametric curves. We rigorously
parametrize the linear isometries that produce (on average) optimally smooth
curves over a given dataset. This parameterization of optimal isometries
reveals many degrees of freedom, and (using recent results on generalized Gauss
sums) we identify a particular member of this set which admits an asymptotic
"tour" property that avoids certain local degeneracies as well. Finally, we
construct unit-length 3D curves (filaments) by numerically solving
Frenet-Serret systems given data from these 3D Andrews plots. We conclude with
examples of filament plots for several standard datasets, illustrating how
filament plots avoid visual clutter. Code and examples available at
https://github.com/n8epi/filaments/ and https://n8epi.github.io/filaments/
