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/