  Andrews plots provide aesthetically pleasant visualizations of
high-dimensional datasets. This work proves that Andrews plots (when defined in
terms of the principal component scores of a dataset) are optimally ``smooth''
on average, and solve an infinite-dimensional quadratic minimization program
over the set of linear isometries from the Euclidean data space to
$L^2([0,1])$. By building technical machinery that characterizes the solutions
to general infinite-dimensional quadratic minimization programs over linear
isometries, we further show that the solution set is (in the generic case) a
manifold. To avoid the ambiguities presented by this manifold of solutions, we
add ``spectral smoothing'' terms to the infinite-dimensional optimization
program to induce Andrews plots with optimal spatial-spectral smoothing. We
characterize the (generic) set of solutions to this program and prove that the
resulting plots admit efficient numerical approximations. These
spatial-spectral smooth Andrews plots tend to avoid some ``visual clutter''
that arises due to the oscillation of trigonometric polynomials.
