# mu-trajectories

Visualize the KKT trajectory of the Euclidean projection of a point `x0` onto an ellipse
`x'Ax + b'x + c = 0`:

    x(mu) = (I + mu A)^{-1} (x0 - mu b / 2)

The figure shows the trajectory in the plane, `f(mu) = psi(x(mu))`, the distance
`||x0 - x(mu)||` and `f'(mu)`, together with the nearest/farthest KKT points `x(mu*)`,
`x(mu**)` and, when `x0` has a zero standardized coordinate, the degenerate point `x^d`.

## Usage

    pip install -r requirements.txt
    python plot_mu_trajectories.py                         # every case, into figures/
    python plot_mu_trajectories.py x_trajectories_mu_2     # a single case
    pytest

Cases (`A`, `x0`, and optionally `b` and `c`) are listed in `CASES` at the top of
`plot_mu_trajectories.py`; each one produces `<name>.png` and `<name>.pdf`.

Text is typeset with LaTeX (`text.usetex`) when `latex` is on the `PATH`, with the `cm-super` and
`type1cm` packages; otherwise matplotlib's Computer Modern math font is used.

## Files

- `plot_mu_trajectories.py`: the figure. The quadric, `x(mu)`, `f`, `f'`, `mu*` and the degenerate
  points come from [quadproj](https://loicvh.gitlab.io/quadproj); the script only adds `mu**`,
  the asymptotes and the ellipse boundary.
- `test_plot_mu_trajectories.py`: checks these helpers against brute-force sampling of the ellipse
  and against `quadproj.project`.