# mu-trajectories

Visualize the KKT trajectory of the Euclidean projection of a point `x0` onto a 2D quadric
`x'Ax + b'x + c = 0` (ellipse, hyperbola, parabola, two lines, a line or a point):

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

The figure shows the trajectory in the plane, the secular function `f(mu)`, the distance
`||x0 - x(mu)||` and `f'(mu)`, together with the KKT points: `x(mu*)`, the other roots of `f`
and, when `x0` has a zero normalized coordinate, the degenerate points `x^d`.

## Setup

Create a virtual environment in `venv/` (ignored by git) and install the dependencies
(tested with Python 3.13):

    python3 -m venv venv
    source venv/bin/activate
    pip install -r requirements.txt

Run `source venv/bin/activate` again in each new shell, and `deactivate` to leave the environment.

## Usage

    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'` and the degenerate
  points come from [quadproj](https://loicvh.gitlab.io/quadproj); the script adds the roots of
  `f`, the asymptotes and the plot limits.
- `test_plot_mu_trajectories.py`: checks, for each type of quadric, that the KKT points contain
  the projection found by brute-force sampling, and that every type can be plotted.