loicvhandClaude Opus 5
8588bc9f30
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
42 lines
1.7 KiB
Markdown
42 lines
1.7 KiB
Markdown
# mu-trajectories
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Visualize the KKT trajectory of the Euclidean projection of a point `x0` onto a 2D quadric
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`x'Ax + b'x + c = 0` (ellipse, hyperbola, parabola, two lines, a line or a point):
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x(mu) = (I + mu A)^{-1} (x0 - mu b / 2)
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The figure shows the trajectory in the plane, the secular function `f(mu)`, the distance
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`||x0 - x(mu)||` and `f'(mu)`, together with the KKT points: `x(mu*)`, the other roots of `f`
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and, when `x0` has a zero normalized coordinate, the degenerate points `x^d`.
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## Setup
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Create a virtual environment in `venv/` (ignored by git) and install the dependencies
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(tested with Python 3.13):
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python3 -m venv venv
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source venv/bin/activate
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pip install -r requirements.txt
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Run `source venv/bin/activate` again in each new shell, and `deactivate` to leave the environment.
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## Usage
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python plot_mu_trajectories.py # every case, into figures/
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python plot_mu_trajectories.py x_trajectories_mu_2 # a single case
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pytest
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Cases (`A`, `x0`, and optionally `b` and `c`) are listed in `CASES` at the top of
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`plot_mu_trajectories.py`; each one produces `<name>.png` and `<name>.pdf`.
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Text is typeset with LaTeX (`text.usetex`) when `latex` is on the `PATH`, with the `cm-super` and
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`type1cm` packages; otherwise matplotlib's Computer Modern math font is used.
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## Files
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- `plot_mu_trajectories.py`: the figure. The quadric, `x(mu)`, `f`, `f'` and the degenerate
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points come from [quadproj](https://loicvh.gitlab.io/quadproj); the script adds the roots of
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`f`, the asymptotes and the plot limits.
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- `test_plot_mu_trajectories.py`: checks, for each type of quadric, that the KKT points contain
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the projection found by brute-force sampling, and that every type can be plotted.
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