loicvhandClaude Opus 5
55cb27c8de
x(mu) only approaches the centre as |mu| -> inf, so the sampling in [mean(poles) - 3, mean(poles) + 3] left a gap around it. The outer branches now continue geometrically up to |mu| ~ 1e4 / lambda_min; the extra samples lie outside the mu axes and only show in the trajectory. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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
quadric.py: theEllipsoidclass (standardization, trajectory, extreme roots, degenerate points).plot_mu_trajectories.py: the figure.test_quadric.py: checks against brute-force sampling of the ellipse.