Use quadproj instead of the local quadric module

The quadric, x(mu), f, f', mu* and the degenerate points now come from
quadproj; the script keeps small helpers for what quadproj does not
provide: mu**, the asymptotes and the ellipse boundary. The tests check
these helpers against brute-force sampling and quadproj.project.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

test_plot_mu_trajectories.py

@@ -0,0 +1,50 @@
+import numpy as np
+import pytest
+from quadproj.fun import Fun
+from quadproj.project import project
+from quadproj.quadrics import Quadric
+
+from plot_mu_trajectories import degenerate_point, ellipse_boundary, kkt_roots, pole_line
+
+ROTATED_ELLIPSE = Quadric(np.array([[3.0, 1.0], [1.0, 2.0]]), np.array([1.0, -1.0]), -4.0)
+
+
+def trajectory(Q, x0):
+    return Fun(Q, Q.to_normalized(x0)).x_not_std
+
+
+def boundary_distances(Q, x0):
+    return np.linalg.norm(ellipse_boundary(Q, 200_000) - x0, axis=1)
+
+
+def test_boundary_is_on_the_ellipse():
+    assert np.all([abs(ROTATED_ELLIPSE.evaluate_point(x)) < 1e-10 for x in ellipse_boundary(ROTATED_ELLIPSE)])
+
+
+@pytest.mark.parametrize('x0', [[0.3, 0.2], [2.0, -1.5], [-0.1, 0.4]])
+def test_kkt_roots_give_nearest_and_farthest_points(x0):
+    Q, x0 = ROTATED_ELLIPSE, np.array(x0)
+    mu_star, mu_star_star = kkt_roots(Q, x0)
+    x = trajectory(Q, x0)
+
+    distances = boundary_distances(Q, x0)
+    assert abs(Q.evaluate_point(x(mu_star_star))) < 1e-8
+    assert np.linalg.norm(x(mu_star) - x0) == pytest.approx(distances.min(), abs=1e-6)
+    assert np.linalg.norm(x(mu_star_star) - x0) == pytest.approx(distances.max(), abs=1e-6)
+
+
+@pytest.mark.parametrize('eig, x0', [([2.0, 1.0], [0.0, -0.1]), ([2.0, 0.75], [0.0, -0.6])])
+def test_degenerate_point_is_the_projection(eig, x0):
+    Q, x0 = Quadric(np.diag(eig), np.zeros(2), -1.0), np.array(x0)
+    xd = degenerate_point(Q, x0, 0)
+    np.testing.assert_allclose(xd, project(Q, x0))
+    assert np.linalg.norm(xd - x0) == pytest.approx(boundary_distances(Q, x0).min(), abs=1e-6)
+    assert degenerate_point(Q, x0, 1) is None
+
+
+def test_pole_line_is_the_asymptote():
+    Q, x0 = ROTATED_ELLIPSE, np.array([0.3, 0.2])
+    point, direction = pole_line(Q, x0, 1)
+    x = trajectory(Q, x0)(-1 / Q.eig[1] + 1e-9)
+    residual = (x - point) - ((x - point) @ direction) * direction
+    np.testing.assert_allclose(residual, 0, atol=1e-6)