research: reuse quadrature and clarify snapshot and VIV coordinates

tests/unit/test_duetsch_protocol.py

@@ -82,6 +82,35 @@ def test_interval_bases_and_morison_fit_recover_known_coefficients():
     assert p.morison_fit(f, b[:, 0], b[:, 1]) == pytest.approx((2.09, 1.45), abs=1e-9)
 
 
+def test_quadrature_rule_reused_across_steps_and_periods(monkeypatch):
+    original = np.polynomial.legendre.leggauss
+    orders = []
+
+    def counted(points):
+        orders.append(points)
+        return original(points)
+
+    p._quadrature_rule.cache_clear()
+    monkeypatch.setattr(np.polynomial.legendre, "leggauss", counted)
+    try:
+        for points in (16, 8, 16):
+            for period in (T, 2 * T):
+                for n in (0, 123, period + 123):
+                    p.interval_bases(n, period, points)
+        assert orders == [16, 8]
+    finally:
+        p._quadrature_rule.cache_clear()
+
+
+def test_interval_sine_basis_matches_exact_integral():
+    for period in (T, 2 * T):
+        half_width = math.pi / period
+        for n in (0, period // 4, period // 2, period - 1, 7 * period + 123):
+            midpoint = 2 * math.pi * ((n % period) + .5) / period - math.pi / 2
+            expected = math.sin(midpoint) * math.sin(half_width) / half_width
+            assert p.interval_bases(n, period)[1] == pytest.approx(expected, rel=0, abs=2e-15)
+
+
 def test_geometry_repeats_exactly_across_cycles():
     A = p.amplitude(20)
     for m in (0, 1, T // 4, T // 2 - 1, 7 * T // 12):