"""Bölüm 22: kuadrik yüzeyler laboratuvarı. Uygulama, bilinen merkez, yarı eksen ve yönelime sahip sentetik bir üç eksenli elipsoidi genel ikinci derece katsayılarına dönüştürür; ardından merkezi, yarı eksenleri ve ana yönleri yalnız bu katsayılardan geri kazanır. Veriler ölçülmüş değildir. Çalışma bir uyumlama, üretim toleransı veya kabul modeli içermez. """ from __future__ import annotations import math import numpy as np from agbook import ( canonical_array_sha256, quadric_diagnostics, quadric_plane_trace, quadric_residuals, quadric_tangent_plane_diagnostics, quadratic_conic_diagnostics, transform_quadric, ) def rotation_xyz(ax: float, ay: float, az: float) -> np.ndarray: """Sağ elli ``Rz(az) Ry(ay) Rx(ax)`` dönmesini kurar.""" cx, sx = math.cos(ax), math.sin(ax) cy, sy = math.cos(ay), math.sin(ay) cz, sz = math.cos(az), math.sin(az) rx = np.array([[1, 0, 0], [0, cx, -sx], [0, sx, cx]], dtype=float) ry = np.array([[cy, 0, sy], [0, 1, 0], [-sy, 0, cy]], dtype=float) rz = np.array([[cz, -sz, 0], [sz, cz, 0], [0, 0, 1]], dtype=float) return rz @ ry @ rx def world_coefficients( canonical_coefficients: np.ndarray, rotation: np.ndarray, center: np.ndarray, ) -> np.ndarray: """``X=center+rotation@Y`` modelini dünya katsayılarına çevirir.""" return transform_quadric( canonical_coefficients, rotation.T, -rotation.T @ center, ) def main() -> None: length_unit = "mm" center_mm = np.array([120.0, -80.0, 45.0]) semi_axes_mm = np.array([60.0, 35.0, 20.0]) rotation = rotation_xyz( math.radians(15.0), math.radians(-20.0), math.radians(35.0), ) canonical_coefficients = np.array( [ 1.0 / semi_axes_mm[0] ** 2, 0.0, 0.0, 1.0 / semi_axes_mm[1] ** 2, 0.0, 1.0 / semi_axes_mm[2] ** 2, 0.0, 0.0, 0.0, -1.0, ] ) coefficients_mm = world_coefficients( canonical_coefficients, rotation, center_mm, ) diagnostics_mm = quadric_diagnostics( coefficients_mm, coordinate_scale=100.0, ) alignment = np.abs(rotation.T @ diagnostics_mm.principal_directions) principal_trace_types: list[str] = [] principal_trace_axes: list[np.ndarray] = [] for indices in ((0, 1), (0, 2), (1, 2)): basis = rotation[:, indices] trace = quadric_plane_trace(coefficients_mm, center_mm, basis) conic = quadratic_conic_diagnostics( trace.conic_coefficients, coordinate_scale=100.0, ) principal_trace_types.append(conic.locus_type) principal_trace_axes.append(conic.semi_axes) surface_point_mm = center_mm + semi_axes_mm[0] * rotation[:, 0] tangent = quadric_tangent_plane_diagnostics( coefficients_mm, surface_point_mm, coordinate_scale=100.0, gradient_tolerance=1.0e-12, residual_tolerance=1.0e-12, ) parameters = np.array( [ [1.0, 0.0, 0.0], [-1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, 1.0], [0.0, 0.0, -1.0], ] ) surface_points_mm = center_mm + (parameters * semi_axes_mm) @ rotation.T residuals_mm = quadric_residuals( coefficients_mm, surface_points_mm, coordinate_scale=100.0, ) motion = rotation_xyz(-0.19, 0.27, -0.41) translation_mm = np.array([-35.0, 70.0, 25.0]) moved_coefficients = transform_quadric( coefficients_mm, motion.T, -motion.T @ translation_mm, ) moved_diagnostics = quadric_diagnostics( moved_coefficients, coordinate_scale=100.0, ) mm_per_m = 1000.0 coefficients_m = coefficients_mm * np.array( [ mm_per_m**2, mm_per_m**2, mm_per_m**2, mm_per_m**2, mm_per_m**2, mm_per_m**2, mm_per_m, mm_per_m, mm_per_m, 1.0, ] ) diagnostics_m = quadric_diagnostics( coefficients_m, coordinate_scale=0.1, ) assert diagnostics_mm.locus_type == "ellipsoid" np.testing.assert_allclose(diagnostics_mm.center, center_mm, atol=3.0e-13) np.testing.assert_allclose(diagnostics_mm.semi_axes, semi_axes_mm, atol=3.0e-13) np.testing.assert_allclose(alignment, np.eye(3), atol=3.0e-14) assert principal_trace_types == ["ellipse", "ellipse", "ellipse"] np.testing.assert_allclose(principal_trace_axes[0], [60.0, 35.0]) np.testing.assert_allclose(principal_trace_axes[1], [60.0, 20.0]) np.testing.assert_allclose(principal_trace_axes[2], [35.0, 20.0]) assert tangent.regular assert tangent.point_on_surface np.testing.assert_allclose(residuals_mm, np.zeros(6), atol=3.0e-15) assert moved_diagnostics.locus_type == "ellipsoid" np.testing.assert_allclose(moved_diagnostics.semi_axes, semi_axes_mm, atol=4.0e-13) np.testing.assert_allclose( moved_diagnostics.center, motion @ center_mm + translation_mm, atol=4.0e-13, ) assert diagnostics_m.locus_type == "ellipsoid" np.testing.assert_allclose(diagnostics_m.center, center_mm / mm_per_m, atol=4.0e-16) np.testing.assert_allclose( diagnostics_m.semi_axes, semi_axes_mm / mm_per_m, atol=4.0e-16, ) signature_values = np.concatenate( [ center_mm, semi_axes_mm, rotation.ravel(), canonical_coefficients, coefficients_mm, diagnostics_mm.center, diagnostics_mm.semi_axes, diagnostics_mm.principal_directions.ravel(), np.concatenate(principal_trace_axes), surface_point_mm, tangent.gradient, tangent.plane_coefficients, surface_points_mm.ravel(), residuals_mm, moved_coefficients, diagnostics_m.center, diagnostics_m.semi_axes, ] ) print("uzunluk_birimi:", length_unit) print("model: sentetik_uc_eksenli_elipsoid") print("olculmus_veri_mi: False") print("uyumlama_yapildi_mi: False") print("uretim_toleransi_modeli_mi: False") print("merkez_mm:", diagnostics_mm.center) print("yari_eksenler_mm:", diagnostics_mm.semi_axes) print("sinif:", diagnostics_mm.locus_type) print("kuadratik_rank:", diagnostics_mm.quadratic_rank) print("atalet:", diagnostics_mm.inertia) print("ana_yon_hizalama_mutlak:") print(alignment) print("asal_iz_turleri:", principal_trace_types) print("asal_iz_yari_eksenleri_mm:", principal_trace_axes) print("teget_noktasi_mm:", surface_point_mm) print("teget_duzlem_katsayilari:", tangent.plane_coefficients) print("en_buyuk_normalize_artik:", f"{np.max(np.abs(residuals_mm)):.12e}") print("rijit_harekette_sinif_korundu_mu:", moved_diagnostics.locus_type == "ellipsoid") print("metrede_yari_eksenler:", diagnostics_m.semi_axes) print("goreli_siniflandirma_esigi:", diagnostics_mm.relative_tolerance) print("sayisal_belirsizlik_var_mi:", diagnostics_mm.numerically_ambiguous) print("bilimsel_imza:", canonical_array_sha256(signature_values)) if __name__ == "__main__": main()