Uzayda Doğrular ve Düzlemler

Bu bölümün laboratuvarı, kitaptaki örnekleri yeniden üretmek ve sayısal sonuçları incelemek için hazırlanmıştır. Matematiksel tanımlar, ispatlar ve sorular kitapta yer alır.

Laboratuvar kodu

"""Bölüm 18: uzayda doğru, düzlem ve rank tanısı laboratuvarı.

Uygulama, sentetik bir CAD datum düzlemi ile delik ekseninin eşdeğer
gösterimlerini sınar. Ölçülmüş veri, üretim toleransı, delik kalitesi veya
kabul kararı içermez.
"""

from __future__ import annotations

import numpy as np

from agbook import (
    affine_system_diagnostics_3d,
    canonical_array_sha256,
    canonical_plane_coefficients,
    parametric_line_points_3d,
    plane_from_points_3d,
    plane_membership_diagnostics_3d,
    plane_residuals_3d,
)


def main() -> None:
    length_unit = "mm"
    rank_relative_tolerance = 1.0e-12
    reference_scale_mm = 200.0

    point_a = np.array([120.0, 80.0, 35.0])
    point_b = np.array([156.0, 128.0, 35.0])
    point_c = np.array([96.0, 98.0, 75.0])
    datum_points_mm = np.vstack([point_a, point_b, point_c])

    datum_plane_mm = plane_from_points_3d(point_a, point_b, point_c)
    expected_normal = np.array([0.64, -0.48, 0.60])
    scaled_plane = canonical_plane_coefficients(-25.0 * datum_plane_mm)

    # Yön uzunluk birimindedir; t boyutsuzdur ve ||direction|| = 25 mm'dir.
    axis_direction_mm = np.array([16.0, -12.0, 15.0])
    axis_parameters = np.array([-2.0, -1.0, 0.0, 1.0, 2.0])
    axis_points_mm = parametric_line_points_3d(
        point_a,
        axis_direction_mm,
        axis_parameters,
    )

    # İki birim normalin düzlemleri delik eksenini ortak çözüm olarak verir.
    tangent_1 = np.array([0.60, 0.80, 0.0])
    tangent_2 = np.array([-0.48, 0.36, 0.80])
    axis_matrix = np.vstack([tangent_1, tangent_2])
    axis_rhs_mm = axis_matrix @ point_a
    axis_diagnosis_mm = affine_system_diagnostics_3d(
        axis_matrix,
        axis_rhs_mm,
        rank_relative_tolerance=rank_relative_tolerance,
        reference_scale=reference_scale_mm,
    )
    axis_plane_residuals = np.column_stack(
        [
            plane_residuals_3d(
                np.concatenate([axis_matrix[index], [-axis_rhs_mm[index]]]),
                axis_points_mm,
            )
            for index in range(2)
        ]
    )

    full_matrix = np.vstack([axis_matrix, expected_normal])
    full_rhs_mm = full_matrix @ point_a
    anchor_diagnosis_mm = affine_system_diagnostics_3d(
        full_matrix,
        full_rhs_mm,
        rank_relative_tolerance=rank_relative_tolerance,
        reference_scale=reference_scale_mm,
    )

    membership_b = plane_membership_diagnostics_3d(
        datum_plane_mm,
        point_b,
        absolute_tolerance=1.0e-12,
        relative_tolerance=0.0,
        reference_scale=reference_scale_mm,
    )
    membership_c = plane_membership_diagnostics_3d(
        datum_plane_mm,
        point_c,
        absolute_tolerance=1.0e-12,
        relative_tolerance=0.0,
        reference_scale=reference_scale_mm,
    )

    datum_points_m = datum_points_mm * 1.0e-3
    datum_plane_m = plane_from_points_3d(*datum_points_m)
    axis_rhs_m = axis_matrix @ (point_a * 1.0e-3)
    axis_diagnosis_m = affine_system_diagnostics_3d(
        axis_matrix,
        axis_rhs_m,
        rank_relative_tolerance=rank_relative_tolerance,
        reference_scale=reference_scale_mm * 1.0e-3,
    )

    # Aynı tam verinin iki açık rank eşiğinde farklı sayısal çözünürlüğü.
    perturbation = 1.0e-10
    near_matrix = np.vstack([tangent_1, tangent_1 + perturbation * tangent_2])
    near_rhs_mm = near_matrix @ point_a
    coarse_rank = affine_system_diagnostics_3d(
        near_matrix,
        near_rhs_mm,
        rank_relative_tolerance=1.0e-8,
        reference_scale=reference_scale_mm,
    )
    fine_rank = affine_system_diagnostics_3d(
        near_matrix,
        near_rhs_mm,
        rank_relative_tolerance=1.0e-12,
        reference_scale=reference_scale_mm,
    )

    np.testing.assert_allclose(
        datum_plane_mm,
        [0.64, -0.48, 0.60, -59.4],
        atol=2.0e-14,
    )
    np.testing.assert_allclose(scaled_plane, datum_plane_mm, atol=2.0e-14)
    np.testing.assert_allclose(axis_direction_mm / 25.0, expected_normal)
    np.testing.assert_allclose(axis_points_mm[2], point_a)
    np.testing.assert_allclose(axis_plane_residuals, 0.0, atol=3.0e-14)
    assert axis_diagnosis_mm.classification == "line"
    assert axis_diagnosis_mm.coefficient_rank == 2
    assert axis_diagnosis_mm.solution_dimension == 1
    assert anchor_diagnosis_mm.classification == "point"
    assert anchor_diagnosis_mm.coefficient_rank == 3
    np.testing.assert_allclose(
        anchor_diagnosis_mm.least_squares_point,
        point_a,
        atol=4.0e-14,
    )
    assert membership_b.on_plane and membership_c.on_plane
    np.testing.assert_allclose(datum_plane_m[:3], datum_plane_mm[:3], atol=2.0e-15)
    np.testing.assert_allclose(datum_plane_m[3], 1.0e-3 * datum_plane_mm[3], atol=2.0e-16)
    assert axis_diagnosis_m.classification == axis_diagnosis_mm.classification
    np.testing.assert_allclose(
        axis_diagnosis_m.least_squares_point,
        1.0e-3 * axis_diagnosis_mm.least_squares_point,
        atol=3.0e-16,
    )
    assert coarse_rank.classification == "plane"
    assert fine_rank.classification == "line"

    signature_values = np.concatenate(
        [
            datum_points_mm.ravel(),
            datum_plane_mm,
            axis_direction_mm,
            axis_parameters,
            axis_points_mm.ravel(),
            axis_matrix.ravel(),
            axis_rhs_mm,
            axis_diagnosis_mm.coefficient_singular_values,
            axis_diagnosis_mm.augmented_singular_values,
            full_matrix.ravel(),
            full_rhs_mm,
            anchor_diagnosis_mm.least_squares_point,
            datum_points_m.ravel(),
            datum_plane_m,
            axis_rhs_m,
            near_matrix.ravel(),
            near_rhs_mm,
            coarse_rank.coefficient_singular_values,
            fine_rank.coefficient_singular_values,
            np.array(
                [
                    rank_relative_tolerance,
                    reference_scale_mm,
                    perturbation,
                    membership_b.normalized_residual,
                    membership_c.normalized_residual,
                ]
            ),
        ]
    )

    print("uzunluk_birimi:", length_unit)
    print("model: sentetik_cad_datum_duzlemi_ve_delik_ekseni")
    print("olculmus_veri_mi: False")
    print("uretim_toleransi_hesabi_mi: False")
    print("uretim_kabul_hesabi_mi: False")
    print("datum_noktalari_mm:")
    print(datum_points_mm)
    print("datum_duzlemi_kanonik:", datum_plane_mm)
    print("olceklenmis_denklem_ayni_duzlem_mi:", bool(np.allclose(scaled_plane, datum_plane_mm)))
    print("delik_ekseni_baslangici_mm:", point_a)
    print("delik_ekseni_yonu_mm:", axis_direction_mm)
    print("delik_ekseni_iki_duzlem_ranki:", axis_diagnosis_mm.coefficient_rank)
    print("delik_ekseni_cozum_boyutu:", axis_diagnosis_mm.solution_dimension)
    print("delik_ekseni_sinifi:", axis_diagnosis_mm.classification)
    print("uc_bagimsiz_duzlem_ranki:", anchor_diagnosis_mm.coefficient_rank)
    print("uc_bagimsiz_duzlem_sinifi:", anchor_diagnosis_mm.classification)
    print("tek_nokta_cozumu_mm:", anchor_diagnosis_mm.least_squares_point)
    print("B_ve_C_datum_duzleminde_mi:", membership_b.on_plane and membership_c.on_plane)
    print("metreye_geciste_normal_korundu_mu:", bool(np.allclose(datum_plane_m[:3], datum_plane_mm[:3])))
    print("metreye_geciste_sabit_terim_olcegi:", f"{datum_plane_m[3] / datum_plane_mm[3]:.12f}")
    print("metreye_geciste_rank_sinifi_korundu_mu:", axis_diagnosis_m.classification == axis_diagnosis_mm.classification)
    print("yakin_bagimli_sistem_kaba_esik:", coarse_rank.classification)
    print("yakin_bagimli_sistem_ince_esik:", fine_rank.classification)
    print("rank_bagil_esigi:", f"{rank_relative_tolerance:.12e}")
    print("referans_uzunlugu_mm:", f"{reference_scale_mm:.12f}")
    print("bilimsel_imza:", canonical_array_sha256(signature_values))


if __name__ == "__main__":
    main()

uzunluk_birimi: mm
model: sentetik_cad_datum_duzlemi_ve_delik_ekseni
olculmus_veri_mi: False
uretim_toleransi_hesabi_mi: False
uretim_kabul_hesabi_mi: False
datum_noktalari_mm:
[[120.  80.  35.]
 [156. 128.  35.]
 [ 96.  98.  75.]]
datum_duzlemi_kanonik: [  0.64  -0.48   0.6  -59.4 ]
olceklenmis_denklem_ayni_duzlem_mi: True
delik_ekseni_baslangici_mm: [120.  80.  35.]
delik_ekseni_yonu_mm: [ 16. -12.  15.]
delik_ekseni_iki_duzlem_ranki: 2
delik_ekseni_cozum_boyutu: 1
delik_ekseni_sinifi: line
uc_bagimsiz_duzlem_ranki: 3
uc_bagimsiz_duzlem_sinifi: point
tek_nokta_cozumu_mm: [120.  80.  35.]
B_ve_C_datum_duzleminde_mi: True
metreye_geciste_normal_korundu_mu: True
metreye_geciste_sabit_terim_olcegi: 0.001000000000
metreye_geciste_rank_sinifi_korundu_mu: True
yakin_bagimli_sistem_kaba_esik: plane
yakin_bagimli_sistem_ince_esik: line
rank_bagil_esigi: 1.000000000000e-12
referans_uzunlugu_mm: 200.000000000000
bilimsel_imza: fd88271e849c8c7abfcd0d8f05ad93b4216d1330321da8a8ccc9982c58db876e

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Üç noktadan datum ve iki düzlemden eksen ---
import numpy as np
from agbook import (
    affine_system_diagnostics_3d,
    canonical_plane_coefficients,
    plane_from_points_3d,
)

A = np.array([120.0, 80.0, 35.0])
B = np.array([156.0, 128.0, 35.0])
C = np.array([96.0, 98.0, 75.0])

datum = plane_from_points_3d(A, B, C)
same = canonical_plane_coefficients(-25.0 * datum)

r1 = np.array([0.60, 0.80, 0.0])
r2 = np.array([-0.48, 0.36, 0.80])
M = np.vstack([r1, r2])
b = M @ A
diagnosis = affine_system_diagnostics_3d(
    M,
    b,
    rank_relative_tolerance=1e-12,
    reference_scale=200.0,
)

print(datum)
print(np.allclose(datum, same))
print(diagnosis.coefficient_rank)
print(diagnosis.solution_dimension)
print(diagnosis.classification)
print(diagnosis.maximum_equation_residual)

1. Üç noktadan datum ve iki düzlemden eksen
[  0.64  -0.48   0.6  -59.4 ]
True
2
1
line
2.842170943040401e-14

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Parametrik doğru değişmezleri ---
import numpy as np
import pytest
from agbook import parametric_line_points_3d

P = np.array([2.0, -1.0, 3.0])
v = np.array([4.0, 2.0, -2.0])
t = np.array([-2.0, -0.5, 0.0, 1.0, 3.0])
points = parametric_line_points_3d(P, v, t)

for delta in points - P:
    np.testing.assert_allclose(np.cross(delta, v), 0.0, atol=1e-13)

with pytest.raises(ValueError):
    parametric_line_points_3d(P, np.zeros(3), t)

# --- 2. Düzlem ölçek eşdeğerliği ---
import numpy as np
from agbook import canonical_plane_coefficients, plane_residuals_3d

plane = np.array([2.0, -4.0, 4.0, -10.0])
factors = np.array([-7.0, -0.5, 0.25, 3.0, 20.0])
points = np.array([
    [0.0, 0.0, 0.0],
    [1.0, 0.0, 2.0],
    [5.0, 0.0, 0.0],
    [2.0, -1.0, 1.0],
])
target = canonical_plane_coefficients(plane)
base_residual = plane_residuals_3d(plane, points)

for factor in factors:
    scaled = factor * plane
    np.testing.assert_allclose(
        canonical_plane_coefficients(scaled), target, atol=1e-14
    )
    np.testing.assert_allclose(
        plane_residuals_3d(scaled, points),
        factor * base_residual,
        atol=1e-13,
    )

# --- 3. Beş afin çözüm sınıfı ---
import numpy as np
from agbook import affine_system_diagnostics_3d

systems = {
    "space": (np.empty((0, 3)), np.empty(0)),
    "plane": (np.array([[1.0, 0.0, 0.0]]), np.array([2.0])),
    "line": (
        np.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0]]),
        np.array([2.0, 3.0]),
    ),
    "point": (np.eye(3), np.array([2.0, 3.0, 4.0])),
    "empty": (
        np.array([[1.0, 0.0, 0.0], [1.0, 0.0, 0.0]]),
        np.array([0.0, 1.0]),
    ),
}

for expected, (A, b) in systems.items():
    result = affine_system_diagnostics_3d(
        A, b, rank_relative_tolerance=1e-12, reference_scale=10.0
    )
    assert result.classification == expected

# --- 4. Yakın bağımlılık ve açık rank eşiği ---
import numpy as np
from agbook import affine_system_diagnostics_3d

r1 = np.array([0.60, 0.80, 0.0])
r2 = np.array([-0.48, 0.36, 0.80])
P = np.array([120.0, 80.0, 35.0])

for delta in [0.0, 1e-14, 1e-10, 1e-6]:
    A = np.vstack([r1, r1 + delta * r2])
    b = A @ P
    for tolerance in [1e-8, 1e-12]:
        result = affine_system_diagnostics_3d(
            A,
            b,
            rank_relative_tolerance=tolerance,
            reference_scale=200.0,
        )
        print(delta, tolerance, result.classification,
              result.coefficient_singular_values,
              result.maximum_equation_residual)

1. Parametrik doğru değişmezleri

2. Düzlem ölçek eşdeğerliği

3. Beş afin çözüm sınıfı

4. Yakın bağımlılık ve açık rank eşiği
0.0 1e-08 plane [1.41421356 0.        ] 0.0
0.0 1e-12 plane [1.41421356 0.        ] 0.0
1e-14 1e-08 plane [1.41421356e+00 7.06903370e-15] 0.0
1e-14 1e-12 plane [1.41421356e+00 7.06903370e-15] 0.0
1e-10 1e-08 plane [1.41421356e+00 7.07106802e-11] 4.0046188587439246e-11
1e-10 1e-12 line [1.41421356e+00 7.07106802e-11] 2.842170943040401e-14
1e-06 1e-08 line [1.41421356e+00 7.07106781e-07] 5.684341886080802e-14
1e-06 1e-12 line [1.41421356e+00 7.07106781e-07] 5.684341886080802e-14

cd analitik-geometri-python-v1.0
python3 -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"
python labs/bolum_18_uzayda_dogrular_duzlemler.py
python -m pytest

cd analitik-geometri-python-v1.0
py -3.13 -m venv .venv
.venv\Scripts\python.exe -m pip install -e ".[dev]"
.venv\Scripts\python.exe labs/bolum_18_uzayda_dogrular_duzlemler.py
.venv\Scripts\python.exe -m pytest