Uzayda Kesişim, Açı ve Uzaklık

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

Laboratuvar kodu

"""Bölüm 19: uzayda kesişim, açı, izdüşüm ve uzaklık laboratuvarı.

Uygulama sentetik bir CAD datum düzlemi, delik ekseni ve ikinci doğru
adayını sınar. Ölçülmüş veri, üretim toleransı veya kabul kararı içermez.
"""

from __future__ import annotations

import numpy as np

from agbook import (
    canonical_array_sha256,
    line_line_diagnostics_3d,
    line_plane_diagnostics_3d,
    plane_plane_diagnostics_3d,
    project_point_to_line_3d,
    project_point_to_plane_3d,
)


def main() -> None:
    length_unit = "mm"
    angular_tolerance = 1.0e-12
    absolute_tolerance_mm = 1.0e-9
    relative_tolerance = 1.0e-12
    reference_scale_mm = 200.0
    relation_options = {
        "angular_tolerance": angular_tolerance,
        "absolute_tolerance": absolute_tolerance_mm,
        "relative_tolerance": relative_tolerance,
        "reference_scale": reference_scale_mm,
    }

    point_a = np.array([120.0, 80.0, 35.0])
    axis_direction = np.array([16.0, -12.0, 15.0])
    point_q = np.array([115.2, 83.6, 43.0])
    second_direction = np.array([12.0, 16.0, 0.0])
    connector = point_q - point_a

    datum_plane = np.array([0.64, -0.48, 0.60, -59.4])
    first_axis_normal = np.array([0.60, 0.80, 0.0])
    second_axis_normal = np.array([-0.48, 0.36, 0.80])
    first_axis_plane = np.append(first_axis_normal, -first_axis_normal @ point_a)
    second_axis_plane = np.append(second_axis_normal, -second_axis_normal @ point_a)

    line_pair = line_line_diagnostics_3d(
        point_a,
        axis_direction,
        point_q,
        second_direction,
        **relation_options,
    )
    q_on_axis_projection = project_point_to_line_3d(
        point_a,
        axis_direction,
        point_q,
    )
    a_on_second_projection = project_point_to_line_3d(
        point_q,
        second_direction,
        point_a,
    )
    axis_datum = line_plane_diagnostics_3d(
        point_a,
        axis_direction,
        datum_plane,
        **relation_options,
    )
    second_datum = line_plane_diagnostics_3d(
        point_q,
        second_direction,
        datum_plane,
        **relation_options,
    )
    axis_plane_pair = plane_plane_diagnostics_3d(
        first_axis_plane,
        second_axis_plane,
        **relation_options,
    )

    probe_point = point_a + 12.0 * datum_plane[:3]
    probe_projection = project_point_to_plane_3d(datum_plane, probe_point)

    scale = 1.0e-3
    metre_options = {
        "angular_tolerance": angular_tolerance,
        "absolute_tolerance": scale * absolute_tolerance_mm,
        "relative_tolerance": relative_tolerance,
        "reference_scale": scale * reference_scale_mm,
    }
    metre_pair = line_line_diagnostics_3d(
        scale * point_a,
        scale * axis_direction,
        scale * point_q,
        scale * second_direction,
        **metre_options,
    )

    loose_near_parallel = line_line_diagnostics_3d(
        [0.0, 0.0, 0.0],
        [1.0, 0.0, 0.0],
        [0.0, 1.0, 0.0],
        [1.0, 1.0e-8, 0.0],
        angular_tolerance=1.0e-7,
        absolute_tolerance=1.0e-7,
        relative_tolerance=0.0,
        reference_scale=1.0,
    )
    strict_near_parallel = line_line_diagnostics_3d(
        [0.0, 0.0, 0.0],
        [1.0, 0.0, 0.0],
        [0.0, 1.0, 0.0],
        [1.0, 1.0e-8, 0.0],
        angular_tolerance=1.0e-10,
        absolute_tolerance=1.0e-7,
        relative_tolerance=0.0,
        reference_scale=1.0,
    )

    np.testing.assert_allclose(np.dot(connector, axis_direction), 0.0, atol=3.0e-13)
    np.testing.assert_allclose(np.dot(connector, second_direction), 0.0, atol=3.0e-13)
    assert line_pair.relation == "skew"
    np.testing.assert_allclose(line_pair.closest_point_first, point_a, atol=3.0e-14)
    np.testing.assert_allclose(line_pair.closest_point_second, point_q, atol=3.0e-14)
    np.testing.assert_allclose(line_pair.distance, 10.0, atol=2.0e-14)
    np.testing.assert_allclose(q_on_axis_projection.foot, point_a, atol=3.0e-14)
    np.testing.assert_allclose(a_on_second_projection.foot, point_q, atol=3.0e-14)
    assert axis_datum.relation == "intersecting"
    np.testing.assert_allclose(axis_datum.intersection_point, point_a, atol=2.0e-14)
    assert second_datum.relation == "contained"
    assert axis_plane_pair.relation == "intersecting"
    np.testing.assert_allclose(
        axis_plane_pair.line_direction,
        axis_direction / np.linalg.norm(axis_direction),
        atol=2.0e-15,
    )
    np.testing.assert_allclose(probe_projection.foot, point_a, atol=3.0e-14)
    np.testing.assert_allclose(probe_projection.signed_distance, 12.0, atol=2.0e-14)
    assert metre_pair.relation == line_pair.relation
    np.testing.assert_allclose(metre_pair.distance, scale * line_pair.distance, atol=3.0e-17)
    assert loose_near_parallel.relation == "parallel"
    assert loose_near_parallel.numerically_ambiguous
    assert strict_near_parallel.relation == "intersecting"
    assert strict_near_parallel.condition_number > 1.0e8

    signature_values = np.concatenate(
        [
            point_a,
            axis_direction,
            point_q,
            second_direction,
            connector,
            datum_plane,
            first_axis_plane,
            second_axis_plane,
            line_pair.closest_point_first,
            line_pair.closest_point_second,
            line_pair.separation_vector,
            np.array(
                [
                    line_pair.distance,
                    line_pair.small_angle,
                    line_pair.condition_number,
                    q_on_axis_projection.distance,
                    a_on_second_projection.distance,
                    axis_datum.line_plane_angle,
                    second_datum.signed_anchor_distance,
                    axis_plane_pair.small_angle,
                    probe_projection.signed_distance,
                    metre_pair.distance,
                    loose_near_parallel.condition_number,
                    strict_near_parallel.condition_number,
                    angular_tolerance,
                    absolute_tolerance_mm,
                    relative_tolerance,
                    reference_scale_mm,
                ]
            ),
        ]
    )

    print("uzunluk_birimi:", length_unit)
    print("model: sentetik_cad_aykiri_dogrular_ve_datum_duzlemi")
    print("olculmus_veri_mi: False")
    print("uretim_toleransi_hesabi_mi: False")
    print("uretim_kabul_hesabi_mi: False")
    print("dogru_dogru_sinifi:", line_pair.relation)
    print("en_yakin_birinci_nokta_mm:", line_pair.closest_point_first)
    print("en_yakin_ikinci_nokta_mm:", line_pair.closest_point_second)
    print("ortak_dikme_uzunlugu_mm:", f"{line_pair.distance:.12f}")
    print("dogru_dogru_kucuk_aci_derece:", f"{np.degrees(line_pair.small_angle):.12f}")
    print("Q_noktasinin_eksen_izdusumu_mm:", q_on_axis_projection.foot)
    print("A_noktasinin_ikinci_dogru_izdusumu_mm:", a_on_second_projection.foot)
    print("delik_ekseni_datum_iliskisi:", axis_datum.relation)
    print("ikinci_dogru_datum_iliskisi:", second_datum.relation)
    print("eksen_duzlemleri_iliskisi:", axis_plane_pair.relation)
    print("eksen_duzlemleri_kesisim_yonu:", axis_plane_pair.line_direction)
    print("sonda_nokta_datum_uzakligi_mm:", f"{probe_projection.distance:.12f}")
    print("metreye_geciste_sinif_korundu_mu:", metre_pair.relation == line_pair.relation)
    print("metreye_geciste_uzaklik_orani:", f"{metre_pair.distance / line_pair.distance:.12f}")
    print("yakin_paralel_kaba_esik:", loose_near_parallel.relation)
    print("yakin_paralel_ince_esik:", strict_near_parallel.relation)
    print("yakin_paralel_kosul_sayisi:", f"{strict_near_parallel.condition_number:.6e}")
    print("acisal_esik:", f"{angular_tolerance:.12e}")
    print("mutlak_uzaklik_esigi_mm:", f"{absolute_tolerance_mm:.12e}")
    print("bagil_uzaklik_esigi:", f"{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_aykiri_dogrular_ve_datum_duzlemi
olculmus_veri_mi: False
uretim_toleransi_hesabi_mi: False
uretim_kabul_hesabi_mi: False
dogru_dogru_sinifi: skew
en_yakin_birinci_nokta_mm: [120.  80.  35.]
en_yakin_ikinci_nokta_mm: [115.2  83.6  43. ]
ortak_dikme_uzunlugu_mm: 10.000000000000
dogru_dogru_kucuk_aci_derece: 90.000000000000
Q_noktasinin_eksen_izdusumu_mm: [120.  80.  35.]
A_noktasinin_ikinci_dogru_izdusumu_mm: [115.2  83.6  43. ]
delik_ekseni_datum_iliskisi: intersecting
ikinci_dogru_datum_iliskisi: contained
eksen_duzlemleri_iliskisi: intersecting
eksen_duzlemleri_kesisim_yonu: [ 0.64 -0.48  0.6 ]
sonda_nokta_datum_uzakligi_mm: 12.000000000000
metreye_geciste_sinif_korundu_mu: True
metreye_geciste_uzaklik_orani: 0.001000000000
yakin_paralel_kaba_esik: parallel
yakin_paralel_ince_esik: intersecting
yakin_paralel_kosul_sayisi: 2.000000e+08
acisal_esik: 1.000000000000e-12
mutlak_uzaklik_esigi_mm: 1.000000000000e-09
bagil_uzaklik_esigi: 1.000000000000e-12
referans_uzunlugu_mm: 200.000000000000
bilimsel_imza: 22ef46e88e98e64c196e3480d6e5793b73b8e3b41c79d39990ee138a071e6029

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Uzay ilişkileri ve izdüşüm için yinelenebilir denetim ---
import numpy as np
from agbook import (
    line_line_diagnostics_3d,
    project_point_to_plane_3d,
)

opts = dict(
    angular_tolerance=1e-12,
    absolute_tolerance=1e-10,
    relative_tolerance=0.0,
    reference_scale=1.0,
)
pair = line_line_diagnostics_3d(
    [0, 0, 0], [1, 0, 0],
    [0, 1, 1], [0, 1, 0],
    **opts,
)
projection = project_point_to_plane_3d(
    [0, 0, 1, -2], [1, 3, 5]
)

assert pair.relation == "skew"
np.testing.assert_allclose(pair.distance, 1.0)
np.testing.assert_allclose(
    pair.separation_vector, [0, 0, 1]
)
np.testing.assert_allclose(projection.foot, [1, 3, 2])
np.testing.assert_allclose(projection.distance, 3.0)
print(pair.relation, pair.distance)
print(projection.foot, projection.distance)

1. Uzay ilişkileri ve izdüşüm için yinelenebilir denetim
skew 1.0
[1. 3. 2.] 3.0

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Dört doğru sınıfının otomatik sınanması ---
import numpy as np
from agbook import line_line_diagnostics_3d

opts = dict(
    angular_tolerance=1e-12,
    absolute_tolerance=1e-10,
    relative_tolerance=0.0,
    reference_scale=1.0,
)
cases = [
    ([0,0,0], [1,0,0], [2,0,0], [-3,0,0], "coincident"),
    ([0,0,0], [1,0,0], [0,2,0], [-3,0,0], "parallel"),
    ([0,0,0], [1,0,0], [1,-1,0], [0,1,0], "intersecting"),
    ([0,0,0], [1,0,0], [0,1,1], [0,1,0], "skew"),
]
for p, u, q, v, expected in cases:
    result = line_line_diagnostics_3d(p, u, q, v, **opts)
    assert result.relation == expected
    np.testing.assert_allclose(result.first_line_residual, 0, atol=1e-12)
    np.testing.assert_allclose(result.second_line_residual, 0, atol=1e-12)
    np.testing.assert_allclose(result.first_orthogonality_residual, 0, atol=1e-12)
    np.testing.assert_allclose(result.second_orthogonality_residual, 0, atol=1e-12)

1. Dört doğru sınıfının otomatik sınanması

cd analitik-geometri-python-v1.0
python3 -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"
python labs/bolum_19_uzayda_kesisim_aci_uzaklik.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_19_uzayda_kesisim_aci_uzaklik.py
.venv\Scripts\python.exe -m pytest