Uzayda Vektörler

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

Laboratuvar kodu

"""Bölüm 17: uzayda vektörler ve yön tanısı laboratuvarı.

Uygulama, döndürülmüş ve ötelenmiş sentetik bir CAD tetrahedronudur.
Ölçülmüş veri, ağ kalitesi, üretim toleransı veya kabul kararı içermez.
"""

from __future__ import annotations

import numpy as np

from agbook import (
    canonical_array_sha256,
    scalar_triple_product,
    spatial_orientation_diagnostics,
    tetrahedron_volume,
    triangle_area_3d,
    unit_normal_3d,
)


def main() -> None:
    length_unit = "mm"

    # Sütunları sağ yönlü, ortonormal bir yerel CAD çerçevesidir.
    rotation = np.array(
        [
            [3.0 / 5.0, -12.0 / 25.0, 16.0 / 25.0],
            [4.0 / 5.0, 9.0 / 25.0, -12.0 / 25.0],
            [0.0, 20.0 / 25.0, 15.0 / 25.0],
        ]
    )
    origin_mm = np.array([120.0, 80.0, 35.0])
    edge_lengths_mm = np.array([60.0, 50.0, 75.0])
    vertices_mm = np.vstack(
        [
            origin_mm,
            origin_mm + edge_lengths_mm[0] * rotation[:, 0],
            origin_mm + edge_lengths_mm[1] * rotation[:, 1],
            origin_mm + edge_lengths_mm[2] * rotation[:, 2],
        ]
    )
    point_a, point_b, point_c, point_d = vertices_mm
    edge_ab = point_b - point_a
    edge_ac = point_c - point_a
    edge_ad = point_d - point_a

    face_normal = unit_normal_3d(edge_ab, edge_ac)
    face_area_mm2 = triangle_area_3d(point_a, point_b, point_c)
    signed_box_volume_mm3 = scalar_triple_product(edge_ab, edge_ac, edge_ad)
    tetrahedron_volume_mm3 = tetrahedron_volume(
        point_a,
        point_b,
        point_c,
        point_d,
    )
    diagnosis = spatial_orientation_diagnostics(
        edge_ab,
        edge_ac,
        edge_ad,
        relative_tolerance=1e-12,
    )
    swapped_diagnosis = spatial_orientation_diagnostics(
        edge_ac,
        edge_ab,
        edge_ad,
        relative_tolerance=1e-12,
    )

    vertices_m = vertices_mm / 1000.0
    point_a_m, point_b_m, point_c_m, point_d_m = vertices_m
    face_area_m2 = triangle_area_3d(point_a_m, point_b_m, point_c_m)
    tetrahedron_volume_m3 = tetrahedron_volume(
        point_a_m,
        point_b_m,
        point_c_m,
        point_d_m,
    )

    # İlk iki kenarın düzlemine çok yakın, fakat tam düzlemde olmayan aday.
    near_edge = (
        15.0 * rotation[:, 0]
        + 15.0 * rotation[:, 1]
        + 1e-10 * rotation[:, 2]
    )
    near_diagnosis = spatial_orientation_diagnostics(
        edge_ab,
        edge_ac,
        near_edge,
        relative_tolerance=1e-10,
    )

    np.testing.assert_allclose(rotation.T @ rotation, np.eye(3), atol=2e-15)
    np.testing.assert_allclose(np.linalg.det(rotation), 1.0, atol=2e-15)
    np.testing.assert_allclose(
        vertices_mm,
        np.array(
            [
                [120.0, 80.0, 35.0],
                [156.0, 128.0, 35.0],
                [96.0, 98.0, 75.0],
                [168.0, 44.0, 80.0],
            ]
        ),
        atol=2e-14,
    )
    np.testing.assert_allclose(face_normal, rotation[:, 2], atol=2e-15)
    np.testing.assert_allclose(face_area_mm2, 1500.0, atol=2e-12)
    np.testing.assert_allclose(signed_box_volume_mm3, 225000.0, atol=2e-10)
    np.testing.assert_allclose(tetrahedron_volume_mm3, 37500.0, atol=2e-11)
    np.testing.assert_allclose(face_area_m2, face_area_mm2 * 1e-6, atol=2e-18)
    np.testing.assert_allclose(
        tetrahedron_volume_m3,
        tetrahedron_volume_mm3 * 1e-9,
        atol=2e-18,
    )
    assert diagnosis.classification == "right_handed"
    assert not diagnosis.numerically_ambiguous
    assert swapped_diagnosis.classification == "left_handed"
    assert near_diagnosis.classification == "uncertain"
    assert near_diagnosis.numerically_ambiguous

    signature_values = np.concatenate(
        [
            rotation.ravel(),
            origin_mm,
            edge_lengths_mm,
            vertices_mm.ravel(),
            edge_ab,
            edge_ac,
            edge_ad,
            face_normal,
            np.array(
                [
                    face_area_mm2,
                    signed_box_volume_mm3,
                    tetrahedron_volume_mm3,
                    diagnosis.normalized_triple_product,
                    swapped_diagnosis.normalized_triple_product,
                ]
            ),
            vertices_m.ravel(),
            np.array([face_area_m2, tetrahedron_volume_m3]),
            near_edge,
            np.array(
                [
                    near_diagnosis.scalar_triple_product,
                    near_diagnosis.normalized_triple_product,
                    near_diagnosis.relative_tolerance,
                ]
            ),
        ]
    )

    print("uzunluk_birimi:", length_unit)
    print("model: sentetik_dondurulmus_cad_tetrahedronu")
    print("olculmus_veri_mi: False")
    print("ag_kalitesi_hesabi_mi: False")
    print("uretim_kabul_hesabi_mi: False")
    print("yerel_cerceve_ortogonal_mi: True")
    print("yerel_cerceve_determinanti:", f"{np.linalg.det(rotation):.12f}")
    print("kose_noktalari_mm:")
    print(vertices_mm)
    print("ABC_birim_normali:", face_normal)
    print("ABC_ucgen_alani_mm2:", f"{face_area_mm2:.12f}")
    print("yonlu_paralelyuz_hacmi_mm3:", f"{signed_box_volume_mm3:.12f}")
    print("tetrahedron_hacmi_mm3:", f"{tetrahedron_volume_mm3:.12f}")
    print("yon_sinifi:", diagnosis.classification)
    print("boyutsuz_normalize_karma_carpim:", f"{diagnosis.normalized_triple_product:.12f}")
    print("ilk_iki_kenar_degistirilince_yon:", swapped_diagnosis.classification)
    print(
        "metre_milimetre_alan_hacim_olcegi_korundu_mu:",
        bool(
            np.isclose(face_area_m2, face_area_mm2 * 1e-6)
            and np.isclose(tetrahedron_volume_m3, tetrahedron_volume_mm3 * 1e-9)
        ),
    )
    print("duzleme_yakin_aday_sinifi:", near_diagnosis.classification)
    print("duzleme_yakin_aday_belirsiz_mi:", near_diagnosis.numerically_ambiguous)
    print(
        "duzleme_yakin_aday_normalize_karma_carpim:",
        f"{near_diagnosis.normalized_triple_product:.12e}",
    )
    print("bilimsel_imza:", canonical_array_sha256(signature_values))


if __name__ == "__main__":
    main()

uzunluk_birimi: mm
model: sentetik_dondurulmus_cad_tetrahedronu
olculmus_veri_mi: False
ag_kalitesi_hesabi_mi: False
uretim_kabul_hesabi_mi: False
yerel_cerceve_ortogonal_mi: True
yerel_cerceve_determinanti: 1.000000000000
kose_noktalari_mm:
[[120.  80.  35.]
 [156. 128.  35.]
 [ 96.  98.  75.]
 [168.  44.  80.]]
ABC_birim_normali: [ 0.64 -0.48  0.6 ]
ABC_ucgen_alani_mm2: 1500.000000000000
yonlu_paralelyuz_hacmi_mm3: 225000.000000000000
tetrahedron_hacmi_mm3: 37500.000000000000
yon_sinifi: right_handed
boyutsuz_normalize_karma_carpim: 1.000000000000
ilk_iki_kenar_degistirilince_yon: left_handed
metre_milimetre_alan_hacim_olcegi_korundu_mu: True
duzleme_yakin_aday_sinifi: uncertain
duzleme_yakin_aday_belirsiz_mi: True
duzleme_yakin_aday_normalize_karma_carpim: 4.714117984861e-12
bilimsel_imza: bc87febcaf544499f48f465892f62361bfe85ae42d32454b019b85910e901cac

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Uzay üçgeni ve tetrahedron tanısı ---
import numpy as np
from agbook import (
    spatial_orientation_diagnostics,
    tetrahedron_volume,
    triangle_area_3d,
    unit_normal_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])
D = np.array([168.0, 44.0, 80.0])

u, v, w = B - A, C - A, D - A
normal = unit_normal_3d(u, v)
area = triangle_area_3d(A, B, C)
volume = tetrahedron_volume(A, B, C, D)
diagnosis = spatial_orientation_diagnostics(
    u, v, w, relative_tolerance=1e-12
)

print(normal)
print(area, volume)
print(diagnosis.scalar_triple_product)
print(diagnosis.normalized_triple_product)
print(diagnosis.classification)

1. Uzay üçgeni ve tetrahedron tanısı
[ 0.64 -0.48  0.6 ]
1500.0 37500.0
225000.0
1.0
right_handed

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Vektörel çarpım karşılaştırması ---
import numpy as np
from agbook import cross_product_3d

pairs = [
    ([1, 0, 0], [0, 1, 0]),
    ([2, -1, 3], [4, 5, -2]),
    ([1, 2, 0], [-3, 1, 4]),
    ([0, 0, 5], [2, -7, 0]),
    ([1, 2, 3], [2, 4, 6]),
]
for u_raw, v_raw in pairs:
    u = np.asarray(u_raw, dtype=float)
    v = np.asarray(v_raw, dtype=float)
    result = cross_product_3d(u, v)
    np.testing.assert_allclose(result, np.cross(u, v))
    np.testing.assert_allclose(result @ u, 0.0, atol=1e-14)
    np.testing.assert_allclose(result @ v, 0.0, atol=1e-14)

# --- 2. Öteleme değişmezliği ---
import numpy as np
from agbook import triangle_area_3d, unit_normal_3d

points = np.array([[1., 0., 2.], [4., 1., 2.], [2., 5., 4.]])
translations = np.array([
    [0., 0., 0.], [10., -4., 7.], [-3., 8., 2.],
    [1e3, -2e3, 3e3], [-0.25, 0.5, -0.75],
])
area0 = triangle_area_3d(*points)
normal0 = unit_normal_3d(points[1] - points[0], points[2] - points[0])
for shift in translations:
    moved = points + shift
    np.testing.assert_allclose(triangle_area_3d(*moved), area0)
    np.testing.assert_allclose(
        unit_normal_3d(moved[1] - moved[0], moved[2] - moved[0]),
        normal0,
    )

# --- 3. Dönme ve yansıma yasası ---
import numpy as np
from agbook import cross_product_3d

R = np.array([
    [3/5, -12/25, 16/25],
    [4/5,   9/25, -12/25],
    [0,    20/25, 15/25],
], dtype=float)
Q = np.diag([-1., 1., 1.])
pairs = [
    (np.array([1., 2., 3.]), np.array([2., -1., 4.])),
    (np.array([3., 0., -2.]), np.array([1., 5., 1.])),
    (np.array([0., 1., 0.]), np.array([0., 0., 1.])),
]
for transform in (R, Q):
    sign = np.linalg.det(transform)
    for u, v in pairs:
        np.testing.assert_allclose(
            cross_product_3d(transform @ u, transform @ v),
            sign * transform @ cross_product_3d(u, v),
            atol=2e-14,
        )
print(np.linalg.det(R), np.linalg.det(Q))  # 1.0, -1.0

# --- 4. Yakın eşdüzlemlilik tablosu ---
from agbook import spatial_orientation_diagnostics

deltas = [0.0, 1e-16, 1e-14, 1e-10, 1e-6]
tolerances = [0.0, 1e-12, 1e-8]
for delta in deltas:
    for tolerance in tolerances:
        result = spatial_orientation_diagnostics(
            [1., 0., 0.], [0., 1., 0.], [1., 1., delta],
            relative_tolerance=tolerance,
        )
        print(delta, tolerance,
              result.normalized_triple_product,
              result.classification)

1. Vektörel çarpım karşılaştırması

2. Öteleme değişmezliği

3. Dönme ve yansıma yasası
1.0 -1.0

4. Yakın eşdüzlemlilik tablosu
0.0 0.0 0.0 coplanar
0.0 1e-12 0.0 coplanar
0.0 1e-08 0.0 coplanar
1e-16 0.0 7.071067811865474e-17 right_handed
1e-16 1e-12 7.071067811865474e-17 uncertain
1e-16 1e-08 7.071067811865474e-17 uncertain
1e-14 0.0 7.071067811865475e-15 right_handed
1e-14 1e-12 7.071067811865475e-15 uncertain
1e-14 1e-08 7.071067811865475e-15 uncertain
1e-10 0.0 7.071067811865475e-11 right_handed
1e-10 1e-12 7.071067811865475e-11 right_handed
1e-10 1e-08 7.071067811865475e-11 uncertain
1e-06 0.0 7.071067811863707e-07 right_handed
1e-06 1e-12 7.071067811863707e-07 right_handed
1e-06 1e-08 7.071067811863707e-07 right_handed

cd analitik-geometri-python-v1.0
python3 -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"
python labs/bolum_17_uzayda_vektorler.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_17_uzayda_vektorler.py
.venv\Scripts\python.exe -m pytest