Küre ve Küre Sistemleri

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

Laboratuvar kodu

"""Bölüm 20: küre ve küre sistemleri laboratuvarı.

Uygulama, dört bilinen merkeze olan tam ve gürültüsüz uzaklıklardan sentetik
bir uzay noktasını bulur. İlk üç küre ayna simetrili iki aday bırakır; dördüncü
küre rankı tamamlayıp tek ortak noktayı seçer. Bu dosya GNSS, fiziksel ölçüm
belirsizliği veya kabul kararı modeli değildir.
"""

from __future__ import annotations

import numpy as np

from agbook import (
    canonical_array_sha256,
    radical_plane_diagnostics,
    sphere_plane_section_diagnostics,
    sphere_point_power,
    sphere_sphere_intersection_diagnostics,
    sphere_system_diagnostics,
    tangent_plane_at_sphere_point,
)


def main() -> None:
    length_unit = "m"
    rank_tolerance = 1.0e-12
    absolute_tolerance_m = 1.0e-11
    relative_tolerance = 1.0e-12
    reference_scale_m = 20.0
    system_options = {
        "rank_tolerance": rank_tolerance,
        "absolute_tolerance": absolute_tolerance_m,
        "relative_tolerance": relative_tolerance,
        "reference_scale": reference_scale_m,
    }
    geometry_options = {
        "absolute_tolerance": absolute_tolerance_m,
        "relative_tolerance": relative_tolerance,
        "reference_scale": reference_scale_m,
    }

    target = np.array([4.0, 6.0, 12.0])
    centers = np.array(
        [
            [0.0, 0.0, 0.0],
            [12.0, 0.0, 0.0],
            [0.0, 16.0, 0.0],
            [0.0, 0.0, 20.0],
        ]
    )
    squared_ranges = np.array([196.0, 244.0, 260.0, 116.0])
    ranges = np.sqrt(squared_ranges)

    three_sphere = sphere_system_diagnostics(
        centers[:3],
        ranges[:3],
        **system_options,
    )
    four_sphere = sphere_system_diagnostics(
        centers,
        ranges,
        **system_options,
    )
    first_pair = sphere_sphere_intersection_diagnostics(
        centers[0],
        ranges[0],
        centers[1],
        ranges[1],
        **geometry_options,
    )
    first_radical_plane = radical_plane_diagnostics(
        centers[0],
        ranges[0],
        centers[1],
        ranges[1],
        **geometry_options,
    )
    horizontal_section = sphere_plane_section_diagnostics(
        centers[0],
        ranges[0],
        [0.0, 0.0, 1.0, -12.0],
        **geometry_options,
    )
    tangent_plane = tangent_plane_at_sphere_point(
        centers[0],
        ranges[0],
        target,
        **geometry_options,
    )

    rotation = np.array(
        [
            [0.0, -1.0, 0.0],
            [1.0, 0.0, 0.0],
            [0.0, 0.0, 1.0],
        ]
    )
    translation = np.array([30.0, -10.0, 7.0])
    moved_centers = centers @ rotation.T + translation
    moved_target = rotation @ target + translation
    moved = sphere_system_diagnostics(
        moved_centers,
        ranges,
        rank_tolerance=rank_tolerance,
        absolute_tolerance=absolute_tolerance_m,
        relative_tolerance=relative_tolerance,
        reference_scale=60.0,
    )

    centimetre_scale = 100.0
    centimetre = sphere_system_diagnostics(
        centimetre_scale * centers,
        centimetre_scale * ranges,
        rank_tolerance=rank_tolerance,
        absolute_tolerance=centimetre_scale * absolute_tolerance_m,
        relative_tolerance=relative_tolerance,
        reference_scale=centimetre_scale * reference_scale_m,
    )

    target_powers = np.array(
        [
            sphere_point_power(centers[index], ranges[index], target)
            for index in range(centers.shape[0])
        ]
    )

    assert three_sphere.relation == "line"
    assert three_sphere.rank == 2
    assert three_sphere.common_locus_type == "two_points"
    np.testing.assert_allclose(three_sphere.anchor, [4.0, 6.0, 0.0], atol=2.0e-13)
    np.testing.assert_allclose(
        three_sphere.common_points,
        [[4.0, 6.0, -12.0], [4.0, 6.0, 12.0]],
        atol=2.0e-12,
    )
    assert four_sphere.relation == "point"
    assert four_sphere.rank == 3
    assert four_sphere.common_locus_type == "point"
    np.testing.assert_allclose(four_sphere.anchor, target, atol=2.0e-13)
    np.testing.assert_allclose(target_powers, 0.0, atol=1.0e-13)
    np.testing.assert_allclose(four_sphere.power_values, 0.0, atol=2.0e-12)
    assert first_pair.relation == "intersection_circle"
    assert first_radical_plane.relation == "plane"
    np.testing.assert_allclose(
        first_pair.circle_plane,
        first_radical_plane.plane_coefficients,
        atol=2.0e-15,
    )
    assert horizontal_section.relation == "circle"
    np.testing.assert_allclose(horizontal_section.section_center, [0.0, 0.0, 12.0])
    np.testing.assert_allclose(horizontal_section.section_radius, np.sqrt(52.0))
    np.testing.assert_allclose(tangent_plane[:3], target / ranges[0], atol=2.0e-15)
    np.testing.assert_allclose(
        tangent_plane[:3] @ target + tangent_plane[3],
        0.0,
        atol=3.0e-15,
    )
    np.testing.assert_allclose(moved.anchor, moved_target, atol=3.0e-13)
    np.testing.assert_allclose(
        centimetre.anchor,
        centimetre_scale * target,
        atol=3.0e-11,
    )
    assert centimetre.relation == four_sphere.relation
    np.testing.assert_allclose(
        centimetre.condition_number,
        four_sphere.condition_number,
        rtol=2.0e-15,
    )

    signature_values = np.concatenate(
        [
            target,
            centers.ravel(),
            squared_ranges,
            ranges,
            three_sphere.anchor,
            three_sphere.direction_basis.ravel(),
            three_sphere.common_points.ravel(),
            four_sphere.anchor,
            four_sphere.singular_values,
            four_sphere.equation_residuals,
            four_sphere.power_values,
            first_pair.circle_center,
            np.array([first_pair.circle_radius]),
            first_radical_plane.plane_coefficients,
            horizontal_section.section_center,
            np.array([horizontal_section.section_radius]),
            tangent_plane,
            moved.anchor,
            centimetre.anchor,
            np.array(
                [
                    four_sphere.condition_number,
                    four_sphere.length_threshold,
                    four_sphere.power_threshold,
                    rank_tolerance,
                    absolute_tolerance_m,
                    relative_tolerance,
                    reference_scale_m,
                ]
            ),
        ]
    )

    print("uzunluk_birimi:", length_unit)
    print("model: sentetik_gurultusuz_mesafe_kureleriyle_konumlama")
    print("gercek_gnss_modeli_mi: False")
    print("olculmus_veri_mi: False")
    print("olcum_belirsizligi_hesabi_mi: False")
    print("hedef_nokta_m:", target)
    print("uzakliklar_m:", ranges)
    print("ilk_uc_kure_ranki:", three_sphere.rank)
    print("ilk_uc_kure_esit_guc_yeri:", three_sphere.relation)
    print("ilk_uc_kure_ortak_yeri:", three_sphere.common_locus_type)
    print("ayna_adaylari_m:", three_sphere.common_points)
    print("dort_kure_ranki:", four_sphere.rank)
    print("dort_kure_esit_guc_yeri:", four_sphere.relation)
    print("dort_kure_ortak_yeri:", four_sphere.common_locus_type)
    print("cozum_m:", four_sphere.anchor)
    print("guc_degerleri_m2:", four_sphere.power_values)
    print("guc_yayilimi_m2:", f"{four_sphere.power_spread:.12e}")
    print("kosul_sayisi:", f"{four_sphere.condition_number:.12f}")
    print("birinci_ikili_kesisim:", first_pair.relation)
    print("birinci_radikal_duzlem:", first_radical_plane.plane_coefficients)
    print("z_12_kesit_yaricapi_m:", f"{horizontal_section.section_radius:.12f}")
    print("hedefte_teget_duzlem:", tangent_plane)
    print("rijit_harekette_cozum_korundu_mu:", np.allclose(moved.anchor, moved_target))
    print(
        "metreden_santimetreye_cozum_orani:",
        centimetre.anchor / four_sphere.anchor,
    )
    print("rank_esigi:", f"{rank_tolerance:.12e}")
    print("mutlak_uzunluk_esigi_m:", f"{absolute_tolerance_m:.12e}")
    print("bagil_uzunluk_esigi:", f"{relative_tolerance:.12e}")
    print("referans_uzunlugu_m:", f"{reference_scale_m:.12f}")
    print("bilimsel_imza:", canonical_array_sha256(signature_values))


if __name__ == "__main__":
    main()

uzunluk_birimi: m
model: sentetik_gurultusuz_mesafe_kureleriyle_konumlama
gercek_gnss_modeli_mi: False
olculmus_veri_mi: False
olcum_belirsizligi_hesabi_mi: False
hedef_nokta_m: [ 4.  6. 12.]
uzakliklar_m: [14.         15.62049935 16.1245155  10.77032961]
ilk_uc_kure_ranki: 2
ilk_uc_kure_esit_guc_yeri: line
ilk_uc_kure_ortak_yeri: two_points
ayna_adaylari_m: [[  4.   6. -12.]
 [  4.   6.  12.]]
dort_kure_ranki: 3
dort_kure_esit_guc_yeri: point
dort_kure_ortak_yeri: point
cozum_m: [ 4.  6. 12.]
guc_degerleri_m2: [2.84217094e-14 2.84217094e-14 5.68434189e-14 4.26325641e-14]
guc_yayilimi_m2: 2.842170943040e-14
kosul_sayisi: 1.666666666667
birinci_ikili_kesisim: intersection_circle
birinci_radikal_duzlem: [ 1.  0.  0. -4.]
z_12_kesit_yaricapi_m: 7.211102550928
hedefte_teget_duzlem: [  0.28571429   0.42857143   0.85714286 -14.        ]
rijit_harekette_cozum_korundu_mu: True
metreden_santimetreye_cozum_orani: [100. 100. 100.]
rank_esigi: 1.000000000000e-12
mutlak_uzunluk_esigi_m: 1.000000000000e-11
bagil_uzunluk_esigi: 1.000000000000e-12
referans_uzunlugu_m: 20.000000000000
bilimsel_imza: 2f366b8c5b303a0ee1bb8f1170dfa3558292dec5690fc7e63c325b7254c85aa4

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1.  ---
import numpy as np
from agbook import sphere_system_diagnostics

centers = np.array([
    [0.0, 0.0, 0.0],
    [12.0, 0.0, 0.0],
    [0.0, 16.0, 0.0],
    [0.0, 0.0, 20.0],
])
radii = np.sqrt([196.0, 244.0, 260.0, 116.0])
options = dict(
    rank_tolerance=1e-12,
    absolute_tolerance=1e-11,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)

three = sphere_system_diagnostics(centers[:3], radii[:3], **options)
four = sphere_system_diagnostics(centers, radii, **options)

assert three.relation == "line"
assert three.common_locus_type == "two_points"
np.testing.assert_allclose(
    three.common_points,
    [[4.0, 6.0, -12.0], [4.0, 6.0, 12.0]],
)
assert four.relation == "point"
assert four.common_locus_type == "point"
np.testing.assert_allclose(four.anchor, [4.0, 6.0, 12.0])
np.testing.assert_allclose(four.power_values, 0.0, atol=1e-12)

1. 

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Genel küre sınıflarının ve ölçek değişmezliğinin sınanması ---
import numpy as np
from agbook import general_sphere_diagnostics

cases = [
    ([1, 0, 0, 0, -4], "sphere"),
    ([1, 0, 0, 0,  0], "point"),
    ([1, 0, 0, 0,  1], "empty"),
]
for coefficients, expected in cases:
    base = general_sphere_diagnostics(coefficients)
    scaled = general_sphere_diagnostics(
        -7.0 * np.asarray(coefficients, dtype=float)
    )
    assert base.locus_type == expected
    assert scaled.locus_type == expected
    np.testing.assert_allclose(
        scaled.normalized_coefficients,
        base.normalized_coefficients,
    )

# --- 2. Doğru ve düzlem kesitlerinin artıklarla sınanması ---
import numpy as np
from agbook import (
    line_sphere_intersections_3d,
    sphere_plane_section_diagnostics,
)

opts = dict(
    absolute_tolerance=1e-12,
    relative_tolerance=1e-12,
    reference_scale=10.0,
)
line_cases = [
    (5.0, [0, 3, 0], "secant"),
    (5.0, [0, 5, 0], "tangent"),
    (5.0, [0, 6, 0], "disjoint"),
    (0.0, [0, 0, 0], "point"),
]
for radius, anchor, expected in line_cases:
    out = line_sphere_intersections_3d(
        [0, 0, 0], radius, anchor, [1, 0, 0], **opts
    )
    assert out.relation == expected
    np.testing.assert_allclose(out.line_residuals, 0, atol=1e-12)
    if expected in {"secant", "point"}:
        np.testing.assert_allclose(out.radial_residuals, 0, atol=1e-12)

plane_cases = [
    (5.0, [0, 0, 1,  0], "circle"),
    (5.0, [0, 0, 1, -5], "tangent"),
    (5.0, [0, 0, 1, -6], "disjoint"),
    (0.0, [0, 0, 1,  0], "point"),
]
for radius, plane, expected in plane_cases:
    out = sphere_plane_section_diagnostics(
        [0, 0, 0], radius, plane, **opts
    )
    assert out.relation == expected
    np.testing.assert_allclose(
        out.section_center_plane_residual, 0, atol=1e-12
    )

# --- 3. İki kürenin yedi konumunun sınanması ---
from agbook import sphere_sphere_intersection_diagnostics

opts = dict(
    absolute_tolerance=1e-12,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)
cases = [
    ([0,0,0], 5, [0,0,0], 5, "coincident"),
    ([0,0,0], 5, [0,0,0], 3, "concentric_disjoint"),
    ([0,0,0], 5, [11,0,0], 5, "separate"),
    ([0,0,0], 5, [10,0,0], 5, "externally_tangent"),
    ([0,0,0], 5, [6,0,0], 5, "intersection_circle"),
    ([0,0,0], 5, [2,0,0], 3, "internally_tangent"),
    ([0,0,0], 5, [1,0,0], 3, "contained"),
]
for c1, r1, c2, r2, expected in cases:
    forward = sphere_sphere_intersection_diagnostics(
        c1, r1, c2, r2, **opts
    )
    reverse = sphere_sphere_intersection_diagnostics(
        c2, r2, c1, r1, **opts
    )
    assert forward.relation == reverse.relation == expected

# --- 4. Permütasyon, rijit hareket ve birim dönüşümü ---
import numpy as np
from agbook import sphere_system_diagnostics

C = np.array([
    [0., 0., 0.], [12., 0., 0.],
    [0., 16., 0.], [0., 0., 20.],
])
r = np.sqrt([196., 244., 260., 116.])
opts = dict(
    rank_tolerance=1e-12,
    absolute_tolerance=1e-11,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)
base = sphere_system_diagnostics(C, r, **opts)
order = [2, 0, 3, 1]
permuted = sphere_system_diagnostics(C[order], r[order], **opts)
np.testing.assert_allclose(permuted.anchor, base.anchor)

Q = np.array([[0.,-1.,0.], [1.,0.,0.], [0.,0.,1.]])
t = np.array([30., -10., 7.])
moved = sphere_system_diagnostics(
    C @ Q.T + t, r,
    rank_tolerance=1e-12,
    absolute_tolerance=1e-11,
    relative_tolerance=1e-12,
    reference_scale=60.0,
)
np.testing.assert_allclose(moved.anchor, Q @ base.anchor + t)

c = 100.0
scaled = sphere_system_diagnostics(
    c*C, c*r,
    rank_tolerance=1e-12,
    absolute_tolerance=c*1e-11,
    relative_tolerance=1e-12,
    reference_scale=c*20.0,
)
np.testing.assert_allclose(scaled.anchor, c*base.anchor)
assert scaled.rank == base.rank
np.testing.assert_allclose(
    scaled.condition_number, base.condition_number
)

1. Genel küre sınıflarının ve ölçek değişmezliğinin sınanması

2. Doğru ve düzlem kesitlerinin artıklarla sınanması

3. İki kürenin yedi konumunun sınanması

4. Permütasyon, rijit hareket ve birim dönüşümü

cd analitik-geometri-python-v1.0
python3 -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"
python labs/bolum_20_kure_sistemleri.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_20_kure_sistemleri.py
.venv\Scripts\python.exe -m pytest