Kuvvet, Radikal Eksen ve Çember 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 11: nokta kuvveti, radikal eksen ve güç diyagramı laboratuvarı.

Model, üç sentetik dairesel hizmet bölgesini iki boyutlu Öklid düzleminde
karşılaştırır. Sinyal yayılımı, nüfus, maliyet, engel, ölçüm gürültüsü veya
gerçek hizmet kalitesi içermez.
"""

from __future__ import annotations

import numpy as np

from agbook import (
    canonical_array_sha256,
    circle_pencil_member,
    point_power,
    radical_axis_diagnostics,
    radical_center_diagnostics,
    rotation_matrix_2d,
)


TOLERANCES = {
    "absolute_tolerance": 1e-10,
    "relative_tolerance": 1e-12,
    "reference_scale": 20.0,
}

CENTER_TOLERANCES = {
    "angular_tolerance": 1e-12,
    **TOLERANCES,
}


def monic_circle(center: np.ndarray, radius: float) -> np.ndarray:
    """Merkez-yarıçap verisini ``(1,D,E,F)`` katsayılarına dönüştürür."""

    return np.array(
        [
            1.0,
            -2.0 * center[0],
            -2.0 * center[1],
            float(center @ center - radius**2),
        ]
    )


def main() -> None:
    centers = np.array([[0.0, 0.0], [8.0, 0.0], [0.0, 6.0]])  # km
    radii = np.array([5.0, 3.0, 4.0])  # km
    pairs = ((0, 1), (0, 2), (1, 2))

    axes = [
        radical_axis_diagnostics(
            centers[i], radii[i], centers[j], radii[j], **TOLERANCES
        )
        for i, j in pairs
    ]
    assert all(axis.relation == "line" for axis in axes)
    axis_matrix = np.vstack([axis.line_coefficients for axis in axes])

    radical_center = radical_center_diagnostics(
        centers, radii, **CENTER_TOLERANCES
    )
    assert radical_center.relation == "point"
    np.testing.assert_allclose(radical_center.point, [5.0, 3.75])
    np.testing.assert_allclose(radical_center.power_values, 14.0625, atol=1e-12)
    assert radical_center.max_axis_residual <= radical_center.length_threshold

    x_values = np.linspace(-4.0, 12.0, 65)
    y_values = np.linspace(-4.0, 10.0, 57)
    xx, yy = np.meshgrid(x_values, y_values)
    grid = np.column_stack([xx.ravel(), yy.ravel()])
    powers = np.sum(
        (grid[:, None, :] - centers[None, :, :]) ** 2,
        axis=2,
    ) - radii[None, :] ** 2
    labels = np.argmin(powers, axis=1)
    label_counts = np.bincount(labels, minlength=3)

    first_circle = monic_circle(centers[0], radii[0])
    second_circle = monic_circle(centers[1], radii[1])
    pencil_parameters = np.array([0.0, 0.5, 0.625, 1.0])
    pencil_members = [
        circle_pencil_member(first_circle, second_circle, parameter)
        for parameter in pencil_parameters
    ]
    assert [member.locus_type for member in pencil_members] == [
        "circle",
        "circle",
        "point",
        "circle",
    ]

    rotation = rotation_matrix_2d(np.deg2rad(23.0))
    translation = np.array([40.0, -25.0])  # km
    moved_centers = centers @ rotation.T + translation
    moved_grid = grid @ rotation.T + translation
    moved_center = radical_center_diagnostics(
        moved_centers, radii, **CENTER_TOLERANCES
    )
    expected_center = rotation @ radical_center.point + translation
    np.testing.assert_allclose(moved_center.point, expected_center, atol=2e-13)
    moved_powers = np.sum(
        (moved_grid[:, None, :] - moved_centers[None, :, :]) ** 2,
        axis=2,
    ) - radii[None, :] ** 2
    moved_labels = np.argmin(moved_powers, axis=1)
    np.testing.assert_allclose(moved_powers, powers, atol=2e-12, rtol=0.0)
    sorted_powers = np.sort(powers, axis=1)
    power_gaps = sorted_powers[:, 1] - sorted_powers[:, 0]
    stable_mask = power_gaps > 1e-10  # km^2
    assert np.array_equal(moved_labels[stable_mask], labels[stable_mask])
    boundary_count = int(np.count_nonzero(~stable_mask))

    center_powers = np.array(
        [point_power(center, radius, radical_center.point) for center, radius in zip(centers, radii)]
    )
    signature_values = np.concatenate(
        [
            centers.ravel(),
            radii,
            axis_matrix.ravel(),
            radical_center.point,
            center_powers,
            label_counts.astype(float),
            np.array([member.radius_squared for member in pencil_members]),
        ]
    )

    print("birim: km")
    print("model: sentetik_dairesel_hizmet_bolgeleri")
    print("gercek_kapsama_modeli_mi: False")
    print("merkezler_km:", centers.tolist())
    print("yaricaplar_km:", radii.tolist())
    for pair, axis in zip(pairs, axes):
        print(f"radikal_eksen_{pair}: {axis.line_coefficients}")
    print("radikal_merkez_km:", radical_center.point)
    print("ortak_kuvvet_km2:", center_powers)
    print("en_buyuk_eksen_artigi_km:", f"{radical_center.max_axis_residual:.3e}")
    print("eksen_sistem_kosul_sayisi:", f"{radical_center.condition_number:.9f}")
    print("izgara_nokta_sayisi:", grid.shape[0])
    print("guc_hucre_sayimlari:", label_counts)
    print("sinir_baglama_noktasi_sayisi:", boundary_count)
    print(
        "demet_siniflari:",
        [(parameter, member.locus_type, member.radius_squared) for parameter, member in zip(pencil_parameters, pencil_members)],
    )
    print(
        "rijit_ic_nokta_etiketleri_korundu_mu:",
        bool(np.array_equal(moved_labels[stable_mask], labels[stable_mask])),
    )
    print("bilimsel_imza:", canonical_array_sha256(signature_values))


if __name__ == "__main__":
    main()

birim: km
model: sentetik_dairesel_hizmet_bolgeleri
gercek_kapsama_modeli_mi: False
merkezler_km: [[0.0, 0.0], [8.0, 0.0], [0.0, 6.0]]
yaricaplar_km: [5.0, 3.0, 4.0]
radikal_eksen_(0, 1): [ 1.  0. -5.]
radikal_eksen_(0, 2): [ 0.    1.   -3.75]
radikal_eksen_(1, 2): [ 0.8  -0.6  -1.75]
radikal_merkez_km: [5.   3.75]
ortak_kuvvet_km2: [14.0625 14.0625 14.0625]
en_buyuk_eksen_artigi_km: 0.000e+00
eksen_sistem_kosul_sayisi: 1.000000000
izgara_nokta_sayisi: 3705
guc_hucre_sayimlari: [1184 1368 1153]
sinir_baglama_noktasi_sayisi: 74
demet_siniflari: [(np.float64(0.0), 'circle', 25.0), (np.float64(0.5), 'circle', 1.0), (np.float64(0.625), 'point', 0.0), (np.float64(1.0), 'circle', 9.0)]
rijit_ic_nokta_etiketleri_korundu_mu: True
bilimsel_imza: a9ba85811fb6d413ddcf17ae4be811515ff3187b6958d8003861e0e13399729b

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Bölüm 11: radikal eksen, merkez ve güç hücresi ---
import numpy as np
from agbook import (
    point_power,
    radical_axis_diagnostics,
    radical_center_diagnostics,
)

centers = np.array([[0., 0.], [8., 0.], [0., 6.]])
radii = np.array([5., 3., 4.])  # km
tol = dict(
    absolute_tolerance=1e-10,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)
axis12 = radical_axis_diagnostics(
    centers[0], radii[0], centers[1], radii[1], **tol
)
center = radical_center_diagnostics(
    centers, radii, angular_tolerance=1e-12, **tol
)
powers = [
    point_power(c, r, center.point)
    for c, r in zip(centers, radii)
]
print(axis12.relation, axis12.line_coefficients)
print(center.relation, center.point, powers)

1. Bölüm 11: radikal eksen, merkez ve güç hücresi
line [ 1.  0. -5.]
point [5.   3.75] [14.0625, 14.0625, 14.0625]

# Kitaptaki kod blokları; bu dosya içinde sırayla çalıştırılır.
# --- 1. Kuvvetin işaret, rijit hareket ve ölçek deneyi ---
import numpy as np
from agbook import point_power, rotation_matrix_2d

C = np.array([0., 0.])
r = 3.0
points = np.array([[1., 1.], [0., 3.], [4., 2.]])
before = np.array([point_power(C, r, p) for p in points])

Q = rotation_matrix_2d(np.deg2rad(31.0))
b = np.array([8., -5.])
Cm = Q @ C + b
moved = points @ Q.T + b
after = np.array([point_power(Cm, r, p) for p in moved])

s = 3.0
scaled = np.array([
    point_power(s * C, s * r, s * p) for p in points
])
print(before)
print(np.allclose(after, before))
print(np.allclose(scaled, s**2 * before))

# --- 2. Radikal eksenin beş temel sınıfı ---
import numpy as np
from agbook import point_power, radical_axis_diagnostics

tol = dict(
    absolute_tolerance=1e-12,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)
cases = {
    "kesisen": ((-2., 0.), 3., (2., 0.), 3.),
    "teget": ((0., 0.), 5., (8., 0.), 3.),
    "ayrik": ((0., 0.), 2., (8., 0.), 2.),
    "ozdes": ((1., 2.), 4., (1., 2.), 4.),
    "esmerkezli_farkli": ((1., 2.), 4., (1., 2.), 3.),
}
for name, (c1, r1, c2, r2) in cases.items():
    out = radical_axis_diagnostics(c1, r1, c2, r2, **tol)
    print(name, out.relation, out.line_coefficients)
    if out.line_coefficients is not None:
        a, b, c = out.line_coefficients
        p = -c * np.array([a, b])
        print(point_power(c1, r1, p)
              - point_power(c2, r2, p))

# --- 3. Radikal merkezde nokta, doğru, düzlem ve boş yer ---
import numpy as np
from agbook import radical_center_diagnostics

tol = dict(
    angular_tolerance=1e-12,
    absolute_tolerance=1e-12,
    relative_tolerance=1e-12,
    reference_scale=20.0,
)
datasets = {
    "nokta": (
        [[0., 0.], [8., 0.], [0., 6.]], [5., 3., 4.]
    ),
    "dogru": (
        [[2., 0.], [3., 0.], [4., 0.]], [2., 3., 4.]
    ),
    "duzlem": (
        [[1., 1.], [1., 1.], [1., 1.]], [2., 2., 2.]
    ),
    "bos": (
        [[0., 0.], [0., 0.], [3., 0.]], [2., 1., 2.]
    ),
}
for name, (centers, radii) in datasets.items():
    out = radical_center_diagnostics(centers, radii, **tol)
    print(name, out.relation, out.point,
          out.line_coefficients)

# --- 4. Demette ölçek değişmezliği, nokta ve boş üye ---
import numpy as np
from agbook import circle_pencil_member

tangent_1 = np.array([1., 0., 0., -25.])
tangent_2 = np.array([1., -16., 0., 55.])
point_a = circle_pencil_member(tangent_1, tangent_2, 0.625)
point_b = circle_pencil_member(
    -3. * tangent_1, 7. * tangent_2, 0.625
)
print(point_a.locus_type, point_a.center,
      point_a.radius_squared)
print(np.allclose(point_a.center, point_b.center))

apart_1 = np.array([1., 0., 0., -4.])
apart_2 = np.array([1., -16., 0., 60.])
empty = circle_pencil_member(apart_1, apart_2, 0.5)
print(empty.locus_type, empty.center,
      empty.radius_squared)

1. Kuvvetin işaret, rijit hareket ve ölçek deneyi
[-7.  0. 11.]
True
True

2. Radikal eksenin beş temel sınıfı
kesisen line [1. 0. 0.]
0.0
teget line [ 1.  0. -5.]
0.0
ayrik line [ 1.  0. -4.]
0.0
ozdes all_plane None
esmerkezli_farkli empty None

3. Radikal merkezde nokta, doğru, düzlem ve boş yer
nokta point [5.   3.75] None
dogru line None [1. 0. 0.]
duzlem all_plane None None
bos empty None None

4. Demette ölçek değişmezliği, nokta ve boş üye
point [ 5. -0.] 0.0
True
empty [ 4. -0.] -12.0

cd analitik-geometri-python-v1.0
python3 -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"
python labs/bolum_11_kuvvet_radikal_eksen.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_11_kuvvet_radikal_eksen.py
.venv\Scripts\python.exe -m pytest