# How Lighthouses Work 4, Exercise 3: where a stepped lens starts to save glass
# Mass of a solid plano-convex lens against a flat stepped one of the same aperture.
import math

NM = 1852.0

def transmissivity(visibility_nm):
    return 0.05 ** (1 / visibility_nm)

def luminous_range_nm(intensity_cd, visibility_nm, threshold_lx=2e-7):
    t = transmissivity(visibility_nm)
    lo, hi = 0.01, 200.0
    for _ in range(100):
        mid = (lo + hi) / 2
        e = intensity_cd * t ** mid / (mid * NM) ** 2
        if e > threshold_lx:
            lo = mid
        else:
            hi = mid
    return (lo + hi) / 2

def geographic_range_nm(light_m, eye_m):
    return 3.86 * (math.sqrt(light_m) + math.sqrt(eye_m)) / 1.852

# Exercise 3: where does a stepped lens start to save glass?
N_GLASS = 1.5          # assumed
DENSITY = 2500.0       # assumed
STEP_THICK_M = 0.02    # assumed
EDGE_M = 0.005         # assumed edge thickness of the solid lens

def masses(radius, focal):
    R = (N_GLASS - 1) * focal
    sag = R - math.sqrt(R ** 2 - radius ** 2)
    solid = math.pi * sag ** 2 * (3 * R - sag) / 3 + math.pi * radius ** 2 * EDGE_M
    stepped = math.pi * radius ** 2 * STEP_THICK_M
    return solid * DENSITY, stepped * DENSITY

for focal in (0.5, 0.92, 1.33):
    row = []
    for dia in (0.2, 0.4):
        s, t = masses(dia / 2, focal)
        row.append((dia, round(s, 1), round(t, 1)))
    print("focal", focal, "m:", row)
# bisection: aperture radius at which the two masses are equal (f = 0.92 m)
lo, hi = 0.05, 0.45
for _ in range(60):
    mid = (lo + hi) / 2
    s, t = masses(mid, 0.92)
    if s < t:
        lo = mid
    else:
        hi = mid
print("Break-even diameter:", round(2 * lo, 3), "m")

# Output:
# focal 0.5 m: [(0.2, 1.2, 1.6), (0.4, 18.6, 6.3)]
# focal 0.92 m: [(0.2, 0.8, 1.6), (0.4, 8.9, 6.3)]
# focal 1.33 m: [(0.2, 0.7, 1.6), (0.4, 6.4, 6.3)]
# Break-even diameter: 0.325 m
#
# On the assumptions in the code (glass index 1.5, density 2,500 kg per cubic metre, a stepped
# panel 20 mm thick, a 5 mm edge on the solid lens), the solid lens is lighter at 0.2 m diameter
# (about 0.7 to 1.2 kg against 1.6 kg) and heavier at 0.4 m (6.4 to 18.6 kg against 6.3 kg).
# Bisection puts the break-even at a diameter of about 0.325 m for the 920 mm focal length. The
# shorter the focal length, the more the solid lens bulges and the sooner it loses: at 0.4 m it
# weighs 18.6 kg for a focal length of 0.5 m and 6.4 kg for 1.33 m, while the stepped panel stays
# at 6.3 kg. The model keeps the stepped panel at a flat 20 mm, which a real panel with prisms
# will not be; the conclusion to trust is the trend, not the kilograms.