# How Lighthouses Work 2, Exercise 2: candles and a fire
# 1) Luminous range of 24 candles (about 1 cd each) at visibilities 5, 10, 20 nm.
# 2) Number of candles giving a 10 nm nominal range.
# 3) Ratio of Winstanley's (60 candles) to Smeaton's (24) nominal range.
# 4) A coal fire of an ASSUMED 5000 cd on a 103 m tower, eye 5 m up: which limit wins?
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

# 1. 24 candles (about 1 cd each) at three visibilities
for v in (5, 10, 20):
    print(v, round(luminous_range_nm(24, v), 2))

# 2. candles needed for a 10 nm nominal range (bisection on the count)
lo, hi = 1.0, 1e6
for _ in range(100):
    mid = (lo + hi) / 2
    if luminous_range_nm(mid, 10) < 10:
        lo = mid
    else:
        hi = mid
print(round(hi))

# 3. Winstanley 60 candles against Smeaton 24: ratio of nominal ranges
print(round(luminous_range_nm(60, 10) / luminous_range_nm(24, 10), 2))

# 4. A fire on a 103 m tower, eye 5 m up, with an ASSUMED 5000 cd
lum = luminous_range_nm(5000, 10)
geo = geographic_range_nm(103, 5)
print(round(lum, 1), round(geo, 1), "limit:", "light" if lum < geo else "horizon")

# Output when run:
# 5 2.66
# 10 3.5
# 20 4.29
# 1372
# 1.33
# 12.7 25.8 limit: light
#
# Lines 1-3: 24 candles reach only 2.66, 3.5 and 4.29 nm at visibilities of 5, 10 and
# 20 nm; the 3.5 nm at 10 nm is the nominal range. Line 4: about 1372 candles would be
# needed for a 10 nm nominal range (no real lantern holds that many, and the 1 cd per
# candle is itself only "roughly" right). Line 5: 60 candles reach 1.33 times as far
# as 24, not 2.5 times, the inverse square law again. Line 6: the assumed 5000 cd fire
# reaches 12.7 nm in standard weather, while the 103 m height would allow 25.8 nm, so
# the brightness limits it, not the horizon. For the candle towers, too, the light was
# the limit. The 5000 cd value is an assumption, not a measurement of the Pharos.