# How Lighthouses Work 1, Exercise 3: which range is the limit?
# A light stands 25 m above the sea and is seen by an eye 4 m above the sea.
# For intensities of 100,000 and 2,000,000 cd and meteorological visibilities of
# 2, 10 and 20 nm, find the luminous range and decide whether the luminous or the
# geographic range limits what the observer sees. Then find the intensity that
# gives a nominal range of 15 nm. (Threshold 0.2 microlux, as in the chapter.)
import math

NM = 1852.0

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

def luminous_range_nm(i, vis_nm, threshold=2e-7):
    t = transmissivity(vis_nm)
    lo, hi = 0.01, 200.0
    for _ in range(100):
        mid = (lo + hi) / 2
        if i * t ** mid / (mid * NM) ** 2 > threshold:
            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

geo = geographic_range_nm(25, 4)
print(round(geo, 1))
for i in (100_000, 2_000_000):
    for vis in (2, 10, 20):
        lum = luminous_range_nm(i, vis)
        limit = "luminous" if lum < geo else "geographic"
        print(i, vis, round(lum, 1), round(min(lum, geo), 1), limit)

# intensity needed for a nominal range (visibility 10 nm) of 15 nm
lo, hi = 1.0, 1e9
for _ in range(100):
    mid = (lo + hi) / 2
    if luminous_range_nm(mid, 10) < 15:
        lo = mid
    else:
        hi = mid
print(round(hi))

# Output when run:
# 14.6
# 100000 2 5.6 5.6 luminous
# 100000 10 19.8 14.6 geographic
# 100000 20 32.8 14.6 geographic
# 2000000 2 7.3 7.3 luminous
# 2000000 10 27.6 14.6 geographic
# 2000000 20 47.8 14.6 geographic
# 13805
#
# First line: the geographic range, 14.6 nm. Then each row is intensity,
# visibility (nm), luminous range (nm), the range actually seen (the smaller of
# the two) and which one limits it. In haze (2 nm) both lights are limited by the
# luminous range (5.6 and 7.3 nm). In clear air (10 or 20 nm) both are limited by
# the geographic range, 14.6 nm, so the extra intensity of the 2,000,000 cd light
# is wasted. The last line is the intensity for a 15 nm nominal range: about
# 13,805 cd, to the nearest candela, under these assumptions.