# How Lighthouses Work 4, Exercise 2: flame size and beam width in a first-order lens
# A 10 cd burner at the focus of a lens of 920 mm focal length, flame radius 5, 10 and 20 mm.
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 2: flame size and beam width in a first-order lens
FOCAL_M = 0.92                 # first-order focal length from the Wikipedia table
LUMENS = 10 * 4 * math.pi      # assumed: one 10 cd burner
CAPTURE = 0.85
for radius_mm in (5, 10, 20):
    half = math.atan(radius_mm / 1000 / FOCAL_M)
    omega = 2 * math.pi * (1 - math.cos(half))
    cd = CAPTURE * LUMENS / omega
    print(radius_mm, "mm flame radius", round(math.degrees(2 * half), 2), "deg", round(cd), "cd", round(luminous_range_nm(cd, 10), 1), "nm")
# flame radius that gives a 2 degree beam
r = FOCAL_M * math.tan(math.radians(1.0))
print("Radius for a 2 degree beam:", round(r * 1000, 1), "mm")
# the same flame in the hyper-radial lens (1330 mm)
half = math.atan(0.01 / 1.330)
cd = CAPTURE * LUMENS / (2 * math.pi * (1 - math.cos(half)))
print("Hyper-radial, 10 mm flame:", round(cd), "cd", round(cd / (CAPTURE * LUMENS / (2 * math.pi * (1 - math.cos(math.atan(0.01 / FOCAL_M))))), 2), "x the first order")

# Output:
# 5 mm flame radius 0.62 deg 1151129 cd 26.1 nm
# 10 mm flame radius 1.25 deg 287801 cd 22.4 nm
# 20 mm flame radius 2.49 deg 71969 cd 19.0 nm
# Radius for a 2 degree beam: 16.1 mm
# Hyper-radial, 10 mm flame: 601451 cd 2.09 x the first order
#
# A bigger flame fills more of the lens and spreads the beam: the full beam width goes from 0.62
# degrees at 5 mm radius to 2.49 degrees at 20 mm, and the peak intensity falls from about 1.15
# million cd to about 72,000 cd (the same light in a wider beam), so the nominal range falls from
# 26.1 to 19.0 nm. A 2 degree beam needs a flame radius of about 16.1 mm. The same 10 mm flame in
# the hyper-radial lens (1330 mm) gives about 2.09 times the intensity of the first order. All
# numbers come from the assumed model of a round lens and a 10 cd burner, not from a real light.