# How Lighthouses Work 1, Exercise 2: the geographic range
# For lights 20, 40 and 80 m above the sea, seen by an eye 3 m above the sea,
# find the geographic range with and without refraction (constants 3.86 and
# 3.57 km per square root of a metre). Then find the height of light needed for a
# 20 nautical mile range, and the height needed to double the range of the 40 m
# light.
import math

K = 3.86                             # km per sqrt(metre), with standard refraction
G = 3.57                             # km per sqrt(metre), pure geometry
NM = 1.852                           # km per nautical mile
eye = 3                              # observer's eye height, m (assumed)

for light in (20, 40, 80):
    refracted = K * (math.sqrt(light) + math.sqrt(eye)) / NM
    plain = G * (math.sqrt(light) + math.sqrt(eye)) / NM
    print(light, round(plain, 1), round(refracted, 1), round(refracted - plain, 1))

# light height needed for a 20 nm geographic range to an eye at 3 m
target_nm = 20
root = target_nm * NM / K - math.sqrt(eye)
print(round(root ** 2, 1))

# range doubling: height needed to double the range of a 40 m light
base = K * (math.sqrt(40) + math.sqrt(eye)) / NM
root2 = 2 * base * NM / K - math.sqrt(eye)
print(round(base, 1), round(root2 ** 2, 1))

# Output when run:
# 20 12.0 12.9 1.0
# 40 15.5 16.8 1.3
# 80 20.6 22.3 1.7
# 61.8
# 16.8 206.8
#
# Columns of the first three lines: light height (m), range without refraction
# (nm), range with refraction (nm), and the gain from refraction (nm). The
# refraction gain is about 8% (3.86 / 3.57), 1.0 to 1.7 nm here. A 20 nm range
# needs a light about 61.8 m up. The last line gives the 40 m light's range
# (16.8 nm) and the height for double that range, 206.8 m: because range grows
# with the SQUARE ROOT of height, doubling the range takes roughly four times
# the height (the observer's own height shifts it slightly).