Given two points A and B, a hyperbola is the set of all the
points (C) such that the difference between the distance from Point
C to Point A and the distance from Point C to Point B is constant.
Now, instead of thinking in terms of a curve on a plane, think in
terms of a surface in three dimensions…
A foreign agent has been sending messages to his handlers using
a transmitter and Morse code. You are an engineer working with the
local counterespionage task force and have set up radio receiving
equipment on three Bay Area peaks: Black Mountain, Mount Umunhum,
and Mount Diablo. With this equipment, you can measure the time of
arrival of a radio transmission very precisely. The precise time of
transmission is unknown, but you can compute the differences in
time of arrival between any two receivers, and translate this to a
distance using the speed of light. Each pair of measurements
translates into a hyperbolic surface. Intersect these surfaces with
each other and the surface of the earth, and you have the
location.
Site Data
Black Mountain: N 37 19.123 W 122 8.771 altitude 10,000 feet (The
receiver is on a tethered balloon -- this makes trial and error by
using
FizzyCalc a bad idea.)
Mt. Umunhum: N 37 9.637 W 121 53.777 altitude 3450 feet
Mt. Diablo: N 37 52.900 W 121 55.077 altitude 3600 feet
Test Run
To make sure the system works before it goes operational, you set
up a transmitter at the coordinates at the top of the page and turn
it on at precisely noon. The measured times of arrival in seconds
past noon were:
Black Mountain: 0.000045930
Mt. Umunhum: 0.000051031
Mt. Diablo: 0.000230880
You check out the calculations and declare the system
operational.
The Real Thing
Finally, you collect the agent’s transmission with the following
measured times of arrival in seconds past noon:
Black Mountain: 2184.633346622
Mt. Umunhum: 2184.633360663
Mt. Diablo: 2184.633517022
Here is some additional help that used to be encoded as a
hint:
1. Convert latitude, longitude, and altitude to rectangular
coordinates, and compute distances in those coordinates.
2. For good accuracy, use an ellipsoid earth model. If you use a
spherical model, try it out on the test data to gauge the magnitude
of the error and be prepared to search a larger area.
3. The transformation from geodetic latitude, longitude, and
altitude to geocentric rectangular coordinates, from Bate, Muller,
and White, “Fundamentals of Astrodynamics”, p. 98:
b = er / sqrt(1 – e*e*sin(lat)*sin(lat))
d = (b + h) * cos(lat)
x = d * cos(lon)
y = d * sin(lon)
z = (b *(1 – e*e) + h) * sin(lat)
where:
er = earth equatorial radius = 6378137 meters (WGS-84)
e*e = earth eccentricity squared = 2f – f*f
f = earth flattening = 1/298.257223560 (WGS-84)
h = height above the earth reference ellipsoid
05/25/08 - Moved the math from the hint to the description, changed
the hint
12/21/08 Replaced cache, changed hint