我正在尝试实现飞机的定位功能以获得“纬度/经度方位角” 我附上了 3 张图片用于汇总公式，您可能会看到这是 5 步三角方程（步骤 0/4），这是我的编程目标。 image1; image2image3
要查找飞机坐标，定义了 9 个输入参数（图像1）：站 U 和 S 纬度、经度、高度；飞机高度和 2 个倾斜范围。 在问题结束时（image3），我们将找到 3 个输出：飞机纬度/经度方位角。
【问题讨论】：
标签： geopositioning
此代码实现了为How can I triangulate a position using two DMEs? on aviation.se 解释的解决方案。代码在 Python 中，我碰巧使用它而不是 C，很容易根据需要转换成另一种语言。我已将计算分解为更小的单位，以使代码更清晰易读并便于您理解。
问题涉及 3D 空间中的 3 个点：U 和 S 是DMEs，A 是飞机。
由于我们只需要 U 和 S 的坐标来确定 A 坐标，因此我使用了 3 个知名 DME 站的坐标。这将允许检查结果是否正确。基于Low Altitude Enroute Chart查看：
程序运行时，输出为：
CAN: lat 49.17319, lon -0.4552778, alt 82
EVX: lat 49.03169, lon 1.220861, alt 152
P north: lat 49.386910325692874, lon 0.646650777948733, alt 296
P south: lat 48.78949175956114, lon 0.5265322105880027, alt 296
首先是我们输入的点 U (CAN DME) 和 S (EVX DME) 的坐标，以及 然后用两条线表示飞机的两个可能位置。
我用 DME 进行了另一次更远距离的测试（ARE 为 1241 公里，GLA 为 557.1 公里），效果也很好：
ARE: lat 48.33264, lon -3.602472, alt 50
GLA: lat 46.40861, lon 6.244222, alt 1000
P north: lat 48.082101174246304, lon 13.210754399535269, alt 10
P south: lat 41.958725412109445, lon 9.470999690780628, alt 10
飞机的实际位置应该是 SZA 导航台，在法国南部：纬度 41.937°，经度 9.399°。
from math import asin, sqrt, cos, sin, atan2, acos, pi, radians, degrees
# Earth radius in meters (https://rechneronline.de/earth-radius/)
E_RADIUS = 6367 * 1000 # at 45°N - Adjust as required
def step_0(r_e, h_u, h_s, h_a, d_ua, d_sa):
# Return angular distance between each station U/S and aircraft
# Triangle UCA and SCA: The three sides are known,
a = (d_ua - h_a + h_u) * (d_ua + h_a - h_u)
b = (r_e + h_u) * (r_e + h_a)
theta_ua = 2 * asin(.5 * sqrt(a / b))
a = (d_sa - h_a + h_s) * (d_sa + h_a - h_s)
b = (r_e + h_s) * (r_e + h_a)
theta_sa = 2 * asin(.5 * sqrt(a / b))
# Return angular distances between stations and aircraft
return theta_ua, theta_sa
def step_1(lat_u, lon_u, lat_s, lon_s):
# Determine angular distance between the two stations
# and the relative azimuth of one to the other.
a = sin(.5 * (lat_s - lat_u))
b = sin(.5 * (lon_s - lon_u))
c = sqrt(a * a + cos(lat_s) * cos(lat_u) * b * b)
theta_us = 2 * asin(c)
a = lon_s - lon_u
b = cos(lat_s) * sin(a)
c = sin(lat_s) * cos(lat_u)
d = cos(lat_s) * sin(lat_u) * cos(a)
psi_su = atan2(b, c - d)
return theta_us, psi_su
def step_2(theta_us, theta_ua, theta_sa):
# Determine whether DME spheres intersect
if (theta_ua + theta_sa) < theta_us:
# Spheres are too remote to intersect
return False
if abs(theta_ua - theta_sa) > theta_us:
# Spheres are concentric and don't intersect
return False
# Spheres intersect
return True
def step_3(theta_us, theta_ua, theta_sa):
# Determine one angle of the USA triangle
a = cos(theta_sa) - cos(theta_us) * cos(theta_ua)
b = sin(theta_us) * sin(theta_ua)
beta_u = acos(a / b)
return beta_u
def step_4(ac_south, lat_u, lon_u, beta_u, psi_su, theta_ua):
# Determine aircraft coordinates
psi_au = psi_su
if ac_south:
psi_au += beta_u
else:
psi_au -= beta_u
# Determine aircraft latitude
a = sin(lat_u) * cos(theta_ua)
b = cos(lat_u) * sin(theta_ua) * cos(psi_au)
lat_a = asin(a + b)
# Determine aircraft longitude
a = sin(psi_au) * sin(theta_ua)
b = cos(lat_u) * cos(theta_ua)
c = sin(lat_u) * sin(theta_ua) * cos(psi_au)
lon_a = atan2(a, b - c) + lon_u
return lat_a, lon_a
def main():
# Program entry point
# -------------------
# For this test, I'm using three locations in France:
# VOR Caen, VOR Evreux and VOR L'Aigle.
# The angles and horizontal distances between them are known
# by looking at the low-altitude enroute chart which I've posted
# here: https://i.stack.imgur.com/m8Wmw.png
# We know there coordinates and altitude by looking at the AIP France too.
# For DMS <--> Decimal degrees, this tool is handy:
# https://www.rapidtables.com/convert/number/degrees-minutes-seconds-to-degrees.html
# Let's pretend the aircraft is at LGL
# lat = 48.79061, lon = 0.5302778
# Stations U and S are:
u = {'label': 'CAN', 'lat': 49.17319, 'lon': -0.4552778, 'alt': 82}
s = {'label': 'EVX', 'lat': 49.03169, 'lon': 1.220861, 'alt': 152}
# We know the aircraft altitude
a_alt = 296 # meters
# We know the approximate slant ranges to stations U and S
au_range = 45 * 1852 # 1 NM = 1,852 m
as_range = 31 * 1852
# Compute angles station - earth center - aircraft for U and S
# Expected values UA: 0.0130890288 rad
# SA: 0.0090168045 rad
theta_ua, theta_sa = step_0(
r_e=E_RADIUS, # Earth
h_u=u['alt'], # Station U altitude
h_s=s['alt'], # Station S altitude
h_a=a_alt, d_ua=au_range, d_sa=as_range # aircraft data
)
# Compute angle between station, and their relative azimuth
# We need to convert angles into radians
theta_us, psi_su = step_1(
lat_u=radians(u['lat']), lon_u=radians(u['lon']), # Station U coordinates
lat_s=radians(s['lat']), lon_s=radians(s['lon'])) # Station S coordinates
# Check validity of ranges
if not step_2(
theta_us=theta_us,
theta_ua=theta_ua,
theta_sa=theta_sa):
# Cannot compute, spheres don't intersect
print('Cannot compute, ranges are not consistant')
return 1
# Solve one angle of the USA triangle
beta_u = step_3(
theta_us=theta_us,
theta_ua=theta_ua,
theta_sa=theta_sa)
# Compute aircraft coordinates.
# The first parameter is whether the aircraft is south of the line
# between U and S. If you don't know, then you need to compute
# both, once with ac_south = False, once with ac_south = True.
# You will get the two possible positions, one must be eliminated.
# North position
lat_n, lon_n = step_4(
ac_south=False, # See comment above
lat_u=radians(u['lat']), lon_u=radians(u['lon']), # Station U
beta_u=beta_u, psi_su=psi_su, theta_ua=theta_ua # previously computed
)
pn = {'label': 'P north', 'lat': degrees(lat_n), 'lon': degrees(lon_n), 'alt': a_alt}
# South position
lat_s, lon_s = step_4(
ac_south=True,
lat_u=radians(u['lat']), lon_u=radians(u['lon']),
beta_u=beta_u, psi_su=psi_su, theta_ua=theta_ua)
ps = {'label': 'P south', 'lat': degrees(lat_s), 'lon': degrees(lon_s), 'alt': a_alt}
# Print results
fmt = '{}: lat {}, lon {}, alt {}'
for p in u, s, pn, ps:
print(fmt.format(p['label'], p['lat'], p['lon'], p['alt']))
# The expected result is about:
# CAN: lat 49.17319, lon -0.4552778, alt 82
# EVX: lat 49.03169, lon 1.220861, alt 152
# P north: lat ??????, lon ??????, alt 296
# P south: lat 48.79061, lon 0.5302778, alt 296
if __name__ == '__main__':
main()
【讨论】：