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geodetic_problems.py
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import math
class DirectProblem(object):
'''
General class to perform the direct geodetic problem with ellipsoidal chords.
Direct (first or forward) task: computing position given azimuth and distance from a known location.
Takes input geodetic coordinates (ϕ, λ, h).
Parameters
----------
lat1: float
Geodetic latitude in decimal degrees (°) at point A in range [-90, 90].
lon1: float
Geodetic longitude in decimal degrees (°) at point A in range [0, 360). If negative,
converts to proper range. Examples: -75°E --> 285°, -195°W --> 165°
height1: float
Geodetic height in meters (m) at point A.
chord: float
Ellipsoidal chord in meters (m), distance from point A to point B.
azimuth: float
Forward (direct) azimuth in decimal degrees (°) from point A to point B.
zenith: float
Chords zenith distance in decimal degrees (°).
a: float, optional
Semi-major axis (equatorial radius) of GRS80 in meters (m). Defaults to 6378137.0.
b: float, optional
Semi-minor axis (polar radius) of GRS80 in meters (m). Defaults to 6356752.3141.
dec_degs: bool, optional
Whether to express angular fractions as decimal degrees or DMS.
Defaults to True.
'''
def __init__(self, lat1: float, lon1: float, height1: float,
chord: float, azimuth: float, zenith: float,
a: float = 6378137.0, b: float = 6356752.3141, dec_degs: bool = True):
self.lat1 = lat1
self.lon1 = lon1
self.height1 = height1
self.chord = chord
self.azimuth = azimuth
self.zenith = zenith
self.a = a
self.b = b
self.dec_degs = dec_degs
# latitude check
if not (-90 <= self.lat1 <= 90):
raise ValueError('Latitude must be in range [-90, 90]')
# longitude checks
if self.lon1 > 360:
raise ValueError('Longitude must be in range (-180, 180)')
# normalize longitude to the range (-180, 180)
if self.lon1 > 180:
self.lon1 -= 360
# convert degrees to radians
self.lat1, self.lon1, self.azimuth, self.zenith \
= map(math.radians, [self.lat1, self.lon1, self.azimuth, self.zenith])
# first eccentricity squared
self.e2 = (self.a**2 - self.b**2) / self.a**2
# normal radius of curvature in prime vertical
self.N1 = self.a / math.sqrt(1 - self.e2 * math.sin(self.lat1)**2)
# chord's direction cosines
self.l = math.cos(self.lat1) * math.cos(self.lon1) * math.cos(self.zenith) \
- math.sin(self.lat1) * math.cos(self.lon1) * math.sin(self.zenith) * math.cos(self.azimuth) \
- math.sin(self.lon1) * math.sin(self.zenith) * math.sin(self.azimuth)
self.m = math.cos(self.lat1) * math.sin(self.lon1) * math.cos(self.zenith) \
- math.sin(self.lat1) * math.sin(self.lon1) * math.sin(self.zenith) * math.cos(self.azimuth) \
+ math.cos(self.lon1) * math.sin(self.zenith) * math.sin(self.azimuth)
self.n = math.sin(self.lat1) * math.cos(self.zenith) \
+ math.cos(self.lat1) * math.sin(self.zenith) * math.cos(self.azimuth)
def latitude(self):
'''
Computes geodetic latitude in decimal degrees (°) and radius of curvature in prime
vertical in meters (m) at point B.
Returns
-------
Value of reduced ellipsoid chord in meters (m).
'''
# initial approximate: latitude
numerator = (self.N1 + self.height1) * math.sin(self.lat1) + self.chord * self.n
denominator = math.sqrt(
((self.N1 + self.height1) * math.cos(self.lat1) * math.cos(self.lon1) + self.chord * self.l)**2 \
+ ((self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1) + self.chord * self.m)**2
)
lat2 = math.atan(numerator / denominator)
# initial approximate: radius of curvature in prime vertical
N2 = self.a / math.sqrt(1 - self.e2 * math.sin(lat2)**2)
# repeat calculations 15 times
for _ in range(0, 15):
numerator = (self.N1 + self.height1) * math.sin(self.lat1) + self.chord * self.n \
+ self.e2 * (N2 * math.sin(lat2) - self.N1 * math.sin(self.lat1))
denominator = math.sqrt(
((self.N1 + self.height1) * math.cos(self.lat1) * math.cos(self.lon1) + self.chord * self.l)**2 \
+ ((self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1) + self.chord * self.m)**2
)
lat2 = math.atan(numerator / denominator)
N2 = self.a / math.sqrt(1 - self.e2 * math.sin(lat2)**2)
return math.degrees(lat2), N2
def longitude(self):
'''
Computes geodetic longitude in decimal degrees (°) at point B.
Returns
-------
Value of longitude in decimal degrees (°).
'''
X = (self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1) + self.chord * self.m
Y = (self.N1 + self.height1) * math.cos(self.lat1) * math.cos(self.lon1) + self.chord * self.l
lon2 = math.degrees(math.atan2(X, Y))
# normalize longitude to (-180, 180)
if lon2 > 180:
lon2 -= 360
elif lon2 < -180:
lon2 += 360
return lon2
def height(self):
'''
Computes geodetic height in meters (m) at point B.
Returns
-------
Value of height in meters (m).
'''
# convert degrees to radians
lat2, lon2 = map(math.radians, [self.latitude()[0], self.longitude()])
# normal radius of curvature at point B
N2 = self.latitude()[1]
# sum of radius of curvature in prime vertical and height at point B
N2_H2 = ((self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1) + self.chord * self.m) \
/ ((math.cos(lat2)) * math.sin(lon2))
return N2_H2 - N2
def reduced_distance(self):
'''
Computes reduced ellipsoid chord (heights are not taken into account).
Returns
-------
Value of reduced ellipsoid chord in meters (m).
'''
# convert degrees to radians
lat2 = math.radians(self.latitude()[0])
# radius of curvature in prime vertical at point B
N2 = self.latitude()[1]
# height at point B
height2 = self.height()
k = (self.height1 / self.N1) + (height2 / N2) + ((self.height1 * height2) / (self.N1 * N2))
mi = (self.a**4 - self.b**4) / self.a**4
tau = 2 * (N2 - self.N1) * (height2 - self.height1) \
+ k * mi * (N2 * math.sin(lat2) - self.N1 * math.sin(self.lat1))**2 \
- 2 * self.e2 * (N2 * math.sin(lat2) - self.N1 * math.sin(self.lat1)) \
* (height2 * math.sin(lat2) - self.height1 * math.sin(self.lat1))
p = (1 / (1 + k)) * (k + ((height2 - self.height1)**2 / self.chord**2) + (tau / self.chord**2))
return self.chord - self.chord * (p / (1 + math.sqrt(1 - p)))
def reverse_zenith_distance(self):
'''
Computes angular distance from the zenith above point B to point A: △ZBA.
Returns
-------
Value of reverse zenith distance in decimal degrees (°).
'''
# convert degrees to radians
lat2, lon2 = map(math.radians, [self.latitude()[0], self.longitude()])
# radius of curvature in prime vertical at point B
N2 = self.latitude()[1]
# height at point B
height2 = self.height()
# cosine of the central angle between two points
cos_fi = math.sin(self.lat1) * math.sin(lat2) \
+ math.cos(self.lat1) * math.cos(lat2) * math.cos(lon2 - self.lon1)
# reverse zenith distance
cos_zen2 = math.acos(
((self.N1 + self.height1) * cos_fi - (N2 + height2) \
+ self.e2 * (N2 * math.sin(lat2) \
- self.N1 * math.sin(self.lat1)) * math.sin(lat2)) / self.chord
)
return math.degrees(cos_zen2)
def convert_to_xyz(self):
'''
Converts geodetic coordinates (ϕ, λ, h) to Cartesian coordinates (X, Y, Z).
Returns
-------
Cartesian coordinates in meters (m).
'''
# convert degrees to radians
lat2, lon2 = map(math.radians, [self.latitude()[0], self.longitude()])
# radius of curvature in prime vertical at point B
N2 = self.latitude()[1]
# height at point B
height2 = self.height()
# get Cartesian coordinates
X2 = (N2 + height2) * math.cos(lat2) * math.cos(lon2)
Y2 = (N2 + height2) * math.cos(lat2) * math.sin(lon2)
Z2 = (N2 * (1 - self.e2) + height2) * math.sin(lat2)
return {'X': X2, 'Y': Y2, 'Z': Z2}
def decimal_to_dms(self, decimal_degrees):
'''
Converts decimal degrees into DMS. Example: 30.5° --> 30°30'00".
Returns
-------
Angular values expressed in DMS notation.
'''
# get degrees (integer part)
degrees = int(decimal_degrees)
# get the fractional part of the degrees and convert it to minutes
minutes_float = (abs(decimal_degrees) - abs(degrees)) * 60
minutes = int(minutes_float)
# get the fractional part of the minutes and convert it to seconds
seconds = (minutes_float - minutes) * 60
return f'{degrees}°{minutes}\'{seconds:.8f}"'
def display_measures(self):
'''
Displays all computed quantities.
Returns
-------
quantities: dict
'''
if self.dec_degs:
return {
'Normal radius of curvature': f'{self.latitude()[1]} m',
'Latitude': f'{self.latitude()[0]}°',
'Longitude': f'{self.longitude()}°',
'Height': f'{self.height()} m',
'Reduced chord': f'{self.reduced_distance()} m',
'Reverse zenith distance': f'{self.reverse_zenith_distance()}°',
'XYZ 2': f'{self.convert_to_xyz()} m',
}
else:
return {
'Normal radius of curvature': f'{self.latitude()[1]} m',
'Latitude': self.decimal_to_dms(self.latitude()[0]),
'Longitude': self.decimal_to_dms(self.longitude()),
'Height': f'{self.height()} m',
'Reduced chord': f'{self.reduced_distance()} m',
'Reverse zenith distance': self.decimal_to_dms(self.reverse_zenith_distance()),
'XYZ 2': f'{self.convert_to_xyz()} m',
}
class InverseProblem(object):
'''
General class to perform the inverse geodetic problem with ellipsoidal chords.
Inverse (second or reverse) task: computing azimuth and distance between known positions.
Takes input geodetic coordinates (ϕ, λ, h).
Parameters
----------
lat1: float
Geodetic latitude in decimal degrees (°) at point A in range [-90, 90].
Positive for north hemisphere and negative for south hemisphere.
lon1: float
Geodetic longitude in decimal degrees (°) at point A in range (-180, 180).
Negative for west hemisphere and positive for east hemisphere.
height1: float
Geodetic height in meters (m) at point A.
lat2: float
Geodetic latitude in decimal degrees (°) at point B in range [-90, 90].
Positive for north hemisphere and negative for south hemisphere.
lon2: float
Geodetic longitude in decimal degrees (°) at point A in range (-180, 180).
Negative for west hemisphere and positive for east hemisphere.
height2: float
Geodetic height in meters (m) at point B.
a: float, optional
Semi-major axis (equatorial radius) of GRS80 in meters (m). Defaults to 6378137.0.
b: float, optional
Semi-minor axis (polar radius) of GRS80 in meters (m). Defaults to 6356752.3141.
dec_degs: bool, optional
Whether to express angular fractions as decimal degrees or DMS.
Defaults to True.
'''
def __init__(self, lat1: float, lon1: float, height1: float,
lat2: float, lon2: float, height2: float,
a: float = 6378137., b: float = 6356752.3141, dec_degs: bool = True):
self.lat1 = lat1
self.lon1 = lon1
self.height1 = height1
self.lat2 = lat2
self.lon2 = lon2
self.height2 = height2
self.a = a
self.b = b
self.dec_degs = dec_degs
# latitude checks
if not (-90 <= self.lat1 <= 90):
raise ValueError('Latitude must be in range [-90, 90]')
if not (-90 <= self.lat2 <= 90):
raise ValueError('Latitude must be in range [-90, 90]')
# longitude checks
if self.lon1 > 360 or self.lon2 > 360:
raise ValueError('Longitude must be in range [0, 360)')
# normalize longitudes to the range (-180, 180)
if self.lon1 > 180:
self.lon1 -= 360
if self.lon2 > 180:
self.lon2 -= 360
# convert degrees to radians
self.lat1, self.lon1, self.lat2, self.lon2 \
= map(math.radians, [self.lat1, self.lon1, self.lat2, self.lon2])
# first eccentricity squared
self.e2 = (self.a**2 - self.b**2) / self.a**2
# normal radius of curvature in prime vertical
self.N1 = self.a / math.sqrt(1 - self.e2 * math.sin(self.lat1)**2)
self.N2 = self.a / math.sqrt(1 - self.e2 * math.sin(self.lat2)**2)
def convert_to_xyz(self):
'''
Converts geodetic coordinates (ϕ, λ, h) to Cartesian coordinates (X, Y, Z).
Returns
-------
Cartesian coordinates in meters (m).
'''
X1 = (self.N1 + self.height1) * math.cos(self.lat1) * math.cos(self.lon1)
Y1 = (self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1)
Z1 = (self.N1 * (1 - self.e2) + self.height1) * math.sin(self.lat1)
X2 = (self.N2 + self.height2) * math.cos(self.lat2) * math.cos(self.lon2)
Y2 = (self.N2 + self.height2) * math.cos(self.lat2) * math.sin(self.lon2)
Z2 = (self.N2 * (1 - self.e2) + self.height2) * math.sin(self.lat2)
return {'X': X1, 'Y': Y1, 'Z': Z1}, {'X': X2, 'Y': Y2, 'Z': Z2}
def chord_distance(self):
'''
Computes ellipsoid chord (slant distance from A to B, not arc).
Returns
-------
Value of ellipsoid chord in meters (m).
'''
mi = (self.a**4 - self.b**4) / self.a**4
# cosine of the central angle between two points
cos_fi = math.sin(self.lat1) * math.sin(self.lat2) \
+ math.cos(self.lat1) * math.cos(self.lat2) * math.cos(self.lon2 - self.lon1)
# chord length
chord = (self.N2 + self.height2)**2 + (self.N1 + self.height1)**2 \
- 2 * (self.N2 + self.height2) * (self.N1 + self.height1) * cos_fi \
- mi * (self.N2 * math.sin(self.lat2) - self.N1 * math.sin(self.lat1))**2 \
- 2 * self.e2 * (self.N2 * math.sin(self.lat2) - self.N1 * math.sin(self.lat1))\
* (self.height2 * math.sin(self.lat2) - self.height1 * math.sin(self.lat1))
return math.sqrt(chord)
def cartesian_distance(self):
'''
Computes ellipsoid chord using Cartesian coordinates. Should return the same
value as chord_distance() method.
Returns
-------
Value of distance (ellipsoid chord) in meters (m).
'''
# get Cartesian coordinates
X1 = (self.N1 + self.height1) * math.cos(self.lat1) * math.cos(self.lon1)
Y1 = (self.N1 + self.height1) * math.cos(self.lat1) * math.sin(self.lon1)
Z1 = (self.N1 * (1 - self.e2) + self.height1) * math.sin(self.lat1)
X2 = (self.N2 + self.height2) * math.cos(self.lat2) * math.cos(self.lon2)
Y2 = (self.N2 + self.height2) * math.cos(self.lat2) * math.sin(self.lon2)
Z2 = (self.N2 * (1 - self.e2) + self.height2) * math.sin(self.lat2)
# Euclidean distance
chord = (X2 - X1)**2 + (Y2 - Y1)**2 + (Z2 - Z1)**2
return math.sqrt(chord)
def reduced_distance(self):
'''
Computes reduced ellipsoid chord (heights are not taken into account).
Returns
-------
Value of reduced ellipsoid chord in meters (m).
'''
mi = (self.a**4 - self.b**4) / self.a**4
# sine of the central angle between two points
sin_2fi = math.sin((self.lat2 - self.lat1)/2)**2 \
+ math.cos(self.lat1) * math.cos(self.lat2) * math.sin((self.lon2 - self.lon1)/2)**2
# chord length (without heights)
reduced_chord = 4 * self.N1 * self.N2 * sin_2fi + (self.N2 - self.N1)**2 \
- mi * (self.N2 * math.sin(self.lat2) - self.N1 * math.sin(self.lat1))**2
return math.sqrt(reduced_chord)
def forward_azimuth(self):
'''
Computes foward (direct) azimuth from point A to point B.
Returns
-------
Value of foward azimuth in decimal degrees (°).
'''
# calculate 1st term
ctga1 = (math.sin(self.lat2) * math.cos(self.lat1) -
math.cos(self.lat2) * math.sin(self.lat1) *
math.cos(self.lon2 - self.lon1)) / (math.cos(self.lat2) *
math.sin(self.lon2 - self.lon1))
# calculate 2nd term
ctgA1 = ctga1 - self.e2 * ((self.N2 * math.sin(self.lat2) \
- self.N1 * math.sin(self.lat1)) *
math.cos(self.lat1)) / ((self.N2 + self.height2) *
math.cos(self.lat2) * math.sin(self.lon2 - self.lon1))
# calculate azimuth using atan2 to avoid quadrant issues
azimuth = math.degrees(math.atan2(1, ctgA1))
# normalize the azimuth to [0, 360)
if azimuth > 180:
azimuth -= 360
return azimuth
def reverse_azimuth(self):
'''
Computes reverse (backward) azimuth from point A to point B.
Returns
-------
Value of reverse azimuth in decimal degrees (°).
'''
# calculate 1st term
ctga2 = -(math.sin(self.lat1) * math.cos(self.lat2) -
math.cos(self.lat1) * math.sin(self.lat2) *
math.cos(self.lon2 - self.lon1)) / (math.cos(self.lat1) *
math.sin(self.lon2 - self.lon1))
# calculate 2nd term
ctgA2 = ctga2 - self.e2 * ((self.N2 * math.sin(self.lat2) \
- self.N1 * math.sin(self.lat1)) *
math.cos(self.lat2)) / ((self.N1 + self.height1) *
math.cos(self.lat1) * math.sin(self.lon2 - self.lon1))
# calculate azimuth using atan2 to avoid quadrant issues
azimuth = math.degrees(math.atan2(1, ctgA2))
# normalize the azimuth to [0, 360)
if azimuth > 180:
azimuth -= 360
# reverse azimuth is typically the forward azimuth ± 180°
azimuth = (azimuth + 180) % 360
return azimuth
def forward_zenith_distance(self):
'''
Computes angular distance from the zenith above point A to point B: △ZAB.
Returns
-------
Value of forward zenith distance in decimal degrees (°).
'''
# cosine of the central angle between two points
cos_fi = math.sin(self.lat1) * math.sin(self.lat2) \
+ math.cos(self.lat1) * math.cos(self.lat2) * math.cos(self.lon2 - self.lon1)
# forward zenith distance
cos_zen1 = math.acos(
((self.N2 + self.height2) * cos_fi - (self.N1 + self.height1) \
- self.e2 * (self.N2 * math.sin(self.lat2) \
- self.N1 * math.sin(self.lat1)) * math.sin(self.lat1)) / self.chord_distance()
)
return math.degrees(cos_zen1)
def reverse_zenith_distance(self):
'''
Computes angular distance from the zenith above point B to point A: △ZBA.
Returns
-------
Value of reverse zenith distance in decimal degrees (°).
'''
# cosine of the central angle between two points
cos_fi = math.sin(self.lat1) * math.sin(self.lat2) \
+ math.cos(self.lat1) * math.cos(self.lat2) * math.cos(self.lon2 - self.lon1)
# reverse zenith distance
cos_zen2 = math.acos(
((self.N1 + self.height1) * cos_fi - (self.N2 + self.height2) \
+ self.e2 * (self.N2 * math.sin(self.lat2) \
- self.N1 * math.sin(self.lat1)) * math.sin(self.lat2)) / self.chord_distance()
)
return math.degrees(cos_zen2)
def decimal_to_dms(self, decimal_degrees):
'''
Converts decimal degrees into DMS. Example: 30.5° --> 30°30'00".
Returns
-------
Angular values expressed in DMS notation.
'''
# get degrees (integer part)
degrees = int(decimal_degrees)
# get the fractional part of the degrees and convert it to minutes
minutes_float = (abs(decimal_degrees) - abs(degrees)) * 60
minutes = int(minutes_float)
# get the fractional part of the minutes and convert it to seconds
seconds = (minutes_float - minutes) * 60
return f'{degrees}°{minutes}\'{seconds:.8f}"'
def display_measures(self):
'''
Displays all computed quantities.
Returns
-------
quantities: dict
'''
if self.dec_degs:
return {
'XYZ 1': f'{self.convert_to_xyz()[0]} m',
'XYZ 2': f'{self.convert_to_xyz()[1]} m',
'Chord (distance)': f'{self.chord_distance()} m',
'Cartesian distance': f'{self.cartesian_distance()} m',
'Reduced chord': f'{self.reduced_distance()} m',
'Forward azimuth': f'{self.forward_azimuth()}°',
'Reverse azimuth': f'{self.reverse_azimuth()}°',
'Forward zenith distance': f'{self.forward_zenith_distance()}°',
'Reverse zenith distance': f'{self.reverse_zenith_distance()}°',
}
else:
return {
'XYZ 1': f'{self.convert_to_xyz()[0]} m',
'XYZ 2': f'{self.convert_to_xyz()[1]} m',
'Chord (distance)': f'{self.chord_distance()} m',
'Cartesian distance': f'{self.cartesian_distance()} m',
'Reduced chord': f'{self.reduced_distance()} m',
'Forward azimuth': self.decimal_to_dms(self.forward_azimuth()),
'Reverse azimuth': self.decimal_to_dms(self.reverse_azimuth()),
'Forward zenith distance': self.decimal_to_dms(self.forward_zenith_distance()),
'Reverse zenith distance': self.decimal_to_dms(self.reverse_zenith_distance()),
}