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from math import asin, atan, cos, radians, sin, sqrt, tan
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AXIS_A = 6378137.0
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AXIS_B = 6356752.314245
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RADIUS = 6378137
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from math import asin, cos, radians, sin, sqrt
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def haversine_distance(lat1: float, lon1: float, lat2: float, lon2: float) -> float:
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"""
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Calculate great circle distance between two points in a sphere,
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given longitudes and latitudes https://en.wikipedia.org/wiki/Haversine_formula
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We know that the globe is "sort of" spherical, so a path between two points
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isn't exactly a straight line. We need to account for the Earth's curvature
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when calculating distance from point A to B. This effect is negligible for
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small distances but adds up as distance increases. The Haversine method treats
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the earth as a sphere which allows us to "project" the two points A and B
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onto the surface of that sphere and approximate the spherical distance between
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them. Since the Earth is not a perfect sphere, other methods which model the
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Earth's ellipsoidal nature are more accurate but a quick and modifiable
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computation like Haversine can be handy for shorter range distances.
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Calculate the great-circle distance between two points
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on the Earth specified by latitude and longitude using
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the Haversine formula.
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Args:
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lat1, lon1: latitude and longitude of coordinate 1
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lat2, lon2: latitude and longitude of coordinate 2
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lat1, lon1: Latitude and longitude of point 1 in decimal degrees.
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lat2, lon2: Latitude and longitude of point 2 in decimal degrees.
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Returns:
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geographical distance between two points in metres
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Distance between the two points in meters.
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>>> from collections import namedtuple
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>>> point_2d = namedtuple("point_2d", "lat lon")
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>>> SAN_FRANCISCO = point_2d(37.774856, -122.424227)
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>>> YOSEMITE = point_2d(37.864742, -119.537521)
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>>> f"{haversine_distance(*SAN_FRANCISCO, *YOSEMITE):0,.0f} meters"
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'254,352 meters'
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'254,033 meters'
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"""
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# CONSTANTS per WGS84 https://en.wikipedia.org/wiki/World_Geodetic_System
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# Distance in metres(m)
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# Equation parameters
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# Equation https://en.wikipedia.org/wiki/Haversine_formula#Formulation
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flattening = (AXIS_A - AXIS_B) / AXIS_A
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phi_1 = atan((1 - flattening) * tan(radians(lat1)))
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phi_2 = atan((1 - flattening) * tan(radians(lat2)))
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lambda_1 = radians(lon1)
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lambda_2 = radians(lon2)
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# Equation
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sin_sq_phi = sin((phi_2 - phi_1) / 2)
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sin_sq_lambda = sin((lambda_2 - lambda_1) / 2)
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# Square both values
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sin_sq_phi *= sin_sq_phi
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sin_sq_lambda *= sin_sq_lambda
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h_value = sqrt(sin_sq_phi + (cos(phi_1) * cos(phi_2) * sin_sq_lambda))
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return 2 * RADIUS * asin(h_value)
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radius = 6378137 # earth radius (meters)
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lat1_rad = radians(lat1)
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lat2_rad = radians(lat2)
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delta_lat = radians(lat2 - lat1)
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delta_lon = radians(lon2 - lon1)
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# Haversine formula
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a = (
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sin(delta_lat / 2) ** 2
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+ cos(lat1_rad) * cos(lat2_rad) * sin(delta_lon / 2) ** 2
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)
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c = 2 * asin(sqrt(a))
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# Great-Circle Distance
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return radius * c
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if __name__ == "__main__":
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two points on the surface of earth given longitudes and latitudes
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https://en.wikipedia.org/wiki/Geographical_distance#Lambert's_formula_for_long_lines
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NOTE: This algorithm uses geodesy/haversine_distance.py to compute central angle,
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sigma
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NOTE: This algorithm uses geodesy/haversine_distance.py to compute the
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central angle, sigma
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Representing the earth as an ellipsoid allows us to approximate distances between
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points on the surface much better than a sphere. Ellipsoidal formulas treat the
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Earth as an oblate ellipsoid which means accounting for the flattening that happens
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at the North and South poles. Lambert's formulae provide accuracy on the order of
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10 meteres over thousands of kilometeres. Other methods can provide
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millimeter-level accuracy but this is a simpler method to calculate long range
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at the North and South poles. Lambert's formulas provide accuracy on the order of
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10 meters over thousands of kilometers. Other methods can provide
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millimeter-level accuracy, but this is a simpler method to calculate long-range
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distances without increasing computational intensity.
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Args:
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lat1, lon1: latitude and longitude of coordinate 1
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lat2, lon2: latitude and longitude of coordinate 2
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Returns:
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geographical distance between two points in metres
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geographical distance between two points in meters
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>>> from collections import namedtuple
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>>> point_2d = namedtuple("point_2d", "lat lon")
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>>> NEW_YORK = point_2d(40.713019, -74.012647)
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>>> VENICE = point_2d(45.443012, 12.313071)
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>>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *YOSEMITE):0,.0f} meters"
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'254,351 meters'
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'254,032 meters'
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>>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *NEW_YORK):0,.0f} meters"
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'4,138,992 meters'
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'4,133,295 meters'
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>>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *VENICE):0,.0f} meters"
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'9,737,326 meters'
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'9,719,525 meters'
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"""
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# CONSTANTS per WGS84 https://en.wikipedia.org/wiki/World_Geodetic_System
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# Distance in metres(m)
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# Equation Parameters
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# https://en.wikipedia.org/wiki/Geographical_distance#Lambert's_formula_for_long_lines
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flattening = (AXIS_A - AXIS_B) / AXIS_A
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# Parametric latitudes
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# https://en.wikipedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude
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b_lat1 = atan((1 - flattening) * tan(radians(lat1)))
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b_lat2 = atan((1 - flattening) * tan(radians(lat2)))
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# Compute central angle between two points
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# using haversine theta. sigma = haversine_distance / equatorial radius
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# Compute the central angle between two points using the haversine function
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sigma = haversine_distance(lat1, lon1, lat2, lon2) / EQUATORIAL_RADIUS
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# Intermediate P and Q values
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q_value = (b_lat2 - b_lat1) / 2
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# Intermediate X value
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# X = (sigma - sin(sigma)) * sin^2Pcos^2Q / cos^2(sigma/2)
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x_numerator = (sin(p_value) ** 2) * (cos(q_value) ** 2)
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x_demonimator = cos(sigma / 2) ** 2
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x_value = (sigma - sin(sigma)) * (x_numerator / x_demonimator)
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x_denominator = cos(sigma / 2) ** 2
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x_value = (sigma - sin(sigma)) * (x_numerator / x_denominator)
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# Intermediate Y value
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# Y = (sigma + sin(sigma)) * cos^2Psin^2Q / sin^2(sigma/2)
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y_numerator = (cos(p_value) ** 2) * (sin(q_value) ** 2)
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y_denominator = sin(sigma / 2) ** 2
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y_value = (sigma + sin(sigma)) * (y_numerator / y_denominator)
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3
geodesy/temp_code_runner_file.py
Normal file
3
geodesy/temp_code_runner_file.py
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# if __name__ == "__main__":
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# import doctest
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# doctest.testmod()
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