from math import atan, cos, radians, sin, tan from .haversine_distance import haversine_distance AXIS_A = 6378137.0 AXIS_B = 6356752.314245 EQUATORIAL_RADIUS = 6378137 def lamberts_ellipsoidal_distance( lat1: float, lon1: float, lat2: float, lon2: float ) -> float: """ Calculate the shortest distance along the surface of an ellipsoid between two points on the surface of earth given longitudes and latitudes https://en.wikipedia.org/wiki/Geographical_distance#Lambert's_formula_for_long_lines NOTE: This algorithm uses geodesy/haversine_distance.py to compute central angle, sigma Representing the earth as an ellipsoid allows us to approximate distances between points on the surface much better than a sphere. Ellipsoidal formulas treat the Earth as an oblate ellipsoid which means accounting for the flattening that happens at the North and South poles. Lambert's formulae provide accuracy on the order of 10 meteres over thousands of kilometeres. Other methods can provide millimeter-level accuracy but this is a simpler method to calculate long range distances without increasing computational intensity. Args: lat1, lon1: latitude and longitude of coordinate 1 lat2, lon2: latitude and longitude of coordinate 2 Returns: geographical distance between two points in metres >>> from collections import namedtuple >>> point_2d = namedtuple("point_2d", "lat lon") >>> SAN_FRANCISCO = point_2d(37.774856, -122.424227) >>> YOSEMITE = point_2d(37.864742, -119.537521) >>> NEW_YORK = point_2d(40.713019, -74.012647) >>> VENICE = point_2d(45.443012, 12.313071) >>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *YOSEMITE):0,.0f} meters" '254,351 meters' >>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *NEW_YORK):0,.0f} meters" '4,138,992 meters' >>> f"{lamberts_ellipsoidal_distance(*SAN_FRANCISCO, *VENICE):0,.0f} meters" '9,737,326 meters' """ # CONSTANTS per WGS84 https://en.wikipedia.org/wiki/World_Geodetic_System # Distance in metres(m) # Equation Parameters # https://en.wikipedia.org/wiki/Geographical_distance#Lambert's_formula_for_long_lines flattening = (AXIS_A - AXIS_B) / AXIS_A # Parametric latitudes # https://en.wikipedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude b_lat1 = atan((1 - flattening) * tan(radians(lat1))) b_lat2 = atan((1 - flattening) * tan(radians(lat2))) # Compute central angle between two points # using haversine theta. sigma = haversine_distance / equatorial radius sigma = haversine_distance(lat1, lon1, lat2, lon2) / EQUATORIAL_RADIUS # Intermediate P and Q values p_value = (b_lat1 + b_lat2) / 2 q_value = (b_lat2 - b_lat1) / 2 # Intermediate X value # X = (sigma - sin(sigma)) * sin^2Pcos^2Q / cos^2(sigma/2) x_numerator = (sin(p_value) ** 2) * (cos(q_value) ** 2) x_demonimator = cos(sigma / 2) ** 2 x_value = (sigma - sin(sigma)) * (x_numerator / x_demonimator) # Intermediate Y value # Y = (sigma + sin(sigma)) * cos^2Psin^2Q / sin^2(sigma/2) y_numerator = (cos(p_value) ** 2) * (sin(q_value) ** 2) y_denominator = sin(sigma / 2) ** 2 y_value = (sigma + sin(sigma)) * (y_numerator / y_denominator) return EQUATORIAL_RADIUS * (sigma - ((flattening / 2) * (x_value + y_value))) if __name__ == "__main__": import doctest doctest.testmod()