diff --git a/geodesy/lamberts_ellipsoidal_distance.py b/geodesy/lamberts_ellipsoidal_distance.py new file mode 100644 index 000000000..224b9404a --- /dev/null +++ b/geodesy/lamberts_ellipsoidal_distance.py @@ -0,0 +1,83 @@ +from math import atan, cos, radians, sin, tan +from haversine_distance import haversine_distance + + +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) + AXIS_A = 6378137.0 + AXIS_B = 6356752.314245 + EQUATORIAL_RADIUS = 6378137 + + # 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()