#### Lamberts Ellipsoidal Distance

p
```from math import atan, cos, radians, sin, tan

from .haversine_distance import haversine_distance

AXIS_A = 6378137.0
AXIS_B = 6356752.314245

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()
```