Compute theoretical maximal visibility on earth with the assumption of the earth being a perfect sphere

compute_max_visibility_on_earth.py

"""Compute theoretical maximal visibility on earth with the assumption of the earth being a perfect sphere and the absence of refraction.

Some interesting links:
https://aty.sdsu.edu/explain/atmos_refr/horizon.html
https://sites.math.washington.edu/~conroy/m120-general/horizon.pdf
"""
import numpy as np

RADIAN_EARTH = 6371e3 # meters

def compute_direct_distance(h_obs, h_obj, r=RADIAN_EARTH):
    """
    Compute maximum distance an observer with height h1 can see an object with height h2.
    The distance is direct (straight line) between h1 and h2.

    Pythagoras can be used here since at maximal possible distance, the line connecting h1 and h2
    will be tangential to min(h1,h2) forming a right triangle.

    :param h_obs: observation height in m
    :param h_obj: object height in m
    :param r: radian of earth in m
    :return: distance of line connecting top of h1 and h2 in km
    """
    hypotenuse = max(h_obs, h_obj) + r
    adjacent = min(h_obs, h_obj) + r
    return np.sqrt(hypotenuse**2 - adjacent**2) * 1e-3

def get_angle(h_obs, h_obj, r=RADIAN_EARTH):
    """
    Compute angle between h1+r and h2+r at maximum distance an observer with height h1
    can see an object with height h2.

    :param h_obs: observation height in m
    :param h_obj: object height in m
    :param r: radian of earth in m
    :return: distance of line connecting top of h1 and h2 in km
    """
    d_direct = compute_direct_distance(h_obs, h_obj, r)
    hypotenuse = max(h_obs, h_obj) + r
    return np.arctan2(d_direct, hypotenuse)

def compute_surface_distance(h_obs, h_obj, r=RADIAN_EARTH):
    """
    Compute maximum distance an observer with height h1 can see an object with height h2.
    The computed distance is the distance over the earth surface

    Pythagoras can be used here since at maximal possible distance, the line connecting h1 and h2
    will be tangential to min(h1,h2) forming a right triangle.

    :param h_obs: observation height in m
    :param h_obj: object height in m
    :param r: radian of earth in m
    :return: distance on earth surface between h1 and h2 in km
    """
    return r * get_angle(h_obs, h_obj, r)


if __name__ == "__main__":
    for i in range(1,200):
        h_obs = 2 # m
        h_obj = 1 * i # m

        print(f"""
        Max visibility with an eyes on height {h_obs}[m] and object of height {h_obj}[m]
        direct distance: {compute_direct_distance(h_obs, h_obj)}[km]
        surface distance: {compute_surface_distance(h_obs, h_obj)}[km]
        angle rad: {get_angle(h_obs, h_obj)}[rad]
        angle deg: {np.rad2deg(get_angle(h_obs, h_obj))}[deg]
        """)