Added geodesy functions
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Mark Qvist
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import RNS
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import time
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from math import pi, sin, cos, acos, tan, atan, atan2
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from math import radians, degrees, sqrt
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# Default planetary metrics
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equatorial_radius = 6378.137 *1e3
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polar_radius = 6356.7523142 *1e3
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ellipsoid_flattening = 1-(polar_radius/equatorial_radius)
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eccentricity_squared = 2*ellipsoid_flattening-pow(ellipsoid_flattening,2)
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mean_earth_radius = (1/3)*(2*equatorial_radius+polar_radius)
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def central_angle(c1, c2):
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lat1 = radians(c1[0]); lon1 = radians(c1[1])
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lat2 = radians(c2[0]); lon2 = radians(c2[1])
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d_lat = abs(lat1-lat2)
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d_lon = abs(lon1-lon2)
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ca = acos(
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sin(lat1) * sin(lat2) +
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cos(lat1) * cos(lat2) * cos(d_lon)
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)
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return ca
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def geocentric_latitude(geodetic_latitude):
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e2 = eccentricity_squared
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lat = radians(geodetic_latitude)
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return degrees(atan((1.0 - e2) * tan(lat)))
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def geodetic_latitude(geocentric_latitude):
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e2 = eccentricity_squared
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lat = radians(geocentric_latitude)
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return degrees(atan( (1/(1.0 - e2)) * tan(lat)))
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def ellipsoid_radius_at(latitude):
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lat = radians(latitude)
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a = equatorial_radius; b = polar_radius;
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a2 = pow(a,2); b2 = pow(b,2)
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r = sqrt(
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( pow(a2*cos(lat), 2) + pow(b2*sin(lat), 2) )
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/
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( pow(a*cos(lat), 2) + pow(b*sin(lat), 2) )
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)
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return r
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def euclidian_point(latitude, longtitude, altitude=0, ellipsoid=True):
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# Convert latitude and longtitude to radians
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# and get ellipsoid or sphere radius
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lat = radians(latitude); lon = radians(longtitude)
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r = ellipsoid_radius_at(latitude) if ellipsoid else mean_earth_radius
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# Calculate euclidian coordinates from longtitude
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# and geocentric latitude.
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gclat = radians(geocentric_latitude(latitude)) if ellipsoid else lat
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x = cos(lat)*cos(lon)*r
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y = cos(gclat)*sin(lon)*r
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z = sin(gclat)*r
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# Calculate surface normal of ellipsoid at
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# coordinates to add altitude to point
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normal_x = cos(lat)*cos(lon)
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normal_y = cos(lat)*sin(lon)
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normal_z = sin(lat)
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if altitude != 0:
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x += altitude*normal_x
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y += altitude*normal_y
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z += altitude*normal_z
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return (x,y,z)
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def distance(p1, p2):
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dx = p1[0]-p2[0]
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dy = p1[1]-p2[1]
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dz = p1[2]-p2[2]
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return sqrt(dx*dx+dy*dy+dz*dz)
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def euclidian_distance(c1, c2, ellipsoid=True):
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if len(c1) >= 2 and len(c2) >= 2:
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if len(c1) == 2: c1 += (0,)
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if len(c2) == 2: c2 += (0,)
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return distance(
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euclidian_point(c1[0], c1[1], c1[2], ellipsoid=ellipsoid),
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euclidian_point(c2[0], c2[1], c2[2], ellipsoid=ellipsoid)
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)
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else:
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return None
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def spherical_distance(c1, c2, altitude=0, r=mean_earth_radius):
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d = (r+altitude)*central_angle(c1, c2)
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return d
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def ellipsoid_distance(c1, c2):
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# TODO: Update this to the method described by Karney in 2013
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# instead of using Vincenty's algorithm.
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try:
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if c1[0] == 0.0: c1 = (1e-6, c1[1])
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a = equatorial_radius
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f = ellipsoid_flattening
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b = (1 - f)*a # polar radius
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tolerance = 1e-9 # to stop iteration
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phi1, phi2 = radians(c1[0]), radians(c2[0])
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U1 = atan((1-f)*tan(phi1))
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U2 = atan((1-f)*tan(phi2))
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L1, L2 = radians(c1[1]), radians(c2[1])
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L = L2 - L1
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lambda_old = L + 0
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max_iterations = 10000
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iteration = 0
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timeout = 1.0
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st = time.time()
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while True:
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iteration += 1
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t = (cos(U2)*sin(lambda_old))**2
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t += (cos(U1)*sin(U2) - sin(U1)*cos(U2)*cos(lambda_old))**2
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sin_sigma = t**0.5
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cos_sigma = sin(U1)*sin(U2) + cos(U1)*cos(U2)*cos(lambda_old)
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sigma = atan2(sin_sigma, cos_sigma)
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sin_alpha = cos(U1)*cos(U2)*sin(lambda_old) / sin_sigma
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cos_sq_alpha = 1 - sin_alpha**2
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cos_2sigma_m = cos_sigma - 2*sin(U1)*sin(U2)/cos_sq_alpha
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C = f*cos_sq_alpha*(4 + f*(4-3*cos_sq_alpha))/16
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t = sigma + C*sin_sigma*(cos_2sigma_m + C*cos_sigma*(-1 + 2*cos_2sigma_m**2))
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lambda_new = L + (1 - C)*f*sin_alpha*t
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if abs(lambda_new - lambda_old) <= tolerance:
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break
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else:
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lambda_old = lambda_new
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if iteration%1000 == 0:
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if iteration >= max_iterations:
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return None
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if time.time() > st+timeout:
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return None
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u2 = cos_sq_alpha*((a**2 - b**2)/b**2)
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A = 1 + (u2/16384)*(4096 + u2*(-768+u2*(320 - 175*u2)))
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B = (u2/1024)*(256 + u2*(-128 + u2*(74 - 47*u2)))
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t = cos_2sigma_m + 0.25*B*(cos_sigma*(-1 + 2*cos_2sigma_m**2))
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t -= (B/6)*cos_2sigma_m*(-3 + 4*sin_sigma**2)*(-3 + 4*cos_2sigma_m**2)
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delta_sigma = B * sin_sigma * t
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s = b*A*(sigma - delta_sigma)
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return s
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except Exception as e:
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return None
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def orthodromic_distance(c1, c2, spherical=False):
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if spherical:
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return spherical_distance(c1, c2)
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else:
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return ellipsoid_distance(c1, c2)
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# def tests():
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# from geographiclib.geodesic import Geodesic
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# geod = Geodesic.WGS84
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# coords = [
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# [(57.758793, 22.605194), (43.048838, -9.241343)],
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# [(0.0, 0.0), (0.0, 0.0)],
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# [(-90.0, 0.0), (90.0, 0.0)],
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# [(-90.0, 0.0), (78.0, 0.0)],
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# [(0.0, 0.0), (0.5, 179.5)],
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# [(0.7, 0.0), (0.0, -180.0)],
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# ]
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# for cs in coords:
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# c1 = cs[0]; c2 = cs[1]
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# print("Testing: "+str(c1)+" -> "+str(c2))
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# us = time.time()
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# ld = c1+c2; g = geod.Inverse(*ld)
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# print("Lib computed in "+str(round((time.time()-us)*1e6, 3))+"us")
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# us = time.time()
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# eld = orthodromic_distance(c1,c2,spherical=False)
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# if eld:
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# print("Own computed in "+str(round((time.time()-us)*1e6, 3))+"us")
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# else:
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# print("Own TIMED OUT in "+str(round((time.time()-us)*1e6, 3))+"us")
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# print("Euclidian = "+RNS.prettydistance(euclidian_distance(c1,c2)))
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# print("Spherical = "+RNS.prettydistance(orthodromic_distance(c1,c2)))
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# if eld: print("Ellipsoid = "+RNS.prettydistance(eld))
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# print("EllipLib = "+RNS.prettydistance(g['s12']))
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# if eld: print("Diff = "+RNS.prettydistance(g['s12']-eld))
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# print("")
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