def test_Ex8_position_A_and_azimuth_and_distance_to_B(): # Position A is given as n_EA_E: # Enter elements as lat/long in deg: lat, lon = rad(80), rad(-90) n_EA_E = lat_lon2n_E(lat, lon) # The initial azimuth and great circle distance (s_AB), and Earth # radius (r_Earth) are also given: azimuth = rad(200) s_AB = 1000 # m r_Earth = 6371e3 # m, mean Earth radius # Find the destination point B, as n_EB_E ("The direct/first geodetic # problem" for a sphere) # SOLUTION: # Step1: Convert distance in meter into distance in [rad]: distance_rad = s_AB / r_Earth # Step2: Find n_EB_E: n_EB_E = n_EA_E_distance_and_azimuth2n_EB_E(n_EA_E, distance_rad, azimuth) # When displaying the resulting position for humans, it is more # convenient to see lat, long: lat_EB, long_EB = n_E2lat_lon(n_EB_E) print('Ex8, Destination: lat, long = {0} {1} deg'.format( deg(lat_EB), deg(long_EB))) assert_allclose(deg(lat_EB), 79.99154867) assert_allclose(deg(long_EB), -90.01769837) azimuth1 = n_EA_E_and_n_EB_E2azimuth(n_EA_E, n_EB_E, a=r_Earth, f=0) assert_allclose(azimuth, azimuth1 + 2 * np.pi)
def test_Ex6_interpolated_position(): # Position B at time t0 and t2 is given as n_EB_E_t0 and n_EB_E_t1: # Enter elements as lat/long in deg: n_EB_E_t0 = lat_lon2n_E(rad(89), rad(0)) n_EB_E_t1 = lat_lon2n_E(rad(89), rad(180)) # The times are given as: t0 = 10 t1 = 20 ti = 16 # time of interpolation # Find the interpolated position at time ti, n_EB_E_ti # SOLUTION: # Using standard interpolation: n_EB_E_ti = unit(n_EB_E_t0 + (ti - t0) * (n_EB_E_t1 - n_EB_E_t0) / (t1 - t0)) # When displaying the resulting position for humans, it is more # convenient to see lat, long: lat_EB_ti, long_EB_ti = n_E2lat_lon(n_EB_E_ti) msg = 'Ex6, Interpolated position: lat, long = {} {} deg' print(msg.format(deg(lat_EB_ti), deg(long_EB_ti))) assert_allclose(deg(lat_EB_ti), 89.7999805) assert_allclose(deg(long_EB_ti), 180.)
def test_Ex9_intersect(): # Two paths A and B are given by two pairs of positions: # Enter elements as lat/long in deg: n_EA1_E = lat_lon2n_E(rad(10), rad(20)) n_EA2_E = lat_lon2n_E(rad(30), rad(40)) n_EB1_E = lat_lon2n_E(rad(50), rad(60)) n_EB2_E = lat_lon2n_E(rad(70), rad(80)) # Find the intersection between the two paths, n_EC_E: n_EC_E_tmp = unit( np.cross(np.cross(n_EA1_E, n_EA2_E, axis=0), np.cross(n_EB1_E, n_EB2_E, axis=0), axis=0)) # n_EC_E_tmp is one of two solutions, the other is -n_EC_E_tmp. Select # the one that is closet to n_EA1_E, by selecting sign from the dot # product between n_EC_E_tmp and n_EA1_E: n_EC_E = np.sign(np.dot(n_EC_E_tmp.T, n_EA1_E)) * n_EC_E_tmp # When displaying the resulting position for humans, it is more # convenient to see lat, long: lat_EC, long_EC = n_E2lat_lon(n_EC_E) msg = 'Ex9, Intersection: lat, long = {} {} deg' print(msg.format(deg(lat_EC), deg(long_EC))) assert_allclose(deg(lat_EC), 40.31864307) assert_allclose(deg(long_EC), 55.90186788)
def test_R2zxy_z90(): x, y, z = rad((0, 0, 90)) R_AB1 = zyx2R(z, y, x) R_AB = [[0.0, -1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]] assert_allclose(R_AB, R_AB1) z1, y1, x1 = R2zyx(R_AB1) assert_allclose((x, y, z), (x1, y1, z1))
def test_R2zxy_0(): x, y, z = rad((0, 0, 0)) R_AB1 = zyx2R(z, y, x) # print(R_AB1.tolist()) R_AB = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]] assert_allclose(R_AB, R_AB1) z1, y1, x1 = R2zyx(R_AB1) assert_allclose((x, y, z), (x1, y1, z1))
def test_R2xyz(): x, y, z = rad((10, 20, 30)) R_AB1 = xyz2R(x, y, z) R_AB = [[0.81379768, -0.46984631, 0.34202014], [0.54383814, 0.82317294, -0.16317591], [-0.20487413, 0.31879578, 0.92541658]] assert_allclose(R_AB, R_AB1) x1, y1, z1 = R2xyz(R_AB1) assert_allclose((x, y, z), (x1, y1, z1))
def test_Ex5_great_circle_distance(): # Position A and B are given as n_EA_E and n_EB_E: # Enter elements as lat/long in deg: n_EA_E = lat_lon2n_E(rad(88), rad(0)) n_EB_E = lat_lon2n_E(rad(89), rad(-170)) r_Earth = 6371e3 # m, mean Earth radius # SOLUTION: s_AB = great_circle_distance(n_EA_E, n_EB_E, radius=r_Earth) d_AB = euclidean_distance(n_EA_E, n_EB_E, radius=r_Earth) msg = 'Ex5, Great circle distance = {} km, Euclidean distance = {} km' print(msg.format(s_AB / 1000, d_AB / 1000)) assert_allclose(s_AB / 1000, 332.45644411) assert_allclose(d_AB / 1000, 332.41872486)
def test_R2xyz_with_vectors(): x, y, z = rad(((10, 10), (20, 20), (30, 30))) R_AB1 = xyz2R(x, y, z) R_AB = np.array([[0.81379768, -0.46984631, 0.34202014], [0.54383814, 0.82317294, -0.16317591], [-0.20487413, 0.31879578, 0.92541658]])[:, :, None] R_AB = np.concatenate((R_AB, R_AB), axis=2) assert_allclose(R_AB, R_AB1) x1, y1, z1 = R2xyz(R_AB1) assert_allclose((x, y, z), (x1, y1, z1))
def test_Ex1_A_and_B_to_delta_in_frame_N(): # Positions A and B are given in (decimal) degrees and depths: lat_EA, lon_EA, z_EA = rad(1), rad(2), 3 lat_EB, lon_EB, z_EB = rad(4), rad(5), 6 # Find the exact vector between the two positions, given in meters # north, east, and down, i.e. find p_AB_N. # SOLUTION: # Step1: Convert to n-vectors (rad() converts to radians): n_EA_E = lat_lon2n_E(lat_EA, lon_EA) n_EB_E = lat_lon2n_E(lat_EB, lon_EB) # Step2: Find p_AB_E (delta decomposed in E). # WGS-84 ellipsoid is default: p_AB_E = n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA, z_EB) # Step3: Find R_EN for position A: R_EN = n_E2R_EN(n_EA_E) # Step4: Find p_AB_N p_AB_N = np.dot(R_EN.T, p_AB_E) # (Note the transpose of R_EN: The "closest-rule" says that when # decomposing, the frame in the subscript of the rotation matrix that # is closest to the vector, should equal the frame where the vector is # decomposed. Thus the calculation np.dot(R_NE, p_AB_E) is correct, # since the vector is decomposed in E, and E is closest to the vector. # In the example we only had R_EN, and thus we must transpose it: # R_EN'=R_NE) # Step5: Also find the direction (azimuth) to B, relative to north: azimuth = np.arctan2(p_AB_N[1], p_AB_N[0]) # positive angle about down-axis print('Ex1, delta north, east, down = {0}, {1}, {2}'.format( p_AB_N[0], p_AB_N[1], p_AB_N[2])) print('Ex1, azimuth = {0} deg'.format(deg(azimuth))) assert_allclose(p_AB_N[0], 331730.23478089) assert_allclose(p_AB_N[1], 332997.87498927) assert_allclose(p_AB_N[2], 17404.27136194) assert_allclose(deg(azimuth), 45.10926324)
def test_Ex4_geodetic_latitude_to_ECEF_vector(): # Position B is given with lat, long and height: lat_EB_deg = 1 long_EB_deg = 2 h_EB = 3 # Find the vector p_EB_E ("ECEF-vector") # SOLUTION: # Step1: Convert to n-vector: n_EB_E = lat_lon2n_E(rad(lat_EB_deg), rad(long_EB_deg)) # Step2: Find the ECEF-vector p_EB_E: p_EB_E = n_EB_E2p_EB_E(n_EB_E, -h_EB) print('Ex4: p_EB_E = {0} m'.format(p_EB_E.ravel())) assert_allclose(p_EB_E.ravel(), [6373290.27721828, 222560.20067474, 110568.82718179])
def test_R2zxy(): x, y, z = rad((10, 20, 30)) R_AB1 = zyx2R(z, y, x) R_AB = [[0.8137976813493738, -0.44096961052988237, 0.37852230636979245], [0.46984631039295416, 0.8825641192593856, 0.01802831123629725], [-0.3420201433256687, 0.16317591116653482, 0.9254165783983234]] assert_allclose(R_AB, R_AB1) z1, y1, x1 = R2zyx(R_AB1) assert_allclose((x, y, z), (x1, y1, z1))
def test_Ex10_cross_track_distance(): # Position A1 and A2 and B are given as n_EA1_E, n_EA2_E, and n_EB_E: # Enter elements as lat/long in deg: n_EA1_E = lat_lon2n_E(rad(0), rad(0)) n_EA2_E = lat_lon2n_E(rad(10), rad(0)) n_EB_E = lat_lon2n_E(rad(1), rad(0.1)) radius = 6371e3 # m, mean Earth radius # Find the cross track distance from path A to position B. # SOLUTION: # Find the unit normal to the great circle: c_E = unit(np.cross(n_EA1_E, n_EA2_E, axis=0)) # Find the great circle cross track distance: sin_theta = -np.dot(c_E.T, n_EB_E) # pylint: disable=invalid-unary-operand-type s_xt = np.arcsin(sin_theta) * radius # ill conditioned for small angles: # s_xt2 = (np.arccos(-sin_theta) - np.pi / 2) * radius # Find the Euclidean cross track distance: d_xt = sin_theta * radius msg = 'Ex10, Cross track distance = {} m, Euclidean = {} m' print(msg.format(s_xt, d_xt)) assert_allclose(s_xt, 11117.79911015) assert_allclose(d_xt, 11117.79346741)
def test_Ex2_B_and_delta_in_frame_B_to_C_in_frame_E(): # delta vector from B to C, decomposed in B is given: p_BC_B = np.r_[3000, 2000, 100].reshape((-1, 1)) # pylint: disable=too-many-function-args # Position and orientation of B is given: n_EB_E = unit([[1], [2], [3]]) # unit to get unit length of vector z_EB = -400 R_NB = zyx2R(rad(10), rad(20), rad(30)) # the three angles are yaw, pitch, and roll # A custom reference ellipsoid is given (replacing WGS-84): a, f = 6378135, 1.0 / 298.26 # (WGS-72) # Find the position of C. # SOLUTION: # Step1: Find R_EN: R_EN = n_E2R_EN(n_EB_E) # Step2: Find R_EB, from R_EN and R_NB: R_EB = np.dot(R_EN, R_NB) # Note: closest frames cancel # Step3: Decompose the delta vector in E: p_BC_E = np.dot(R_EB, p_BC_B) # no transpose of R_EB, since the vector is in B # Step4: Find the position of C, using the functions that goes from one # position and a delta, to a new position: n_EC_E, z_EC = n_EA_E_and_p_AB_E2n_EB_E(n_EB_E, p_BC_E, z_EB, a, f) # When displaying the resulting position for humans, it is more # convenient to see lat, long: lat_EC, long_EC = n_E2lat_lon(n_EC_E) # Here we also assume that the user wants output height (= - depth): msg = 'Ex2, Pos C: lat, long = {},{} deg, height = {} m' print(msg.format(deg(lat_EC), deg(long_EC), -z_EC)) assert_allclose(deg(lat_EC), 53.32637826) assert_allclose(deg(long_EC), 63.46812344) assert_allclose(z_EC, -406.00719607)
def test_Ex7_mean_position(): # Three positions A, B and C are given: # Enter elements as lat/long in deg: n_EA_E = lat_lon2n_E(rad(90), rad(0)) n_EB_E = lat_lon2n_E(rad(60), rad(10)) n_EC_E = lat_lon2n_E(rad(50), rad(-20)) # Find the horizontal mean position: n_EM_E = unit(n_EA_E + n_EB_E + n_EC_E) truth = [0.3841171702926, -0.046602405485689447, 0.9221074857571395] # The result is best viewed with a figure that shows the n-vectors # relative to an Earth-model: # print('Ex7, See figure') # plot_earth_figure(n_EA_E,n_EB_E,n_EC_E,n_EM_E) # print(n_EM_E.ravel().tolist()) assert_allclose(n_EM_E.ravel(), truth) # Alternatively: n_EM_E = mean_horizontal_position(np.hstack((n_EA_E, n_EB_E, n_EC_E))) # print(n_EM_E.ravel()) assert_allclose(n_EM_E.ravel(), truth)
def test_small_and_large_cross_track_distance(): radius = 6371e3 # m, mean Earth radius n_EA1_E = lat_lon2n_E(rad(5), rad(10)) n_EA2_E = lat_lon2n_E(rad(10), rad(10)) n_EB0_E = lat_lon2n_E(rad(7), rad(10.1)) path = (n_EA1_E, n_EA2_E) n_EB1_E = closest_point_on_great_circle(path, n_EB0_E) for s_xt0 in [ np.pi * radius, np.pi / 3 * radius, 10., 0.1, 1e-3, 1e-4, 1e-5, 1e-8 ]: distance_rad = s_xt0 / radius n_EB_E = n_EA_E_distance_and_azimuth2n_EB_E(n_EB1_E, distance_rad, np.pi / 2) n_EB2_E = closest_point_on_great_circle(path, n_EB_E) s_xt = great_circle_distance(n_EB1_E, n_EB_E, radius) c_E = unit(np.cross(n_EA1_E, n_EA2_E, axis=0)) s_xt2 = (np.arccos(np.dot(c_E.T, n_EB_E)) - np.pi / 2) * radius s_xt3 = cross_track_distance(path, n_EB_E, method='greatcircle', radius=radius) # pylint: disable=invalid-unary-operand-type s_xt4 = np.arctan2( -np.dot(c_E.T, n_EB_E), np.linalg.norm(np.cross(c_E, n_EB_E, axis=0), axis=0)) * radius rtol = 10**(-min(9 + np.log10(s_xt0), 15)) if s_xt0 <= np.pi / 3 * radius: assert_allclose(n_EB2_E, n_EB1_E) assert_allclose(s_xt2, s_xt0, rtol=rtol) assert_allclose(s_xt3, s_xt0, rtol=rtol) assert_allclose(s_xt4, s_xt0, rtol=rtol) assert_allclose(s_xt, s_xt0, rtol=rtol)
def test_lat_lon2n_E(): n_E = lat_lon2n_E(rad([0, 90]), [0, 0]) print(n_E.tolist()) assert_allclose(n_E, [[1.0, 0.0], [0.0, 0.0], [0.0, 1.0]])
def test_rad_and_deg(values): radians = rad(values) assert_allclose(deg(radians), values)