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_array_almost_equal(deg(lat_EB), 79.99154867)
assert_array_almost_equal(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_array_almost_equal(azimuth, azimuth1 + 2 * np.pi)
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_array_almost_equal(deg(lat_EC), 40.31864307)
assert_array_almost_equal(deg(long_EC), 55.90186788)
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_array_almost_equal(deg(lat_EB_ti), 89.7999805)
assert_array_almost_equal(deg(long_EB_ti), 180.)
def plot_mean_position():
"""
Example
-------
>>> plot_mean_position()
Ex7, Average lat=[ 67.23615295], lon=[-6.91751117]
"""
positions = np.array([
(90, 0),
(60, 10),
(50, -20),
])
lats, lons = positions.T
nvecs = lat_lon2n_E(rad(lats), rad(lons))
# Find the horizontal mean position:
n_EM_E = unit(np.sum(nvecs, axis=1).reshape((3, 1)))
lat, lon = n_E2lat_lon(n_EM_E)
lat, lon = deg(lat), deg(lon)
print(('Ex7, Average lat={0}, lon={1}'.format(lat, lon)))
map1 = Basemap(projection='ortho',
lat_0=int(lat),
lon_0=int(lon),
resolution='l')
plot_world(map1)
x, y = map1(lon, lat)
map1.scatter(x, y, linewidth=5, marker='o', color='r')
x1, y1 = map1(lons, lats)
# print(len(lons), x1, y1)
map1.scatter(x1, y1, linewidth=5, marker='o', color='k')
plt.title('Figure of mean position (red dot) compared to positions '
'A, B, and C (black dots).')
def plot_mean_position():
positions = np.array([(90, 0),
(60, 10),
(50, -20),
])
lats, lons = positions.T
nvecs = lat_lon2n_E(rad(lats), rad(lons))
# Find the horizontal mean position:
n_EM_E = unit(np.sum(nvecs, axis=1).reshape((3, 1)))
lat, lon = n_E2lat_lon(n_EM_E)
lat, lon = deg(lat), deg(lon)
print('Ex7, Average lat={0}, lon={1}'.format(lat, lon))
map1 = Basemap(projection='ortho', lat_0=int(lat), lon_0=int(lon),
resolution='l')
plot_world(map1)
x, y = map1(lon, lat)
map1.scatter(x, y, linewidth=5, marker='o', color='r')
x1, y1 = map1(lons, lats)
print(len(lons), x1, y1)
map1.scatter(x1, y1, linewidth=5, marker='o', color='k')
plt.title('Figure of mean position (red dot) compared to positions '
'A, B, and C (black dots).')
def test_Ex9_intersection():
# 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_array_almost_equal(deg(lat_EC), 40.31864307)
assert_array_almost_equal(deg(long_EC), 55.90186788)
def plot_mean_position():
"""
Example
-------
>>> plot_mean_position()
Ex7, Average lat=67.2, lon=-6.9
>>> plt.show() # doctest: +SKIP
>>> plt.close()
"""
positions = np.array([
(90, 0),
(60, 10),
(50, -20),
])
lats, lons = np.transpose(positions)
nvecs = lat_lon2n_E(rad(lats), rad(lons))
# Find the horizontal mean position:
n_EM_E = unit(np.sum(nvecs, axis=1).reshape((3, 1)))
lat, lon = n_E2lat_lon(n_EM_E)
lat, lon = deg(lat), deg(lon)
print('Ex7, Average lat={0:2.1f}, lon={1:2.1f}'.format(lat[0], lon[0]))
plotter = _init_plotter(lat, lon)
plotter(lon, lat, linewidth=5, marker='o', color='r')
plotter(lons, lats, linewidth=5, marker='o', color='k')
plt.title('Figure of mean position (red dot) compared to \npositions '
'A, B, and C (black dots).')
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_array_almost_equal(deg(lat_EB_ti), 89.7999805)
assert_array_almost_equal(deg(long_EB_ti), 180.)
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_array_almost_equal(deg(lat_EB), 79.99154867)
assert_array_almost_equal(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_array_almost_equal(azimuth, azimuth1+2*np.pi)
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_array_almost_equal(p_AB_N[0], 331730.23478089)
assert_array_almost_equal(p_AB_N[1], 332997.87498927)
assert_array_almost_equal(p_AB_N[2], 17404.27136194)
assert_array_almost_equal(deg(azimuth), 45.10926324)
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_array_almost_equal(p_AB_N[0], 331730.23478089)
assert_array_almost_equal(p_AB_N[1], 332997.87498927)
assert_array_almost_equal(p_AB_N[2], 17404.27136194)
assert_array_almost_equal(deg(azimuth), 45.10926324)
def test_Ex3_ECEF_vector_to_geodetic_latitude():
# Position B is given as p_EB_E ("ECEF-vector")
p_EB_E = 6371e3 * np.vstack((0.9, -1, 1.1)) # m
# Find position B as geodetic latitude, longitude and height
# SOLUTION:
# Find n-vector from the p-vector:
n_EB_E, z_EB = p_EB_E2n_EB_E(p_EB_E)
# Convert to lat, long and height:
lat_EB, long_EB = n_E2lat_lon(n_EB_E)
h_EB = -z_EB
msg = 'Ex3, Pos B: lat, long = {} {} deg, height = {} m'
print(msg.format(deg(lat_EB), deg(long_EB), h_EB))
assert_array_almost_equal(deg(lat_EB), 39.37874867)
assert_array_almost_equal(deg(long_EB), -48.0127875)
assert_array_almost_equal(h_EB, 4702059.83429485)
def test_Ex3_ECEF_vector_to_geodetic_latitude():
# Position B is given as p_EB_E ("ECEF-vector")
p_EB_E = 6371e3 * np.vstack((0.9, -1, 1.1)) # m
# Find position B as geodetic latitude, longitude and height
# SOLUTION:
# Find n-vector from the p-vector:
n_EB_E, z_EB = p_EB_E2n_EB_E(p_EB_E)
# Convert to lat, long and height:
lat_EB, long_EB = n_E2lat_lon(n_EB_E)
h_EB = -z_EB
msg = 'Ex3, Pos B: lat, long = {} {} deg, height = {} m'
print(msg.format(deg(lat_EB), deg(long_EB), h_EB))
assert_array_almost_equal(deg(lat_EB), 39.37874867)
assert_array_almost_equal(deg(long_EB), -48.0127875)
assert_array_almost_equal(h_EB, 4702059.83429485)
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))
# 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_array_almost_equal(deg(lat_EC), 53.32637826)
assert_array_almost_equal(deg(long_EC), 63.46812344)
assert_array_almost_equal(z_EC, -406.00719607)
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))
# 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_array_almost_equal(deg(lat_EC), 53.32637826)
assert_array_almost_equal(deg(long_EC), 63.46812344)
assert_array_almost_equal(z_EC, -406.00719607)
def getNEDElevationAngle (self, other_point):
"""
Returns north-east-down elevation angle to other point, measured from tangential plane to other
(self as NED origin).
Parameters
----------
other_point: GeoPosition
containing other position
Returns
-------
float
elevation angle(pointing up) in degrees
"""
assert isinstance(other_point, GeoPosition)
d_r2g = self.__geopoint.delta_to(other_point.__geopoint) # type: nv.Pvector
return nv.deg(-d_r2g.elevation[0])
def getNEDAzimuth (self, other_point):
"""
Returns north-east-down azimuth to other point, measured from north to other (self as NED origin).
Parameters
----------
other_point: GeoPosition
containing other position
Returns
-------
float
azimuth in degrees
"""
assert isinstance(other_point, GeoPosition)
d_r2g = self.__geopoint.delta_to(other_point.__geopoint) # type: nv.Pvector
return nv.deg(d_r2g.azimuth[0])
def bearing(self,gridB):
_instanceTypeCheck(gridB,Grid)
_dist, azia, _azib = self.__point.distance_and_azimuth(gridB.__point)
return deg(azia)
def test_distance_and_azimuth(self):
wgs84 = FrameE(name='WGS84')
point1 = wgs84.GeoPoint(latitude=-30, longitude=0, degrees=True)
point2 = wgs84.GeoPoint(latitude=29.9, longitude=179.8, degrees=True)
s_12, azi1, azi2 = point1.distance_and_azimuth(point2)
n_a = point1.to_nvector()
n_b = point2.to_nvector()
s_ab, azia, azib = nv.geodesic_distance(n_a.normal, n_b.normal,
wgs84.a, wgs84.f)
assert_allclose(s_12, 19989832.82761)
point1 = wgs84.GeoPoint(latitude=0, longitude=0, degrees=True)
point2 = wgs84.GeoPoint(latitude=0.5, longitude=179.5, degrees=True)
s_12, azi1, azi2 = point1.distance_and_azimuth(point2)
assert_allclose(s_12, 19936288.578965)
n_a = point1.to_nvector()
n_b = point2.to_nvector()
s_ab, azia, azib = nv.geodesic_distance(n_a.normal, n_b.normal,
wgs84.a, wgs84.f)
assert_allclose(s_ab, 19936288.578965)
point1 = wgs84.GeoPoint(latitude=88, longitude=0, degrees=True)
point2 = wgs84.GeoPoint(latitude=89, longitude=-170, degrees=True)
s_12, azi1, azi2 = point1.distance_and_azimuth(point2)
assert_allclose(s_12, 333947.509468)
n_a = point1.to_nvector()
n_b = point2.to_nvector()
s_ab, azia, azib = nv.geodesic_distance(n_a.normal, n_b.normal,
wgs84.a, wgs84.f)
# n_EA_E = nv.lat_lon2n_E(0,0)
# n_EB_E = nv.lat_lon2n_E(*nv.rad(0.5, 179.5))
# np.allclose(nv.geodesic_distance(n_EA_E, n_EB_E), 19909099.44101977)
assert_allclose(nv.deg(azi1, azi2),
(-3.3309161604062467, -173.327884597742))
p3, azib = point1.displace(s_12, azi1)
assert_allclose(nv.deg(azib), -173.327884597742)
assert_allclose(p3.latlon_deg, (89, -170, 0))
p4, azia = point2.displace(s_12, azi2 + np.pi)
assert_allclose(nv.deg(azia), -3.3309161604062467 + 180)
truth = (88, 0, 0)
assert_allclose(p4.latlon_deg, truth, atol=1e-12) # pylint: disable=redundant-keyword-arg
# ------ greatcircle --------
s_12, azi1, azi2 = point1.distance_and_azimuth(point2,
method='greatcircle')
assert_allclose(s_12, 331713.817039)
assert_allclose(nv.deg(azi1, azi2), (-3.330916, -173.327885))
p3, azib = point1.displace(s_12, azi1, method='greatcircle')
assert_allclose(nv.deg(azib), -173.32784)
assert_allclose(p3.latlon_deg, (89.000005, -169.999949, 0))
p4, azia = point2.displace(s_12, azi2 + np.pi, method='greatcircle')
_assert_allclose(p4.latlon_deg, truth,
atol=1e-4) # Less than 0.4 meters
assert_allclose(nv.deg(azia), -3.3309161604062467 + 180)
