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 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 find_c_point_from_precalc(to_point, point1, point2, c12, p1h, p2h, dp1p2):
tpn = to_point.nv
ctp = cross(tpn, c12, axis=0)
try:
c = unit(cross(ctp, c12, axis=0))
except Exception:
print((to_point, point1, point2))
raise
sutable_c = None
for co in (c, 0 - c):
co_rs = co.reshape((3, ))
dp1co = arccos(arccos_limit(dot(p1h, co_rs)))
dp2co = arccos(arccos_limit(dot(p2h, co_rs)))
if abs(dp1co + dp2co - dp1p2) < 0.000001:
sutable_c = co
break
if sutable_c is not None:
c_point_lat, c_point_lng = n_E2lat_lon(sutable_c)
c_point = Point(lat=rad2deg(c_point_lat[0]),
lng=rad2deg(c_point_lng[0]))
c_dist = distance(to_point, c_point)
else:
c_dist, c_point = min(
((distance(to_point, p), p) for p in (point1, point2)),
key=itemgetter(0))
return find_c_point_result(c_dist, c_point)
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 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 find_closest_point_pair(points,
to_point,
req_min_dist=20,
stop_after_dist=50):
tpn = to_point.nv
min_distance = None
min_point_pair = None
min_c_point = None
for point1, point2 in pairs(points):
p1 = point1.nv
p2 = point2.nv
c12 = cross(p1, p2, axis=0)
ctp = cross(tpn, c12, axis=0)
c = unit(cross(ctp, c12, axis=0))
p1h = p1.reshape((3, ))
p2h = p2.reshape((3, ))
dp1p2 = arccos(dot(p1h, p2h))
sutable_c = None
for co in (c, 0 - c):
co_rs = co.reshape((3, ))
dp1co = arccos(dot(p1h, co_rs))
dp2co = arccos(dot(p2h, co_rs))
if abs(dp1co + dp2co - dp1p2) < 0.000001:
sutable_c = co
break
if sutable_c is not None:
c_point_lat, c_point_lng = n_E2lat_lon(sutable_c)
c_point = Point(lat=rad2deg(c_point_lat[0]),
lng=rad2deg(c_point_lng[0]))
c_dist = distance(to_point, c_point)
else:
c_dist, c_point = min(
((distance(to_point, p), p) for p in (point1, point2)))
if min_distance is None or c_dist < min_distance:
min_distance = c_dist
min_point_pair = (point1, point2)
min_c_point = c_point
if min_distance < req_min_dist and c_dist > stop_after_dist:
break
return min_point_pair, min_c_point, min_distance
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 from_nv(cls, nv, round_digits=6):
lat, lng = n_E2lat_lon(nv)
point = cls(round(rad2deg(lat[0]), round_digits),
round(rad2deg(lng[0]), round_digits))
point._nv = nv
return point
def to_lla(self):
lat, lon = nv.n_E2lat_lon(self.n_EB_E)
return Lla(np.rad2deg(lat[0]), np.rad2deg(lon[0]), -self.depth)
def pvec_b_to_lla(forward, right, down, roll, pitch, yaw, lat, lon, alt):
"""
returns the lat lon and alt corresponding to a p-vec in the UAV's body frame
Parameters
----------
forward: float
The number of meters forward of the UAV
right: float
The number of meters to the right of the UAV
down: float
The number of meters below the UAV
roll: float
The UAV's roll angle in degrees
pitch: float
The UAV's pitch angle in degrees
yaw: float
The UAV's yaw angle in degrees
lat: float
The UAV's latitude in degrees
lon: float
The UAV's longitude in degrees
alt: float
The UAV's altitude in meters
Returns
-------
list
This list holds three floats representing the latitude in degrees, longitude in degrees and altitude in meters (in that order).
"""
# create a p-vector with the forward, right and down values
p_B = np.array([forward, right, down])
# this matrix can transform a pvec in the body frame to a pvec in the NED frame
rot_NB = nv.zyx2R(radians(yaw), radians(pitch), radians(roll))
# calculate the pvec in the NED frame
p_N = rot_NB.dot(p_B)
# create an n-vector for the UAV
n_UAV = nv.lat_lon2n_E(radians(lat), radians(lon))
# this creates a matrix that rotates pvecs from NED to ECEF
rot_EN = nv.n_E2R_EN(n_UAV)
# find the offset vector from the UAV to the point of interest in the ECEF frame
p_delta_E = rot_EN.dot(p_N)
# find the p-vector for the UAV in the ECEF frame
p_EUAV_E = nv.n_EB_E2p_EB_E(n_UAV, -alt).reshape(1, 3)[0]
# find the p-vector for the point of interest. This is the UAV + the offset in the ECEF frame.
p_E = p_EUAV_E + p_delta_E
# find the n-vector for the point of interest given the p-vector in the ECEF frame.
n_result, z_result = nv.p_EB_E2n_EB_E(p_E.reshape(3, 1))
# convert the n-vector to a lat and lon
lat_result, lon_result = nv.n_E2lat_lon(n_result)
lat_result, lon_result = degrees(lat_result), degrees(lon_result)
# convert depth to alt
alt_result = -z_result[0]
return [lat_result, lon_result, alt_result]
def getMiddlePointFromList(positions):
n_EM_E = nv.unit(sum(positions))
lat, lon = nv.n_E2lat_lon(n_EM_E)
lat, lon = degrees(lat[0]), degrees(lon[0])
return lat, lon