"""
This file is part of NavLab and is available from www.navlab.net/nvector
The content of this file is based on the following publication:
Gade, K. (2010). A Nonsingular Horizontal Position Representation, The Journal
of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.
(www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf)
This paper should be cited in publications using this file.
Copyright (c) 2015, Norwegian Defence Research Establishment (FFI)
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
1. Redistributions of source code must retain the above publication
information, copyright notice, this list of conditions and the following
disclaimer.
2. Redistributions in binary form must reproduce the above publication
information, copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials provided with the
distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS
BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
THE POSSIBILITY OF SUCH DAMAGE.
"""
from __future__ import division, print_function
import warnings
import numpy as np
from numpy import rad2deg, deg2rad, arctan2, sin, cos, array, cross, dot, sqrt
from numpy.linalg import norm
from nvector import _examples
from nvector._common import test_docstrings, use_docstring_from
__all__ = ['select_ellipsoid', 'E_rotation',
'unit', 'deg', 'rad', 'nthroot',
'lat_lon2n_E', 'n_E2lat_lon',
'n_EA_E_and_n_EB_E2p_AB_E', 'n_EA_E_and_p_AB_E2n_EB_E',
'p_EB_E2n_EB_E', 'n_EB_E2p_EB_E',
'n_EA_E_distance_and_azimuth2n_EB_E',
'n_EA_E_and_n_EB_E2azimuth',
'great_circle_distance', 'euclidean_distance',
'great_circle_normal', 'cross_track_distance',
'closest_point_on_great_circle', 'on_great_circle',
'on_great_circle_path', 'interpolate',
'intersect', 'mean_horizontal_position',
'R2xyz', 'xyz2R', 'R2zyx', 'zyx2R',
'n_E_and_wa2R_EL', 'n_E2R_EN', 'R_EL2n_E', 'R_EN2n_E']
E_ROTATION_MATRIX = dict(e=array([[0, 0, 1],
[0, 1, 0],
[-1, 0, 0]]),
E=np.eye(3))
_EPS = np.finfo(float).eps # machine precision (machine epsilon)
_TINY = np.finfo(float).tiny
ELLIPSOID = {1: ({'a': 6377563.3960, 'f': 1.0 / 299.3249646}, 'Airy 1858'),
2: ({'a': 6377340.189, 'f': 1.0 / 299.3249646}, 'Airy Modified'),
3: ({'a': 6378160, 'f': 1.0 / 298.25}, 'Australian National'),
4: ({'a': 6377397.155, 'f': 1.0 / 299.1528128}, 'Bessel 1841'),
5: ({'a': 6378249.145, 'f': 1.0 / 293.465}, 'Clarke 1880'),
6: ({'a': 6377276.345, 'f': 1.0 / 300.8017}, 'Everest 1830'),
7: ({'a': 6377304.063, 'f': 1.0 / 300.8017}, 'Everest Modified'),
8: ({'a': 6378166.0, 'f': 1.0 / 298.3}, 'Fisher 1960'),
9: ({'a': 6378150.0, 'f': 1.0 / 298.3}, 'Fisher 1968'),
10: ({'a': 6378270.0, 'f': 1.0 / 297}, 'Hough 1956'),
11: ({'a': 6378388.0, 'f': 1.0 / 297},
'International (Hayford)/European Datum (ED50)'),
12: ({'a': 6378245.0, 'f': 1.0 / 298.3}, 'Krassovsky 1938'),
13: ({'a': 6378145., 'f': 1.0 / 298.25}, 'NWL-9D (WGS 66)'),
14: ({'a': 6378160., 'f': 1.0 / 298.25}, 'South American 1969 (SAD69'),
15: ({'a': 6378136, 'f': 1.0 / 298.257},
'Soviet Geod. System 1985'),
16: ({'a': 6378135., 'f': 1.0 / 298.26}, 'WGS 72'),
17: ({'a': 6378206.4, 'f': 1.0 / 294.9786982138},
'Clarke 1866 (NAD27)'),
18: ({'a': 6378137.0, 'f': 1.0 / 298.257223563},
'GRS80 / WGS84 (NAD83)'),
19: ({'a': 6378137, 'f': 298.257222101}, 'ETRS89')
}
ELLIPSOID_IX = {'airy1858': 1,
'airymodified': 2,
'australiannational': 3,
'bessel': 4,
'bessel1841': 4,
'clarke1880': 5,
'everest1830': 6,
'everestmodified': 7,
'fisher1960': 8,
'fisher1968': 9,
'hough1956': 10,
'hough': 10,
'international': 11,
'hayford': 11,
'ed50': 11,
'krassovsky': 12,
'krassovsky1938': 12,
'nwl-9d': 13,
'wgs66': 13,
'southamerican1969': 14,
'sad69': 14,
'sovietgeod.system1985': 15,
'wgs72': 16,
'clarke1866': 17,
'nad27': 17,
'grs80': 18,
'wgs84': 18,
'nad83': 18,
}
[docs]def select_ellipsoid(name):
"""
Return semi-major axis (a), flattening (f) and name of ellipsoid
Parameters
----------
name : string
name of ellipsoid. Valid options are:
1) Airy 1858
2) Airy Modified
3) Australian National
4) Bessel 1841
5) Clarke 1880
6) Everest 1830
7) Everest Modified
8) Fisher 1960
9) Fisher 1968
10) Hough 1956
11) International (Hayford)/European Datum (ED50)
12) Krassovsky 1938
13) NWL-9D (WGS 66)
14) South American 1969
15) Soviet Geod. System 1985
16) WGS 72
17) Clarke 1866 (NAD27)
18) GRS80 / WGS84 (NAD83)
19) ETRS89
Examples
--------
>>> import nvector as nv
>>> nv.select_ellipsoid(name='wgs84')
(6378137.0, 0.0033528106647474805, 'GRS80 / WGS84 (NAD83)')
>>> nv.select_ellipsoid(name='GRS80')
(6378137.0, 0.0033528106647474805, 'GRS80 / WGS84 (NAD83)')
>>> nv.select_ellipsoid(name='NAD83')
(6378137.0, 0.0033528106647474805, 'GRS80 / WGS84 (NAD83)')
>>> nv.select_ellipsoid(name=18)
(6378137.0, 0.0033528106647474805, 'GRS80 / WGS84 (NAD83)')
"""
if isinstance(name, str):
name = name.lower().replace(' ', '')
ellipsoid_id = ELLIPSOID_IX.get(name, name)
ellipsoid, fullname = ELLIPSOID[ellipsoid_id]
return ellipsoid['a'], ellipsoid['f'], fullname
[docs]def E_rotation(axes='e'):
"""
Return rotation matrix R_Ee defining the axes of the coordinate frame E.
Parameters
----------
axes : 'e' or 'E'
defines orientation of the axes of the coordinate frame E. Options are:
'e': z-axis points to the North Pole along the Earth's rotation axis,
x-axis points towards the point where latitude = longitude = 0.
This choice is very common in many fields.
'E': x-axis points to the North Pole along the Earth's rotation axis,
y-axis points towards longitude +90deg (east) and latitude = 0.
(the yz-plane coincides with the equatorial plane).
This choice of axis ensures that at zero latitude and longitude,
frame N (North-East-Down) has the same orientation as frame E.
If roll/pitch/yaw are zero, also frame B (forward-starboard-down)
has this orientation. In this manner, the axes of frame E is
chosen to correspond with the axes of frame N and B.
The functions in this library originally used this option.
Returns
-------
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E as
described in Table 2 in Gade (2010)
R_Ee controls the axes of the coordinate frame E (Earth-Centred,
Earth-Fixed, ECEF) used by the other functions in this library
Examples
--------
>>> import numpy as np
>>> import nvector as nv
>>> np.allclose(nv.E_rotation(axes='e'), [[ 0, 0, 1],
... [ 0, 1, 0],
... [-1, 0, 0]])
True
>>> np.allclose(nv.E_rotation(axes='E'), [[ 1., 0., 0.],
... [ 0., 1., 0.],
... [ 0., 0., 1.]])
True
Reference
---------
Gade, K. (2010). `A Nonsingular Horizontal Position Representation,
The Journal of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.
<www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>`_
"""
return E_ROTATION_MATRIX[axes]
[docs]def nthroot(x, n):
"""
Return the n'th root of x to machine precision
Parameters
x, n
Examples
--------
>>> import numpy as np
>>> import nvector as nv
>>> np.allclose(nv.nthroot(27.0, 3), 3.0)
True
"""
y = x**(1. / n)
return np.where((x != 0) & (_EPS * np.abs(x) < 1),
y - (y**n - x) / (n * y**(n - 1)), y)
[docs]def deg(rad_angle):
"""
Converts angle in radians to degrees.
Parameters
----------
rad_angle:
angle in radians
Returns
-------
deg_angle:
angle in degrees
See also
--------
rad
"""
return rad2deg(rad_angle)
[docs]def rad(deg_angle):
"""
Converts angle in degrees to radians.
Parameters
----------
deg_angle:
angle in degrees
Returns
-------
rad_angle:
angle in radians
See also
--------
deg
"""
return deg2rad(deg_angle)
[docs]def unit(vector, norm_zero_vector=1):
"""
Convert input vector to a vector of unit length.
Parameters
----------
vector : 3 x m array
m column vectors
Returns
-------
unitvector : 3 x m array
normalized unitvector(s) along axis==0.
Notes
-----
The column vector(s) that have zero length will be returned as unit vector(s)
pointing in the x-direction, i.e, [[1], [0], [0]]
Examples
--------
>>> import numpy as np
>>> import nvector as nv
>>> np.allclose(nv.unit([[1, 0],[1, 0],[1, 0]]), [[ 0.57735027, 1],
... [ 0.57735027, 0],
... [ 0.57735027, 0]])
True
"""
current_norm = norm(vector, axis=0)
idx = np.flatnonzero(current_norm == 0)
unit_vector = vector / (current_norm + _TINY)
unit_vector[:, idx] = 0 * norm_zero_vector
unit_vector[0, idx] = 1 * norm_zero_vector
return unit_vector
def mdot(a, b):
"""
Return multiple matrix multiplications of two arrays
i.e. dot(a, b)[i,j,k] = sum(a[i,:,j] * b[:,j,k])
or
np.concatenate([np.dot(a[...,i], b[...,i])[:, :, None]
for i in range(2)], axis=2)
Examples
--------
3 x 3 x 2 times 3 x 3 x 2 array -> 3 x 2 x 2 array
>>> import numpy as np
>>> a = 1.0 * np.arange(18).reshape(3,3,2)
>>> b = - a
>>> t = np.concatenate([np.dot(a[...,i], b[...,i])[:, :, None]
... for i in range(2)], axis=2)
>>> tt = mdot(a, b)
>>> tt.shape
(3, 3, 2)
>>> np.allclose(t, tt)
True
3 x 3 x 2 times 3 x 1 array -> 3 x 1 x 2 array
>>> t1 = np.concatenate([np.dot(a[...,i], b[:,0,0][:,None])[:,:,None]
... for i in range(2)], axis=2)
>>> tt = mdot(a, b[:,0,0].reshape(-1,1))
>>> tt.shape
(3, 1, 2)
>>> np.allclose(t1, tt)
True
3 x 3 times 3 x 3 array -> 3 x 3 array
>>> tt0 = mdot(a[...,0], b[...,0])
>>> tt0.shape
(3, 3)
>>> np.allclose(t[...,0], tt0)
True
3 x 3 times 3 x 1 array -> 3 x 1 array
>>> tt0 = mdot(a[...,0], b[:,0,0][:,None])
>>> tt0.shape
(3, 1)
>>> np.allclose(t[:,0,0][:,None], tt0)
True
3 x 3 times 3 x 2 array -> 3 x 1 x 2 array
>>> tt0 = mdot(a[..., 0], b[:, :2, 0][:, None])
>>> tt0.shape
(3, 1, 2)
>>> np.allclose(t[:,:2,0][:,None], tt0)
True
"""
return np.einsum('ij...,jk...->ik...', a, b)
[docs]def lat_lon2n_E(latitude, longitude, R_Ee=None):
"""
Converts latitude and longitude to n-vector.
Parameters
----------
latitude, longitude: real scalars or vectors of length n.
Geodetic latitude and longitude given in [rad]
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
n_E: 3 x n array
n-vector(s) [no unit] decomposed in E.
See also
--------
n_E2lat_lon
"""
if R_Ee is None:
R_Ee = E_rotation()
# Equation (3) from Gade (2010):
nvec = np.vstack((sin(latitude),
sin(longitude) * cos(latitude),
-cos(longitude) * cos(latitude)))
n_E = dot(R_Ee.T, nvec)
return n_E
[docs]def n_E2lat_lon(n_E, R_Ee=None):
"""
Converts n-vector to latitude and longitude.
Parameters
----------
n_E: 3 x n array
n-vector [no unit] decomposed in E.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
latitude, longitude: real scalars or vectors of length n.
Geodetic latitude and longitude given in [rad]
See also
--------
lat_lon2n_E
"""
if R_Ee is None:
R_Ee = E_rotation()
_check_length_deviation(n_E)
n_E = dot(R_Ee, n_E)
# Equation (5) in Gade (2010):
longitude = arctan2(n_E[1, :], -n_E[2, :])
# Equation (6) in Gade (2010) (Robust numerical solution)
equatorial_component = sqrt(n_E[1, :]**2 + n_E[2, :]**2)
# vector component in the equatorial plane
latitude = arctan2(n_E[0, :], equatorial_component)
# atan() could also be used since latitude is within [-pi/2,pi/2]
# latitude=asin(n_E[0] is a theoretical solution, but close to the Poles
# it is ill-conditioned which may lead to numerical inaccuracies (and it
# will give imaginary results for norm(n_E)>1)
return latitude, longitude
def _check_length_deviation(n_E, limit=0.1):
"""
n-vector should have length=1, i.e. norm(n_E)=1.
A deviation from 1 exceeding this limit gives a warning.
This function only depends of the direction of n-vector, thus the warning
is included only to give a notice in cases where a wrong input is given
unintentionally (i.e. the input is not even approximately a unit vector).
If a matrix of n-vectors is input, only first is controlled to save time
(assuming advanced users input correct n-vectors)
"""
length_deviation = np.abs(norm(n_E[:, 0]) - 1)
if length_deviation > limit:
warnings.warn('n-vector should have unit length: '
'norm(n_E)~=1 ! Error is: {}'.format(length_deviation))
[docs]def n_E2R_EN(n_E, R_Ee=None):
"""
Returns the rotation matrix R_EN from n-vector.
Parameters
----------
n_E: 3 x n array
n-vector [no unit] decomposed in E
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
R_EN: 3 x 3 x n array
The resulting rotation matrix [no unit] (direction cosine matrix).
See also
--------
R_EN2n_E, n_E_and_wa2R_EL, R_EL2n_E
"""
if R_Ee is None:
R_Ee = E_rotation()
_check_length_deviation(n_E)
n_E = unit(dot(R_Ee, n_E))
# N coordinate frame (North-East-Down) is defined in Table 2 in Gade (2010)
# Find z-axis of N (Nz):
Nz_E = -n_E # z-axis of N (down) points opposite to n-vector
# Find y-axis of N (East)(remember that N is singular at Poles)
# Equation (9) in Gade (2010):
# Ny points perpendicular to the plane
Ny_E_direction = cross([[1], [0], [0]], n_E, axis=0)
# formed by n-vector and Earth's spin axis
on_poles = np.flatnonzero(norm(Ny_E_direction, axis=0) == 0)
Ny_E = unit(Ny_E_direction)
Ny_E[:, on_poles] = array([[0], [1], [0]]) # selected y-axis direction
# Find x-axis of N (North):
Nx_E = cross(Ny_E, Nz_E, axis=0) # Final axis found by right hand rule
# Form R_EN from the unit vectors:
# R_EN = dot(R_Ee.T, np.hstack((Nx_E, Ny_E, Nz_E)))
Nxyz_E = np.hstack((Nx_E[:, None, ...],
Ny_E[:, None, ...],
Nz_E[:, None, ...]))
R_EN = mdot(np.rollaxis(R_Ee, 1, 0), Nxyz_E)
return np.squeeze(R_EN)
[docs]def n_E_and_wa2R_EL(n_E, wander_azimuth, R_Ee=None):
"""
Returns rotation matrix R_EL from n-vector and wander azimuth angle.
R_EL = n_E_and_wa2R_EL(n_E,wander_azimuth) Calculates the rotation matrix
(direction cosine matrix) R_EL using n-vector (n_E) and the wander
azimuth angle.
When wander_azimuth=0, we have that N=L (See Table 2 in Gade (2010) for
details)
Parameters
----------
n_E: 3 x n array
n-vector [no unit] decomposed in E
wander_azimuth: real scalar or array of length n
Angle [rad] between L's x-axis and north, positive about L's z-axis.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
R_EL: 3 x 3 x n array
The resulting rotation matrix. [no unit]
See also
--------
R_EL2n_E, R_EN2n_E, n_E2R_EN
"""
if R_Ee is None:
R_Ee = E_rotation()
latitude, longitude = n_E2lat_lon(n_E, R_Ee)
# Reference: See start of Section 5.2 in Gade (2010):
R_EL = mdot(R_Ee.T, xyz2R(longitude, -latitude, wander_azimuth))
return np.squeeze(R_EL)
class _Nvector2ECEFvector(object):
__doc__ = """
Converts n-vector to Cartesian position vector in meters.
Parameters
----------
n_EB_E: 3 x n array
n-vector(s) [no unit] of position B, decomposed in E.
depth: 1 x n array
Depth(s) [m] of system B, relative to the ellipsoid (depth = -height)
a: real scalar, default WGS-84 ellipsoid.
Semi-major axis of the Earth ellipsoid given in [m].
f: real scalar, default WGS-84 ellipsoid.
Flattening [no unit] of the Earth ellipsoid. If f==0 then spherical
Earth with radius a is used in stead of WGS-84.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
p_EB_E: 3 x n array
Cartesian position vector(s) from E to B, decomposed in E.
Notes
-----
The position of B (typically body) relative to E (typically Earth) is
given into this function as n-vector, n_EB_E. The function converts
to cartesian position vector ("ECEF-vector"), p_EB_E, in meters.
The calculation is excact, taking the ellipsity of the Earth into account.
It is also non-singular as both n-vector and p-vector are non-singular
(except for the center of the Earth).
The default ellipsoid model used is WGS-84, but other ellipsoids/spheres
might be specified.
Examples
--------
{0}
See also
--------
p_EB_E2n_EB_E, n_EA_E_and_p_AB_E2n_EB_E, n_EA_E_and_n_EB_E2p_AB_E
""".format(_examples.get_examples([4], OO=False))
[docs]@use_docstring_from(_Nvector2ECEFvector)
def n_EB_E2p_EB_E(n_EB_E, depth=0, a=6378137, f=1.0 / 298.257223563, R_Ee=None):
if R_Ee is None:
R_Ee = E_rotation()
_check_length_deviation(n_EB_E)
n_EB_E = unit(dot(R_Ee, n_EB_E))
b = a * (1 - f) # semi-minor axis
# The following code implements equation (22) in Gade (2010):
scale = np.vstack((1,
(1 - f),
(1 - f)))
denominator = norm(n_EB_E / scale, axis=0)
# We first calculate the position at the origin of coordinate system L,
# which has the same n-vector as B (n_EL_E = n_EB_E),
# but lies at the surface of the Earth (z_EL = 0).
p_EL_E = b / denominator * n_EB_E / scale**2
p_EB_E = dot(R_Ee.T, p_EL_E - n_EB_E * depth)
return p_EB_E
class _ECEFvector2Nvector(object):
__doc__ = """
Converts Cartesian position vector in meters to n-vector.
Parameters
----------
p_EB_E: 3 x n array
Cartesian position vector(s) from E to B, decomposed in E.
a: real scalar, default WGS-84 ellipsoid.
Semi-major axis of the Earth ellipsoid given in [m].
f: real scalar, default WGS-84 ellipsoid.
Flattening [no unit] of the Earth ellipsoid. If f==0 then spherical
Earth with radius a is used in stead of WGS-84.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
n_EB_E: 3 x n array
n-vector(s) [no unit] of position B, decomposed in E.
depth: 1 x n array
Depth(s) [m] of system B, relative to the ellipsoid (depth = -height)
Notes
-----
The position of B (typically body) relative to E (typically Earth) is
given into this function as cartesian position vector p_EB_E, in meters.
("ECEF-vector"). The function converts to n-vector, n_EB_E and its
depth, depth.
The calculation is excact, taking the ellipsity of the Earth into account.
It is also non-singular as both n-vector and p-vector are non-singular
(except for the center of the Earth).
The default ellipsoid model used is WGS-84, but other ellipsoids/spheres
might be specified.
Examples
--------
{0}
See also
--------
n_EB_E2p_EB_E, n_EA_E_and_p_AB_E2n_EB_E, n_EA_E_and_n_EB_E2p_AB_E
""".format(_examples.get_examples([3], OO=False))
[docs]@use_docstring_from(_ECEFvector2Nvector)
def p_EB_E2n_EB_E(p_EB_E, a=6378137, f=1.0 / 298.257223563, R_Ee=None):
if R_Ee is None:
# R_Ee selects correct E-axes, see E_rotation for details
R_Ee = E_rotation()
p_EB_E = dot(R_Ee, p_EB_E)
# e_2 = eccentricity**2
e_2 = 2 * f - f**2 # = 1-b**2/a**2
# The following code implements equation (23) from Gade (2010):
R_2 = p_EB_E[1, :]**2 + p_EB_E[2, :]**2
R = sqrt(R_2) # R = component of p_EB_E in the equatorial plane
p = R_2 / a**2
q = (1 - e_2) / (a**2) * p_EB_E[0, :]**2
r = (p + q - e_2**2) / 6
s = e_2**2 * p * q / (4 * r**3)
t = nthroot((1 + s + sqrt(s * (2 + s))), 3)
# t = (1 + s + sqrt(s * (2 + s)))**(1. / 3)
u = r * (1 + t + 1. / t)
v = sqrt(u**2 + e_2**2 * q)
w = e_2 * (u + v - q) / (2 * v)
k = sqrt(u + v + w**2) - w
d = k * R / (k + e_2)
temp0 = sqrt(d**2 + p_EB_E[0, :]**2)
# Calculate height:
height = (k + e_2 - 1) / k * temp0
temp1 = 1. / temp0
temp2 = temp1 * k / (k + e_2)
n_EB_E_x = temp1 * p_EB_E[0, :]
n_EB_E_y = temp2 * p_EB_E[1, :]
n_EB_E_z = temp2 * p_EB_E[2, :]
n_EB_E = np.vstack((n_EB_E_x, n_EB_E_y, n_EB_E_z))
n_EB_E = unit(dot(R_Ee.T, n_EB_E)) # Ensure unit length
depth = -height
return n_EB_E, depth
class _DeltaFromPositionAtoB(object):
__doc__ = """
Return the delta vector from position A to B.
Parameters
----------
n_EA_E, n_EB_E: 3 x n array
n-vector(s) [no unit] of position A and B, decomposed in E.
z_EA, z_EB: 1 x n array
Depth(s) [m] of system A and B, relative to the ellipsoid.
(z_EA = -height, z_EB = -height)
a: real scalar, default WGS-84 ellipsoid.
Semi-major axis of the Earth ellipsoid given in [m].
f: real scalar, default WGS-84 ellipsoid.
Flattening [no unit] of the Earth ellipsoid. If f==0 then spherical
Earth with radius a is used in stead of WGS-84.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
p_AB_E: 3 x n array
Cartesian position vector(s) from A to B, decomposed in E.
Notes
-----
The n-vectors for positions A (n_EA_E) and B (n_EB_E) are given. The
output is the delta vector from A to B (p_AB_E).
The calculation is excact, taking the ellipsity of the Earth into account.
It is also non-singular as both n-vector and p-vector are non-singular
(except for the center of the Earth).
The default ellipsoid model used is WGS-84, but other ellipsoids/spheres
might be specified.
Examples
--------
{0}
See also
--------
n_EA_E_and_p_AB_E2n_EB_E, p_EB_E2n_EB_E, n_EB_E2p_EB_E
""".format(_examples.get_examples([1], False))
[docs]@use_docstring_from(_DeltaFromPositionAtoB)
def n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA=0, z_EB=0, a=6378137,
f=1.0 / 298.257223563, R_Ee=None):
# Function 1. in Section 5.4 in Gade (2010):
p_EA_E = n_EB_E2p_EB_E(n_EA_E, z_EA, a, f, R_Ee)
p_EB_E = n_EB_E2p_EB_E(n_EB_E, z_EB, a, f, R_Ee)
p_AB_E = p_EB_E - p_EA_E
return p_AB_E
[docs]def n_EA_E_and_p_AB_E2n_EB_E(n_EA_E, p_AB_E, z_EA=0, a=6378137,
f=1.0 / 298.257223563, R_Ee=None):
"""
Return position B from position A and delta.
Parameters
----------
n_EA_E: 3 x n array
n-vector(s) [no unit] of position A, decomposed in E.
p_AB_E: 3 x n array
Cartesian position vector(s) from A to B, decomposed in E.
z_EA: 1 x n array
Depth(s) [m] of system A, relative to the ellipsoid. (z_EA = -height)
a: real scalar, default WGS-84 ellipsoid.
Semi-major axis of the Earth ellipsoid given in [m].
f: real scalar, default WGS-84 ellipsoid.
Flattening [no unit] of the Earth ellipsoid. If f==0 then spherical
Earth with radius a is used in stead of WGS-84.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
n_EB_E: 3 x n array
n-vector(s) [no unit] of position B, decomposed in E.
z_EB: 1 x n array
Depth(s) [m] of system B, relative to the ellipsoid.
(z_EB = -height)
Notes
-----
The n-vector for position A (n_EA_E) and the position-vector from position
A to position B (p_AB_E) are given. The output is the n-vector of position
B (n_EB_E) and depth of B (z_EB).
The calculation is excact, taking the ellipsity of the Earth into account.
It is also non-singular as both n-vector and p-vector are non-singular
(except for the center of the Earth).
The default ellipsoid model used is WGS-84, but other ellipsoids/spheres
might be specified.
See also
--------
n_EA_E_and_n_EB_E2p_AB_E, p_EB_E2n_EB_E, n_EB_E2p_EB_E
"""
if R_Ee is None:
R_Ee = E_rotation()
# Function 2. in Section 5.4 in Gade (2010):
p_EA_E = n_EB_E2p_EB_E(n_EA_E, z_EA, a, f, R_Ee)
p_EB_E = p_EA_E + p_AB_E
n_EB_E, z_EB = p_EB_E2n_EB_E(p_EB_E, a, f, R_Ee)
return n_EB_E, z_EB
[docs]def R2xyz(R_AB):
"""
Returns the angles about new axes in the xyz-order from a rotation matrix.
Parameters
----------
R_AB: 3 x 3 x n array
rotation matrix [no unit] (direction cosine matrix) such that the
relation between a vector v decomposed in A and B is given by:
v_A = mdot(R_AB, v_B)
Returns
-------
x, y, z: real scalars or array of length n.
Angles [rad] of rotation about new axes.
Notes
-----
The x, y, z angles are called Euler angles or Tait-Bryan angles and are
defined by the following procedure of successive rotations:
Given two arbitrary coordinate frames A and B. Consider a temporary frame
T that initially coincides with A. In order to make T align with B, we
first rotate T an angle x about its x-axis (common axis for both A and T).
Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T
is rotated an angle z about its NEWEST z-axis. The final orientation of
T now coincides with the orientation of B.
The signs of the angles are given by the directions of the axes and the
right hand rule.
See also
--------
xyz2R, R2zyx, xyz2R
"""
z = arctan2(-R_AB[0, 1, ...], R_AB[0, 0, ...]) # atan2: [-pi pi]
x = arctan2(-R_AB[1, 2, ...], R_AB[2, 2, ...])
sin_y = R_AB[0, 2, ...]
# cos_y is based on as many elements as possible, to average out
# numerical errors. It is selected as the positive square root since
# y: [-pi/2 pi/2]
cos_y = sqrt((R_AB[0, 0, ...]**2 + R_AB[0, 1, ...]**2
+ R_AB[1, 2, ...]**2 + R_AB[2, 2, ...]**2) / 2)
y = arctan2(sin_y, cos_y)
return x, y, z
[docs]def R2zyx(R_AB):
"""
Returns the angles about new axes in the zxy-order from a rotation matrix.
Parameters
----------
R_AB: 3x3 array
rotation matrix [no unit] (direction cosine matrix) such that the
relation between a vector v decomposed in A and B is given by:
v_A = np.dot(R_AB, v_B)
Returns
-------
z, y, x: real scalars
Angles [rad] of rotation about new axes.
Notes
-----
The z, x, y angles are called Euler angles or Tait-Bryan angles and are
defined by the following procedure of successive rotations:
Given two arbitrary coordinate frames A and B. Consider a temporary frame
T that initially coincides with A. In order to make T align with B, we
first rotate T an angle z about its z-axis (common axis for both A and T).
Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T
is rotated an angle x about its NEWEST x-axis. The final orientation of
T now coincides with the orientation of B.
The signs of the angles are given by the directions of the axes and the
right hand rule.
Note that if A is a north-east-down frame and B is a body frame, we
have that z=yaw, y=pitch and x=roll.
See also
--------
zyx2R, xyz2R, R2xyz
"""
x, y, z = R2xyz(np.rollaxis(R_AB, 1, 0))
return -z, -y, -x
[docs]def R_EL2n_E(R_EL):
"""
Returns n-vector from the rotation matrix R_EL.
Parameters
----------
R_EL: 3 x 3 x n array
Rotation matrix (direction cosine matrix) [no unit]
Returns
-------
n_E: 3 x n array
n-vector(s) [no unit] decomposed in E.
See also
--------
R_EN2n_E, n_E_and_wa2R_EL, n_E2R_EN
"""
# n-vector equals minus the last column of R_EL and R_EN, see Section 5.5
# in Gade (2010)
n_E = mdot(R_EL, np.vstack((0, 0, -1)))
return n_E.reshape(3, -1)
[docs]def R_EN2n_E(R_EN):
"""
Returns n-vector from the rotation matrix R_EN.
Parameters
----------
R_EN: 3 x 3 x n array
Rotation matrix (direction cosine matrix) [no unit]
Returns
-------
n_E: 3 x n array
n-vector [no unit] decomposed in E.
See also
--------
n_E2R_EN, R_EL2n_E, n_E_and_wa2R_EL
"""
# n-vector equals minus the last column of R_EL and R_EN, see Section 5.5
# in Gade (2010)
return R_EL2n_E(R_EN)
def _atleast_3d(x, y, z):
x, y, z = np.atleast_1d(x, y, z)
return x[None, None, :], y[None, None, :], z[None, None, :]
[docs]def xyz2R(x, y, z):
"""
Returns rotation matrix from 3 angles about new axes in the xyz-order.
Parameters
----------
x,y,z: real scalars or array of lengths n
Angles [rad] of rotation about new axes.
Returns
-------
R_AB: 3 x 3 x n array
rotation matrix [no unit] (direction cosine matrix) such that the
relation between a vector v decomposed in A and B is given by:
v_A = mdot(R_AB, v_B)
Notes
-----
The rotation matrix R_AB is created based on 3 angles x,y,z about new axes
(intrinsic) in the order x-y-z. The angles are called Euler angles or
Tait-Bryan angles and are defined by the following procedure of successive
rotations:
Given two arbitrary coordinate frames A and B. Consider a temporary frame
T that initially coincides with A. In order to make T align with B, we
first rotate T an angle x about its x-axis (common axis for both A and T).
Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T
is rotated an angle z about its NEWEST z-axis. The final orientation of
T now coincides with the orientation of B.
The signs of the angles are given by the directions of the axes and the
right hand rule.
See also
--------
R2xyz, zyx2R, R2zyx
"""
x, y, z = _atleast_3d(x, y, z)
cx, sx = cos(x), sin(x)
cy, sy = cos(y), sin(y)
cz, sz = cos(z), sin(z)
R_AB = array(([cy * cz, -cy * sz, sy],
[sy * sx * cz + cx * sz, -sy * sx * sz + cx * cz, -cy * sx],
[-sy * cx * cz + sx * sz, sy * cx * sz + sx * cz, cy * cx]))
return np.squeeze(R_AB)
[docs]def zyx2R(z, y, x):
"""
Returns rotation matrix from 3 angles about new axes in the zyx-order.
Parameters
----------
z, y, x: real scalars or arrays of lenths n
Angles [rad] of rotation about new axes.
Returns
-------
R_AB: 3 x 3 x n array
rotation matrix [no unit] (direction cosine matrix) such that the
relation between a vector v decomposed in A and B is given by:
v_A = mdot(R_AB, v_B)
Notes
-----
The rotation matrix R_AB is created based on 3 angles
z,y,x about new axes (intrinsic) in the order z-y-x. The angles are called
Euler angles or Tait-Bryan angles and are defined by the following
procedure of successive rotations:
Given two arbitrary coordinate frames A and B. Consider a temporary frame
T that initially coincides with A. In order to make T align with B, we
first rotate T an angle z about its z-axis (common axis for both A and T).
Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T
is rotated an angle x about its NEWEST x-axis. The final orientation of
T now coincides with the orientation of B.
The signs of the angles are given by the directions of the axes and the
right hand rule.
Note that if A is a north-east-down frame and B is a body frame, we
have that z=yaw, y=pitch and x=roll.
See also
--------
R2zyx, xyz2R, R2xyz
"""
x, y, z = _atleast_3d(x, y, z)
cx, sx = cos(x), sin(x)
cy, sy = cos(y), sin(y)
cz, sz = cos(z), sin(z)
R_AB = array([[cz * cy, -sz * cx + cz * sy * sx, sz * sx + cz * sy * cx],
[sz * cy, cz * cx + sz * sy * sx, - cz * sx + sz * sy * cx],
[-sy, cy * sx, cy * cx]])
return np.squeeze(R_AB)
def interpolate(path, ti):
"""
Return the interpolated point along the path
Parameters
----------
path: tuple of n-vectors (positionA, positionB)
ti: real scalar
interpolation time assuming position A and B is at t0=0 and t1=1,
respectively.
Returns
-------
point: Nvector
point of interpolation along path
"""
n_EB_E_t0, n_EB_E_t1 = path
n_EB_E_ti = unit(n_EB_E_t0 + ti * (n_EB_E_t1 - n_EB_E_t0),
norm_zero_vector=np.nan)
return n_EB_E_ti
class _Intersect(object):
__doc__ = """Return the intersection(s) between the great circles of the two paths
Parameters
----------
path_a, path_b: tuple of 2 n-vectors
defining path A and path B, respectively.
Path A and B has shape 2 x 3 x n and 2 x 3 x m, respectively.
Returns
-------
n_EC_E : array of shape 3 x max(n, m)
n-vector(s) [no unit] of position C decomposed in E.
point(s) of intersection between paths.
Examples
--------
{0}
""".format(_examples.get_examples([9], OO=False))
[docs]@use_docstring_from(_Intersect)
def intersect(path_a, path_b):
n_EA1_E, n_EA2_E = path_a
n_EB1_E, n_EB2_E = path_b
# Find the intersection between the two paths, n_EC_E:
n_EC_E_tmp = unit(cross(cross(n_EA1_E, n_EA2_E, axis=0),
cross(n_EB1_E, n_EB2_E, axis=0), axis=0),
norm_zero_vector=np.nan)
# 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(dot(n_EC_E_tmp.T, n_EA1_E)) * n_EC_E_tmp
if np.any(np.isnan(n_EC_E)):
warnings.warn('Paths are Equal. Intersection point undefined. '
'NaN returned.')
return n_EC_E
def great_circle_normal(n_EA_E, n_EB_E):
"""
Return the unit normal(s) to the great circle(s)
Parameters
----------
n_EA_E, n_EB_E: 3 x n array
n-vector(s) [no unit] of position A and B, decomposed in E.
"""
return unit(cross(n_EA_E, n_EB_E, axis=0), norm_zero_vector=np.nan)
def _euclidean_cross_track_distance(sin_theta, radius=1):
return sin_theta * radius
def _great_circle_cross_track_distance(sin_theta, radius=1):
return np.arcsin(sin_theta) * radius
# ill conditioned for small angles:
# return (np.arccos(-sin_theta) - np.pi / 2) * radius
class _CrossTrackDistance(object):
__doc__ = """ Return cross track distance between path A and position B.
Parameters
----------
path: tuple of 2 n-vectors
2 n-vectors of positions defining path A, decomposed in E.
n_EB_E: 3 x m array
n-vector(s) of position B to measure the cross track distance to.
method: string
defining distance calculated. Options are: 'greatcircle' or 'euclidean'
radius: real scalar
radius of sphere. (default 6371009.0)
Returns
-------
distance : array of length max(n, m)
cross track distance(s)
Examples
--------
{0}
""".format(_examples.get_examples([10], OO=False))
[docs]@use_docstring_from(_CrossTrackDistance)
def cross_track_distance(path, n_EB_E, method='greatcircle',
radius=6371009.0):
c_E = great_circle_normal(path[0], path[1])
sin_theta = -np.dot(c_E.T, n_EB_E).ravel()
if method[0].lower() == 'e':
return _euclidean_cross_track_distance(sin_theta, radius)
return _great_circle_cross_track_distance(sin_theta, radius)
class _OnGreatCircle(object):
__doc__ = """ True if position B is on great circle through path A.
Parameters
----------
path: tuple of 2 n-vectors
2 n-vectors of positions defining path A, decomposed in E.
n_EB_E: 3 x m array
n-vector(s) of position B to check to.
radius: real scalar
radius of sphere. (default 6371009.0)
rtol, atol: real scalars
defining relative and absolute tolerance
Returns
-------
on : bool array of length max(n, m)
True if position B is on great circle through path A.
Examples
--------
{0}
""".format(_examples.get_examples([10], OO=False))
[docs]@use_docstring_from(_OnGreatCircle)
def on_great_circle(path, n_EB_E, radius=6371009.0, rtol=1e-6, atol=1e-8):
distance = np.abs(cross_track_distance(path, n_EB_E, radius=radius))
return np.isclose(distance, 0, rtol, atol)
class _OnGreatCirclePath(object):
__doc__ = """ True if position B is on great circle and between endpoints of path A.
Parameters
----------
path: tuple of 2 n-vectors
2 n-vectors of positions defining path A, decomposed in E.
n_EB_E: 3 x m array
n-vector(s) of position B to measure the cross track distance to.
radius: real scalar
radius of sphere. (default 6371009.0)
rtol, atol: real scalars
defining relative and absolute tolerance
Returns
-------
on : bool array of length max(n, m)
True if position B is on great circle and between endpoints of path A.
Examples
--------
{0}
""".format(_examples.get_examples([10], OO=False))
[docs]@use_docstring_from(_OnGreatCirclePath)
def on_great_circle_path(path, n_EB_E, radius=6371009.0, rtol=1e-6, atol=1e-8):
n_EA1_E, n_EA2_E = path
scale = norm(n_EA2_E - n_EA1_E, axis=0)
ti1 = norm(n_EB_E - n_EA1_E, axis=0) / scale
ti2 = norm(n_EB_E - n_EA2_E, axis=0) / scale
return (ti1 <= 1) & (ti2 <= 1) & on_great_circle(path, n_EB_E, radius,
rtol, atol)
class _ClosestPointOnGreatCircle(object):
__doc__ = """ Return closest point C on great circle path A to position B.
Parameters
----------
path: tuple of 2 n-vectors of 3 x n arrays
2 n-vectors of positions defining path A, decomposed in E.
n_EB_E: 3 x m array
n-vector(s) of position B to find the closest point to.
Returns
-------
n_EC_E: 3 x max(m, n) array
n-vector(s) of closest position C on great circle path A
Examples
--------
{0}
""".format(_examples.get_examples([10], OO=False))
[docs]@use_docstring_from(_ClosestPointOnGreatCircle)
def closest_point_on_great_circle(path, n_EB_E):
n_EA1_E, n_EA2_E = path
c1 = cross(n_EA1_E, n_EA2_E, axis=0)
c2 = cross(n_EB_E, c1, axis=0)
n_EC_E = unit(cross(c1, c2, axis=0))
return n_EC_E
class _GreatCircleDistance(object):
__doc__ = """ Return great circle distance between positions A and B
Parameters
----------
n_EA_E, n_EB_E: 3 x n array
n-vector(s) [no unit] of position A and B, decomposed in E.
radius: real scalar
radius of sphere.
Formulae is given by equation (16) in Gade (2010) and is well
conditioned for all angles.
Examples
--------
{0}
""".format(_examples.get_examples([5], OO=False))
[docs]@use_docstring_from(_GreatCircleDistance)
def great_circle_distance(n_EA_E, n_EB_E, radius=6371009.0):
s_AB = np.arctan2(norm(np.cross(n_EA_E, n_EB_E, axis=0), axis=0),
dot(n_EA_E.T, n_EB_E)) * radius
# ill conditioned for small angles:
# s_AB_version1 = arccos(dot(n_EA_E,n_EB_E))*radius
# ill-conditioned for angles near pi/2 (and not valid above pi/2)
# s_AB_version2 = arcsin(norm(cross(n_EA_E,n_EB_E)))*radius
return s_AB.ravel()
class _EuclideanDistance(object):
__doc__ = """Return Euclidean distance between positions A and B
Parameters
----------
n_EA_E, n_EB_E: 3 x n array
n-vector(s) [no unit] of position A and B, decomposed in E.
radius: real scalar
radius of sphere.
Examples
--------
{0}
""".format(_examples.get_examples([5], OO=False))
[docs]@use_docstring_from(_EuclideanDistance)
def euclidean_distance(n_EA_E, n_EB_E, radius=6371009.0):
d_AB = norm(n_EB_E - n_EA_E, axis=0) * radius
return d_AB.ravel()
[docs]def n_EA_E_and_n_EB_E2azimuth(n_EA_E, n_EB_E, a=6378137, f=1.0 / 298.257223563,
R_Ee=None):
"""
Return azimuth from A to B, relative to North:
Parameters
----------
n_EA_E, n_EB_E: 3 x n array
n-vector(s) [no unit] of position A and B, respectively,
decomposed in E.
a: real scalar, default WGS-84 ellipsoid.
Semi-major axis of the Earth ellipsoid given in [m].
f: real scalar, default WGS-84 ellipsoid.
Flattening [no unit] of the Earth ellipsoid. If f==0 then spherical
Earth with radius a is used in stead of WGS-84.
R_Ee : 3 x 3 array
rotation matrix defining the axes of the coordinate frame E.
Returns
-------
azimuth: n array
Angle [rad] the line makes with a meridian, taken clockwise from north.
"""
if R_Ee is None:
R_Ee = E_rotation()
# Step2: Find p_AB_E (delta decomposed in E).
p_AB_E = n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA=0, z_EB=0, a=a, f=f,
R_Ee=R_Ee)
# Step3: Find R_EN for position A:
R_EN = n_E2R_EN(n_EA_E, R_Ee=R_Ee)
# Step4: Find p_AB_N
# p_AB_N = dot(R_EN.T, p_AB_E)
p_AB_N = mdot(np.rollaxis(R_EN, 1, 0), p_AB_E[:, None, ...]).reshape(3, -1)
# (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:
return arctan2(p_AB_N[1], p_AB_N[0])
class _PositionBFromAzimuthAndDistanceFromPositionA(object):
__doc__ = """
Return position B from azimuth and distance from position A
Parameters
----------
n_EA_E: 3 x n array
n-vector(s) [no unit] of position A decomposed in E.
distance_rad: n, array
great circle distance [rad] from position A to B
azimuth: n array
Angle [rad] the line makes with a meridian, taken clockwise from north.
Returns
-------
n_EB_E: 3 x n array
n-vector(s) [no unit] of position B decomposed in E.
Examples
--------
{0}
""".format(_examples.get_examples([8], OO=False))
[docs]@use_docstring_from(_PositionBFromAzimuthAndDistanceFromPositionA)
def n_EA_E_distance_and_azimuth2n_EB_E(n_EA_E, distance_rad, azimuth,
R_Ee=None):
if R_Ee is None:
R_Ee = E_rotation()
# Step1: Find unit vectors for north and east:
k_east_E = unit(cross(dot(R_Ee.T, [[1], [0], [0]]), n_EA_E, axis=0))
k_north_E = cross(n_EA_E, k_east_E, axis=0)
# Step2: Find the initial direction vector d_E:
d_E = k_north_E * cos(azimuth) + k_east_E * sin(azimuth)
# Step3: Find n_EB_E:
n_EB_E = n_EA_E * cos(distance_rad) + d_E * sin(distance_rad)
return n_EB_E
class _MeanHorizontalPosition(object):
__doc__ = """
Return the n-vector of the horizontal mean position.
Parameters
----------
n_EB_E: 3 x n array
n-vectors [no unit] of positions Bi, decomposed in E.
Returns
-------
p_EM_E: 3 x 1 array
n-vector [no unit] of the mean positions of all Bi, decomposed in E.
Examples
--------
{0}
""".format(_examples.get_examples([7], OO=False))
[docs]@use_docstring_from(_MeanHorizontalPosition)
def mean_horizontal_position(n_EB_E):
n_EM_E = unit(np.sum(n_EB_E, axis=1).reshape((3, 1)))
return n_EM_E
if __name__ == "__main__":
test_docstrings(__file__)