import numpy
from shapely.geometry import LineString, Point
[docs]def subdivide_great_circle(x, y, z, maxRes, earthRadius):
"""
Subdivide each segment of the transect so the horizontal resolution
approximately matches the requested resolution
Uses a formula for interpolating unit vectors on the sphere from
https://en.wikipedia.org/wiki/Slerp
Parameters
----------
x, y, z : numpy.array
The Cartesian coordinates of a transect, where the number of segments
is ``len(x) - 1``. ``x``, ``y`` and ``z`` are of the same length.
maxRes : float
The maximum allowed spacing in m after subdivision
earthRadius : float
The radius of the Earth in m
Returns
-------
xOut, yOut, zOut : numpy.array
The transect subdivided into segments with segment length at most
``maxRes``. All the points in ``x``, ``y`` and ``z`` are guaranteed
to be included.
dIn : numpy.array
The distance along the transect before subdivision
dOut : numpy.array
The distance along the transect after subdivision
"""
angularDistance = angular_distance(x=x, y=y, z=z)
dx = angularDistance * earthRadius
nSegments = numpy.maximum(
(dx / maxRes + 0.5).astype(int), 1)
dIn = numpy.zeros(x.shape)
dIn[1:] = numpy.cumsum(dx)
frac = []
outIndices = []
delta = []
for index in range(len(dIn) - 1):
n = nSegments[index]
frac.extend(numpy.arange(0, n)/n)
outIndices.extend(index*numpy.ones(n, int))
delta.extend(angularDistance[index]*numpy.ones(n))
frac.append(1.)
outIndices.append(len(dIn) - 2)
delta.append(angularDistance[-1])
frac = numpy.array(frac)
delta = numpy.array(delta)
outIndices = numpy.array(outIndices)
a = numpy.ones(delta.shape)
b = numpy.zeros(delta.shape)
mask = delta > 0.
denom = 1./numpy.sin(delta[mask])
a[mask] = denom*numpy.sin((1.-frac[mask])*delta[mask])
b[mask] = denom*numpy.sin(frac[mask]*delta[mask])
xOut = a*x[outIndices] + b*x[outIndices+1]
yOut = a*y[outIndices] + b*y[outIndices+1]
zOut = a*z[outIndices] + b*z[outIndices+1]
dOut = (-frac + 1.)*dIn[outIndices] + frac*dIn[outIndices+1]
return xOut, yOut, zOut, dIn, dOut
[docs]def cartesian_to_great_circle_distance(x, y, z, earth_radius):
"""
Cartesian transect points to great-circle distance
Parameters
----------
x, y, z : numpy.array
Cartesian coordinates along a transect
earth_radius : float
The radius of the earth
Returns
-------
distance : numpy.array
The distance along the transect
"""
distance = numpy.zeros(x.shape)
for segIndex in range(len(x)-1):
transectv0 = Vector(x[segIndex], y[segIndex], z[segIndex])
transectv1 = Vector(x[segIndex+1], y[segIndex+1], z[segIndex+1])
distance[segIndex+1] = distance[segIndex] + \
earth_radius*angular_distance(first=transectv0, second=transectv1)
return distance
[docs]def subdivide_planar(x, y, maxRes):
"""
Subdivide each segment of the transect so the horizontal resolution
approximately matches the requested resolution
Uses a formula for interpolating unit vectors on the sphere from
https://en.wikipedia.org/wiki/Slerp
Parameters
----------
x, y: numpy.array
The planar coordinates of a transect, where the number of segments
is ``len(x) - 1``. ``x`` and ``y`` are of the same length.
maxRes : float
The maximum allowed spacing in m after subdivision
Returns
-------
xOut, yOut : numpy.array
The transect subdivided into segments with segment length at most
``maxRes``. All the points in ``x`` and ``y`` are guaranteed to be
included.
dIn : numpy.array
The distance along the transect before subdivision
dOut : numpy.array
The distance along the transect after subdivision
"""
dx = numpy.zeros(len(x)-1)
for index in range(len(x)-1):
start = Point(x[index], y[index])
end = Point(x[index+1], y[index+1])
segment = LineString([start, end])
dx[index] = segment.length
nSegments = numpy.maximum(
(dx / maxRes + 0.5).astype(int), 1)
dIn = numpy.zeros(x.shape)
dIn[1:] = numpy.cumsum(dx)
frac = []
outIndices = []
for index in range(len(dIn) - 1):
n = nSegments[index]
frac.extend(numpy.arange(0, n)/n)
outIndices.extend(index*numpy.ones(n, int))
frac.append(1.)
outIndices.append(len(dIn) - 2)
frac = numpy.array(frac)
outIndices = numpy.array(outIndices)
xOut = (-frac + 1.)*x[outIndices] + frac*x[outIndices+1]
yOut = (-frac + 1.)*y[outIndices] + frac*y[outIndices+1]
dOut = (-frac + 1.)*dIn[outIndices] + frac*dIn[outIndices+1]
return xOut, yOut, dIn, dOut
[docs]def lon_lat_to_cartesian(lon, lat, earth_radius, degrees):
"""Convert from lon/lat to Cartesian x, y, z"""
if degrees:
lon = numpy.deg2rad(lon)
lat = numpy.deg2rad(lat)
x = earth_radius * numpy.cos(lat) * numpy.cos(lon)
y = earth_radius * numpy.cos(lat) * numpy.sin(lon)
z = earth_radius * numpy.sin(lat)
return x, y, z
[docs]def cartesian_to_lon_lat(x, y, z, earth_radius, degrees):
"""Convert from Cartesian x, y, z to lon/lat"""
lon = numpy.arctan2(y, x)
lat = numpy.arcsin(z/earth_radius)
if degrees:
lon = numpy.rad2deg(lon)
lat = numpy.rad2deg(lat)
return lon, lat
[docs]def angular_distance(x=None, y=None, z=None, first=None, second=None):
"""
Compute angular distance between points on the sphere, following:
https://en.wikipedia.org/wiki/Great-circle_distance
Parameters
----------
x, y, z : numpy.array, optional
The Cartsian coordinates of a transect, where the number of segments
is ``len(x) - 1``. ``x``, ``y`` and ``z`` are of the same length and
all must be present if ``first`` and ``second`` are not provided.
first, second : Vector
The start and end points of each segment of the transect, where the
``x``, ``y``, and ``z`` attributes of each vector are ``numpy.array``
objects.
Returns
-------
angularDistance : numpy.array
The angular distance (in radians) between segments of the transect.
"""
if first is None or second is None:
first = Vector(x[0:-1], y[0:-1], z[0:-1])
second = Vector(x[1:], y[1:], z[1:])
angularDistance = numpy.arctan2(_mag(_cross(first, second)),
_dot(first, second))
return angularDistance
[docs]def intersects(a1, a2, b1, b2):
"""
Based on https://stackoverflow.com/a/26669130/7728169
Determine if the great circle arc from ``a1`` to ``a2`` intersects that
from ``b1`` to ``b2``.
Parameters
----------
a1, a2, b1, b2 : mpas_tools.transects.Vector
Cartesian coordinates of the end points of two great circle arcs.
The types of the attributes ``x``, ``y``, and ``z`` must either be
``numpy.arrays`` of identical size for all 4 vectors (in which case
intersections are found element-wise), or scalars for
at least one of either ``a1`` and ``a2`` or ``b1`` and ``b2``.
Returns
-------
intersect : numpy.array
A boolean array of the same size as ``a1`` and ``a2`` or ``b1`` and
``b2``, whichever is greater, indicating if the particular pair of arcs
intersects
"""
return numpy.logical_and(_straddles(a1, a2, b1, b2),
_straddles(b1, b2, a1, a2))
[docs]def intersection(a1, a2, b1, b2):
"""
Based on https://stackoverflow.com/a/26669130/7728169
Find the intersection point between great circle arc from ``a1`` to ``a2``
and from ``b1`` to ``b2``. The arcs should have already have been found
to intersect by calling ``intersects()``
Parameters
----------
a1, a2, b1, b2 : mpas_tools.transects.Vector
Cartesian coordinates of the end points of two great circle arcs.
The types of the attributes ``x``, ``y``, and ``z`` must either be
``numpy.arrays`` of identical size for all 4 vectors (in which case
intersections are found element-wise), or scalars for
at least one of either the ``a1`` and ``a2`` or ``b1`` and ``b2``.
Returns
-------
points : mpas_tools.transects.Vector
An array of Cartesian points *on the unit sphere* indicating where the
arcs intersect
"""
points = _cross(_cross(a1, a2), _cross(b1, b2))
s = numpy.sign(_det(a1, b1, b2))/_mag(points)
points = Vector(s*points.x, s*points.y, s*points.z)
return points
[docs]class Vector:
"""
A class for representing Cartesian vectors with ``x``, ``y`` and ``z``
components that are either ``float`` or ``numpy.array`` objects of identical
size.
"""
[docs] def __init__(self, x, y, z):
self.x = x
self.y = y
self.z = z
def _straddles(a1, a2, b1, b2):
"""
Based on https://stackoverflow.com/a/26669130/7728169
Determines if the great circle segment determined by (a1, a2)
straddles the great circle determined by (b1, b2)
Parameters
----------
a1, a2, b1, b2 : Vector
Cartesian coordinates of the end points of two great circle arcs.
The types of the attributes ``x``, ``y``, and ``z`` must either be
``numpy.arrays`` of identical size for all 4 vectors (in which case
intersections are found element-wise), or scalars for
at least one of either the ``a``s or the ``b``s.
Returns
-------
straddle : numpy.array
A boolean array of the same size as the ``a``s or the ``b``s, whichever
is greater, indicating if the great circle segment determined by
(a1, a2) straddles the great circle determined by (b1, b2)
"""
return _det(a1, b1, b2) * _det(a2, b1, b2) < 0
def _dot(v1, v2):
"""The dot product between two ``Vector`` objects ``v1`` and ``v2``"""
return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z
def _cross(v1, v2):
"""The cross product between two ``Vector`` objects ``v1`` and ``v2``"""
return Vector(v1.y * v2.z - v1.z * v2.y,
v1.z * v2.x - v1.x * v2.z,
v1.x * v2.y - v1.y * v2.x)
def _det(v1, v2, v3):
"""The determinant of the matrix defined by the three ``Vector`` objects"""
return _dot(v1, _cross(v2, v3))
def _mag(v):
"""The magnitude of the ``Vector`` object ``v``"""
return numpy.sqrt(_dot(v, v))