# Source code for openquake.hazardlib.geo.utils

```
# -*- coding: utf-8 -*-
# vim: tabstop=4 shiftwidth=4 softtabstop=4
#
# Copyright (C) 2012-2017 GEM Foundation
#
# OpenQuake is free software: you can redistribute it and/or modify it
# under the terms of the GNU Affero General Public License as published
# by the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# OpenQuake is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Affero General Public License for more details.
#
# You should have received a copy of the GNU Affero General Public License
# along with OpenQuake. If not, see <http://www.gnu.org/licenses/>.
"""
Module :mod:`openquake.hazardlib.geo.utils` contains functions that are common
to several geographical primitives and some other low-level spatial operations.
"""
import operator
import collections
try:
import rtree
except:
rtree = None
import numpy
import shapely.geometry
from openquake.hazardlib.geo import geodetic
from openquake.hazardlib.geo.geodetic import (
EARTH_RADIUS, geodetic_distance, min_idx_dst)
from openquake.baselib.slots import with_slots
SphericalBB = collections.namedtuple('SphericalBB', 'west east north south')
[docs]class GeographicObjects(object):
"""
Store a collection of geographic objects, i.e. objects with longitudes
and latitudes. By default extracts the coordinates from the attributes
.lon and .lat, but you can provide your own getters. It is possible
to extract the closest object to a given location by calling the
method .get_closest(lon, lat).
"""
def __init__(self, objects, getlon=operator.attrgetter('lon'),
getlat=operator.attrgetter('lat')):
self.objects = list(objects)
lons, lats = [], []
for i, obj in enumerate(self.objects):
lon, lat = getlon(obj), getlat(obj)
lons.append(lon)
lats.append(lat)
self.lons, self.lats = numpy.array(lons), numpy.array(lats)
if rtree:
self.index = rtree.index.Index()
self.proj = OrthographicProjection.from_lons_lats(lons, lats)
xs, ys = self.proj(self.lons, self.lats)
for i, (x, y) in enumerate(zip(xs, ys)):
self.index.insert(i, (x, y, x, y))
[docs] def get_closest(self, lon, lat):
"""
Get the closest object to the given longitude and latitude
and its distance.
:param lon: longitude in degrees
:param lat: latitude in degrees
:param max_distance: distance in km (or None)
"""
if rtree:
x, y = self.proj(lon, lat)
idx = list(self.index.nearest((x, y, x, y), 1))[0]
min_dist = geodetic_distance(
lon, lat, self.lons[idx], self.lats[idx])
else:
zeros = numpy.zeros_like(self.lons)
idx, min_dist = min_idx_dst(self.lons, self.lats, zeros, lon, lat)
return self.objects[idx], min_dist
[docs]def clean_points(points):
"""
Given a list of :class:`~openquake.hazardlib.geo.point.Point` objects,
return a new list with adjacent duplicate points removed.
"""
if not points:
return points
result = [points[0]]
for point in points:
if point != result[-1]:
result.append(point)
return result
[docs]def line_intersects_itself(lons, lats, closed_shape=False):
"""
Return ``True`` if line of points intersects itself.
Line with the last point repeating the first one considered
intersecting itself.
The line is defined by lists (or numpy arrays) of points'
longitudes and latitudes (depth is not taken into account).
:param closed_shape:
If ``True`` the line will be checked twice: first time with
its original shape and second time with the points sequence
being shifted by one point (the last point becomes first,
the first turns second and so on). This is useful for
checking that the sequence of points defines a valid
:class:`~openquake.hazardlib.geo.polygon.Polygon`.
"""
assert len(lons) == len(lats)
if len(lons) <= 3:
# line can not intersect itself unless there are
# at least four points
return False
west, east, north, south = get_spherical_bounding_box(lons, lats)
proj = get_orthographic_projection(west, east, north, south)
xx, yy = proj(lons, lats)
if not shapely.geometry.LineString(list(zip(xx, yy))).is_simple:
return True
if closed_shape:
xx, yy = proj(numpy.roll(lons, 1), numpy.roll(lats, 1))
if not shapely.geometry.LineString(list(zip(xx, yy))).is_simple:
return True
return False
[docs]def get_longitudinal_extent(lon1, lon2):
"""
Return the distance between two longitude values as an angular measure.
Parameters represent two longitude values in degrees.
:return:
Float, the angle between ``lon1`` and ``lon2`` in degrees. Value
is positive if ``lon2`` is on the east from ``lon1`` and negative
otherwise. Absolute value of the result doesn't exceed 180 for
valid parameters values.
"""
return (lon2 - lon1 + 180) % 360 - 180
[docs]def get_spherical_bounding_box(lons, lats):
"""
Given a collection of points find and return the bounding box,
as a pair of longitudes and a pair of latitudes.
Parameters define longitudes and latitudes of a point collection
respectively in a form of lists or numpy arrays.
:return:
A tuple of four items. These items represent western, eastern,
northern and southern borders of the bounding box respectively.
Values are floats in decimal degrees.
:raises ValueError:
If points collection has the longitudinal extent of more than
180 degrees (it is impossible to define a single hemisphere
bound to poles that would contain the whole collection).
"""
north, south = numpy.max(lats), numpy.min(lats)
west, east = numpy.min(lons), numpy.max(lons)
assert (-180 <= west <= 180) and (-180 <= east <= 180)
if get_longitudinal_extent(west, east) < 0:
# points are lying on both sides of the international date line
# (meridian 180). the actual west longitude is the lowest positive
# longitude and east one is the highest negative.
west = min(lon for lon in lons if lon > 0)
east = max(lon for lon in lons if lon < 0)
if not all((get_longitudinal_extent(west, lon) >= 0
and get_longitudinal_extent(lon, east) >= 0)
for lon in lons):
raise ValueError('points collection has longitudinal extent '
'wider than 180 deg')
return SphericalBB(west, east, north, south)
[docs]@with_slots
class OrthographicProjection(object):
"""
Callable object to compute orthographic projections. See the docstring
of get_orthographic_projection.
"""
_slots_ = ('west east north south lambda0 phi0 '
'cos_phi0 sin_phi0 sin_pi_over_4').split()
[docs] @classmethod
def from_lons_lats(cls, lons, lats):
return cls(*get_spherical_bounding_box(lons, lats))
def __init__(self, west, east, north, south):
self.west = west
self.east = east
self.north = north
self.south = south
self.lambda0, self.phi0 = numpy.radians(
get_middle_point(west, north, east, south))
self.cos_phi0 = numpy.cos(self.phi0)
self.sin_phi0 = numpy.sin(self.phi0)
self.sin_pi_over_4 = (2 ** 0.5) / 2
def __call__(self, lons, lats, reverse=False):
if not reverse:
lambdas, phis = numpy.radians(lons), numpy.radians(lats)
cos_phis = numpy.cos(phis)
lambdas -= self.lambda0
# calculate the sine of the distance between projection center
# and each of the points to project
sin_dist = numpy.sqrt(
numpy.sin((self.phi0 - phis) / 2.0) ** 2.0
+ self.cos_phi0 * cos_phis * numpy.sin(lambdas / 2.0) ** 2.0
)
if (sin_dist > self.sin_pi_over_4).any():
raise ValueError('some points are too far from the projection '
'center lon=%s lat=%s' %
(numpy.degrees(self.lambda0),
numpy.degrees(self.phi0)))
xx = numpy.cos(phis) * numpy.sin(lambdas)
yy = (self.cos_phi0 * numpy.sin(phis) - self.sin_phi0 * cos_phis
* numpy.cos(lambdas))
return xx * EARTH_RADIUS, yy * EARTH_RADIUS
else:
# "reverse" mode, arguments are actually abscissae
# and ordinates in 2d space
xx, yy = lons / EARTH_RADIUS, lats / EARTH_RADIUS
cos_c = numpy.sqrt(1 - (xx ** 2 + yy ** 2))
phis = numpy.arcsin(cos_c * self.sin_phi0 + yy * self.cos_phi0)
lambdas = numpy.arctan2(
xx, self.cos_phi0 * cos_c - yy * self.sin_phi0)
xx = numpy.degrees(self.lambda0 + lambdas)
yy = numpy.degrees(phis)
# shift longitudes greater than 180 back into the western
# hemisphere, that is in range [0, -180], and longitudes
# smaller than -180, to the heastern emisphere [0, 180]
idx = xx >= 180.
xx[idx] = xx[idx] - 360.
idx = xx <= -180.
xx[idx] = xx[idx] + 360.
return xx, yy
[docs]def get_orthographic_projection(west, east, north, south):
"""
Create and return a projection object for a given bounding box.
:returns:
callable OrthographicProjection object that can perform both forward
and reverse projection (converting from longitudes and latitudes to x
and y values on 2d-space and vice versa). The call takes three
arguments: first two are numpy arrays of longitudes and latitudes *or*
abscissae and ordinates of points to project and the third one
is a boolean that allows to choose what operation is requested --
is it forward or reverse one. ``True`` value given to third
positional argument (or keyword argument "reverse") indicates
that the projection of points in 2d space back to earth surface
is needed. The default value for "reverse" argument is ``False``,
which means forward projection (degrees to kilometers).
Raises ``ValueError`` in forward projection
mode if any of the target points is further than 90 degree
(along the great circle arc) from the projection center.
Parameters are given as floats, representing decimal degrees (first two
are longitudes and last two are latitudes). They define a bounding box
in a spherical coordinates of the collection of points that is about
to be projected. The center point of the projection (coordinates (0, 0)
in Cartesian space) is set to the middle point of that bounding box.
The resulting projection is defined for spherical coordinates that are
not further from the bounding box center than 90 degree on the great
circle arc.
The result projection is of type `Orthographic
<http://mathworld.wolfram.com/OrthographicProjection.html>`_.
This projection is prone to distance, area and angle distortions
everywhere outside of the center point, but still can be used for
checking shapes: verifying if line intersects itself (like in
:func:`line_intersects_itself`) or if point is inside of a polygon
(like in :meth:`openquake.hazardlib.geo.polygon.Polygon.discretize`). It
can be also used for measuring distance to an extent of around 700
kilometers (error doesn't exceed 1 km up until then).
"""
return OrthographicProjection(west, east, north, south)
[docs]def get_middle_point(lon1, lat1, lon2, lat2):
"""
Given two points return the point exactly in the middle lying on the same
great circle arc.
Parameters are point coordinates in degrees.
:returns:
Tuple of longitude and latitude of the point in the middle.
"""
if lon1 == lon2 and lat1 == lat2:
return lon1, lat1
dist = geodetic.geodetic_distance(lon1, lat1, lon2, lat2)
azimuth = geodetic.azimuth(lon1, lat1, lon2, lat2)
return geodetic.point_at(lon1, lat1, azimuth, dist / 2.0)
[docs]def spherical_to_cartesian(lons, lats, depths):
"""
Return the position vectors (in Cartesian coordinates) of list of spherical
coordinates.
For equations see: http://mathworld.wolfram.com/SphericalCoordinates.html.
Parameters are components of spherical coordinates in a form of scalars,
lists or numpy arrays. ``depths`` can be ``None`` in which case it's
considered zero for all points.
:returns:
``numpy.array`` of 3d vectors representing points' coordinates in
Cartesian space. The array has the same shape as parameter arrays.
In particular it means that if ``lons`` and ``lats`` are scalars,
the result is a single 3d vector. Vector of length ``1`` represents
distance of 1 km.
See also :func:`cartesian_to_spherical`.
"""
phi = numpy.radians(lons)
theta = numpy.radians(lats)
if depths is None:
rr = EARTH_RADIUS
else:
rr = EARTH_RADIUS - numpy.array(depths)
cos_theta_r = rr * numpy.cos(theta)
xx = cos_theta_r * numpy.cos(phi)
yy = cos_theta_r * numpy.sin(phi)
zz = rr * numpy.sin(theta)
vectors = numpy.array([xx.transpose(), yy.transpose(), zz.transpose()]) \
.transpose()
return vectors
[docs]def cartesian_to_spherical(vectors):
"""
Return the spherical coordinates for coordinates in Cartesian space.
This function does an opposite to :func:`spherical_to_cartesian`.
:param vectors:
Array of 3d vectors in Cartesian space.
:returns:
Tuple of three arrays of the same shape as ``vectors`` representing
longitude (decimal degrees), latitude (decimal degrees) and depth (km)
in specified order.
"""
rr = numpy.sqrt(numpy.sum(vectors * vectors, axis=-1))
xx, yy, zz = vectors.transpose()
xx = xx.transpose()
yy = yy.transpose()
zz = zz.transpose()
lats = numpy.degrees(numpy.arcsin((zz / rr).clip(-1., 1.)))
lons = numpy.degrees(numpy.arctan2(yy, xx))
depths = EARTH_RADIUS - rr
return lons, lats, depths
[docs]def triangle_area(e1, e2, e3):
"""
Get the area of triangle formed by three vectors.
Parameters are three three-dimensional numpy arrays representing
vectors of triangle's edges in Cartesian space.
:returns:
Float number, the area of the triangle in squared units of coordinates,
or numpy array of shape of edges with one dimension less.
Uses Heron formula, see http://mathworld.wolfram.com/HeronsFormula.html.
"""
# calculating edges length
e1_length = numpy.sqrt(numpy.sum(e1 * e1, axis=-1))
e2_length = numpy.sqrt(numpy.sum(e2 * e2, axis=-1))
e3_length = numpy.sqrt(numpy.sum(e3 * e3, axis=-1))
# calculating half perimeter
s = (e1_length + e2_length + e3_length) / 2.0
# applying Heron's formula
return numpy.sqrt(s * (s - e1_length) * (s - e2_length) * (s - e3_length))
[docs]def normalized(vector):
"""
Get unit vector for a given one.
:param vector:
Numpy vector as coordinates in Cartesian space, or an array of such.
:returns:
Numpy array of the same shape and structure where all vectors are
normalized. That is, each coordinate component is divided by its
vector's length.
"""
length = numpy.sum(vector * vector, axis=-1)
length = numpy.sqrt(length.reshape(length.shape + (1, )))
return vector / length
[docs]def point_to_polygon_distance(polygon, pxx, pyy):
"""
Calculate the distance to polygon for each point of the collection
on the 2d Cartesian plane.
:param polygon:
Shapely "Polygon" geometry object.
:param pxx:
List or numpy array of abscissae values of points to calculate
the distance from.
:param pyy:
Same structure as ``pxx``, but with ordinate values.
:returns:
Numpy array of distances in units of coordinate system. Points
that lie inside the polygon have zero distance.
"""
pxx = numpy.array(pxx)
pyy = numpy.array(pyy)
assert pxx.shape == pyy.shape
if pxx.ndim == 0:
pxx = pxx.reshape((1, ))
pyy = pyy.reshape((1, ))
result = numpy.array([
polygon.distance(shapely.geometry.Point(pxx.item(i), pyy.item(i)))
for i in range(pxx.size)
])
return result.reshape(pxx.shape)
[docs]def cross_idl(lon1, lon2):
"""
Return True if two longitude values define line crossing international date
line.
>>> cross_idl(-45, 45)
False
>>> cross_idl(-180, -179)
False
>>> cross_idl(180, 179)
False
>>> cross_idl(45, -45)
False
>>> cross_idl(0, 0)
False
>>> cross_idl(-170, 170)
True
>>> cross_idl(170, -170)
True
>>> cross_idl(-180, 180)
True
"""
# a line crosses the international date line if the end positions
# have different sign and they are more than 180 degrees longitude apart
return lon1 * lon2 < 0 and abs(lon1 - lon2) > 180
[docs]def plane_fit(points):
"""
This fits an n-dimensional plane to a set of points. See
http://stackoverflow.com/questions/12299540/plane-fitting-to-4-or-more-xyz-points
:parameter points:
An instance of :class:~numpy.ndarray. The number of columns must be
equal to three.
:return:
A point on the plane and the normal to the plane.
"""
points = numpy.transpose(points)
points = numpy.reshape(points, (numpy.shape(points)[0], -1))
assert points.shape[0] < points.shape[1], points.shape
ctr = points.mean(axis=1)
x = points - ctr[:, None]
M = numpy.dot(x, x.T)
return ctr, numpy.linalg.svd(M)[0][:, -1]
```