def triangulate(self):
"""
Convert mesh points to vectors in Cartesian space.
:returns:
Tuple of four elements, each being 2d numpy array of 3d vectors
(the same structure and shape as the mesh itself). Those arrays
are:
#. points vectors,
#. vectors directed from each point (excluding the last column)
to the next one in a same row →,
#. vectors directed from each point (excluding the first row)
to the previous one in a same column ↑,
#. vectors pointing from a bottom left point of each mesh cell
to top right one ↗.
So the last three arrays of vectors allow to construct triangles
covering the whole mesh.
"""
points = geo_utils.spherical_to_cartesian(self.lons, self.lats,
self.depths)
# triangulate the mesh by defining vectors of triangles edges:
# →
along_azimuth = points[:, 1:] - points[:, :-1]
# ↑
updip = points[:-1] - points[1:]
# ↗
diag = points[:-1, 1:] - points[1:, :-1]
return points, along_azimuth, updip, diag
def triangulate(self):
"""
Convert mesh points to vectors in Cartesian space.
:returns:
Tuple of four elements, each being 2d numpy array of 3d vectors
(the same structure and shape as the mesh itself). Those arrays
are:
#. points vectors,
#. vectors directed from each point (excluding the last column)
to the next one in a same row →,
#. vectors directed from each point (excluding the first row)
to the previous one in a same column ↑,
#. vectors pointing from a bottom left point of each mesh cell
to top right one ↗.
So the last three arrays of vectors allow to construct triangles
covering the whole mesh.
"""
points = geo_utils.spherical_to_cartesian(self.lons, self.lats,
self.depths)
# triangulate the mesh by defining vectors of triangles edges:
# →
along_azimuth = points[:, 1:] - points[:, :-1]
# ↑
updip = points[:-1] - points[1:]
# ↗
diag = points[:-1, 1:] - points[1:, :-1]
return points, along_azimuth, updip, diag
class SphericalToCartesianAndBackTestCase(unittest.TestCase):
def _test(self, (lons, lats, depths), vectors):
res_cart = utils.spherical_to_cartesian(lons, lats, depths)
self.assertIsInstance(res_cart, numpy.ndarray)
self.assertTrue(numpy.allclose(vectors, res_cart), str(res_cart))
res_sphe = utils.cartesian_to_spherical(res_cart)
self.assertIsInstance(res_sphe, tuple)
self.assertEqual(len(res_sphe), 3)
if depths is None:
depths = numpy.zeros_like(lons)
self.assertEqual(
numpy.array(res_sphe).shape,
numpy.array([lons, lats, depths]).shape)
self.assertTrue(numpy.allclose([lons, lats, depths], res_sphe),
str(res_sphe))
def _init_plane(self):
"""
Prepare everything needed for projecting arbitrary points on a plane
containing the surface.
"""
tl, tr, bl, br = geo_utils.spherical_to_cartesian(
self.corner_lons, self.corner_lats, self.corner_depths
)
# these two parameters define the plane that contains the surface
# (in 3d Cartesian space): a normal unit vector,
self.normal = geo_utils.normalized(numpy.cross(tl - tr, tl - bl))
# ... and scalar "d" parameter from the plane equation (uses
# an equation (3) from http://mathworld.wolfram.com/Plane.html)
self.d = - (self.normal * tl).sum()
# these two 3d vectors together with a zero point represent surface's
# coordinate space (the way to translate 3d Cartesian space with
# a center in earth's center to 2d space centered in surface's top
# left corner with basis vectors directed to top right and bottom left
# corners. see :meth:`_project`.
self.uv1 = geo_utils.normalized(tr - tl)
self.uv2 = numpy.cross(self.normal, self.uv1)
self.zero_zero = tl
def _init_plane(self):
"""
Prepare everything needed for projecting arbitrary points on a plane
containing the surface.
"""
tl, tr, bl, br = geo_utils.spherical_to_cartesian(
self.corner_lons, self.corner_lats, self.corner_depths
)
# these two parameters define the plane that contains the surface
# (in 3d Cartesian space): a normal unit vector,
self.normal = geo_utils.normalized(numpy.cross(tl - tr, tl - bl))
# ... and scalar "d" parameter from the plane equation (uses
# an equation (3) from http://mathworld.wolfram.com/Plane.html)
self.d = - (self.normal * tl).sum()
# these two 3d vectors together with a zero point represent surface's
# coordinate space (the way to translate 3d Cartesian space with
# a center in earth's center to 2d space centered in surface's top
# left corner with basis vectors directed to top right and bottom left
# corners. see :meth:`_project`.
self.uv1 = geo_utils.normalized(tr - tl)
self.uv2 = numpy.cross(self.normal, self.uv1)
self.zero_zero = tl
def _project(self, lons, lats, depths):
"""
Project points to a surface's plane.
Parameters are lists or numpy arrays of coordinates of points
to project.
:returns:
A tuple of three items: distances between original points
and surface's plane in km, "x" and "y" coordinates of points'
projections to the plane (in a surface's coordinate space).
"""
points = geo_utils.spherical_to_cartesian(lons, lats, depths)
# uses method from http://www.9math.com/book/projection-point-plane
dists = (self.normal * points).sum(axis=-1) + self.d
t0 = - dists
projs = points + self.normal * t0.reshape(t0.shape + (1, ))
# translate projected points' to surface's coordinate space
vectors2d = projs - self.zero_zero
xx = (vectors2d * self.uv1).sum(axis=-1)
yy = (vectors2d * self.uv2).sum(axis=-1)
return dists, xx, yy
def _project(self, lons, lats, depths):
"""
Project points to a surface's plane.
Parameters are lists or numpy arrays of coordinates of points
to project.
:returns:
A tuple of three items: distances between original points
and surface's plane in km, "x" and "y" coordinates of points'
projections to the plane (in a surface's coordinate space).
"""
points = geo_utils.spherical_to_cartesian(lons, lats, depths)
# uses method from http://www.9math.com/book/projection-point-plane
dists = (self.normal * points).sum(axis=-1) + self.d
t0 = - dists
projs = points + self.normal * t0.reshape(t0.shape + (1, ))
# translate projected points' to surface's coordinate space
vectors2d = projs - self.zero_zero
xx = (vectors2d * self.uv1).sum(axis=-1)
yy = (vectors2d * self.uv2).sum(axis=-1)
return dists, xx, yy
def test_from_vector(self):
point = geo.Point(12.34, -56.78, 91.011)
vector = spherical_to_cartesian(point.longitude, point.latitude,
point.depth)
self.assertEqual(point, geo.Point.from_vector(vector))
def test_from_vector(self):
point = geo.Point(12.34, -56.78, 91.011)
vector = spherical_to_cartesian(point.longitude, point.latitude,
point.depth)
self.assertEqual(point, geo.Point.from_vector(vector))
