"""
Commonly used mathematical functions.
"""
#-----------------------------------------------------------------------------
# Copyright (c) 2013, yt Development Team.
#
# Distributed under the terms of the Modified BSD License.
#
# The full license is in the file COPYING.txt, distributed with this software.
#-----------------------------------------------------------------------------
import numpy as np
import math
from yt.units.yt_array import \
YTArray
prec_accum = {
np.int: np.int64,
np.int8: np.int64,
np.int16: np.int64,
np.int32: np.int64,
np.int64: np.int64,
np.uint8: np.uint64,
np.uint16: np.uint64,
np.uint32: np.uint64,
np.uint64: np.uint64,
np.float: np.float64,
np.float16: np.float64,
np.float32: np.float64,
np.float64: np.float64,
np.complex: np.complex128,
np.complex64: np.complex128,
np.complex128: np.complex128,
np.dtype('int'): np.int64,
np.dtype('int8'): np.int64,
np.dtype('int16'): np.int64,
np.dtype('int32'): np.int64,
np.dtype('int64'): np.int64,
np.dtype('uint8'): np.uint64,
np.dtype('uint16'): np.uint64,
np.dtype('uint32'): np.uint64,
np.dtype('uint64'): np.uint64,
np.dtype('float'): np.float64,
np.dtype('float16'): np.float64,
np.dtype('float32'): np.float64,
np.dtype('float64'): np.float64,
np.dtype('complex'): np.complex128,
np.dtype('complex64'): np.complex128,
np.dtype('complex128'): np.complex128,
}
[docs]def periodic_position(pos, ds):
r"""Assuming periodicity, find the periodic position within the domain.
Parameters
----------
pos : array
An array of floats.
ds : Dataset
A simulation static output.
Examples
--------
>>> a = np.array([1.1, 0.5, 0.5])
>>> data = {'Density':np.ones([32,32,32])}
>>> ds = load_uniform_grid(data, [32,32,32], 1.0)
>>> ppos = periodic_position(a, ds)
>>> ppos
array([ 0.1, 0.5, 0.5])
"""
off = (pos - ds.domain_left_edge) % ds.domain_width
return ds.domain_left_edge + off
[docs]def periodic_dist(a, b, period, periodicity=(True, True, True)):
r"""Find the Euclidean periodic distance between two sets of points.
Parameters
----------
a : array or list
Either an ndim long list of coordinates corresponding to a single point
or an (ndim, npoints) list of coordinates for many points in space.
b : array of list
Either an ndim long list of coordinates corresponding to a single point
or an (ndim, npoints) list of coordinates for many points in space.
period : float or array or list
If the volume is symmetrically periodic, this can be a single float,
otherwise an array or list of floats giving the periodic size of the
volume for each dimension.
periodicity : An ndim-element tuple of booleans
If an entry is true, the domain is assumed to be periodic along
that direction.
Examples
--------
>>> a = [0.1, 0.1, 0.1]
>>> b = [0.9, 0,9, 0.9]
>>> period = 1.
>>> dist = periodic_dist(a, b, 1.)
>>> dist
0.346410161514
"""
a = np.array(a)
b = np.array(b)
period = np.array(period)
if period.size == 1:
period = np.array([period, period, period])
if a.shape != b.shape:
raise RuntimeError("Arrays must be the same shape.")
if period.shape != a.shape and len(a.shape) > 1:
n_tup = tuple([1 for i in range(a.ndim-1)])
period = np.tile(np.reshape(period, (a.shape[0],)+n_tup), (1,)+a.shape[1:])
elif len(a.shape) == 1:
a = np.reshape(a, (a.shape[0],)+(1,1))
b = np.reshape(b, (a.shape[0],)+(1,1))
period = np.reshape(period, (a.shape[0],)+(1,1))
c = np.empty((2,) + a.shape, dtype="float64")
c[0,:] = np.abs(a - b)
p_directions = [i for i,p in enumerate(periodicity) if p == True]
np_directions = [i for i,p in enumerate(periodicity) if p == False]
for d in p_directions:
c[1,d,:] = period[d,:] - np.abs(a - b)[d,:]
for d in np_directions:
c[1,d,:] = c[0,d,:]
d = np.amin(c, axis=0)**2
r2 = d.sum(axis=0)
if r2.size == 1:
return np.sqrt(r2[0,0])
return np.sqrt(r2)
[docs]def euclidean_dist(a, b):
r"""Find the Euclidean distance between two points.
Parameters
----------
a : array or list
Either an ndim long list of coordinates corresponding to a single point
or an (ndim, npoints) list of coordinates for many points in space.
b : array or list
Either an ndim long list of coordinates corresponding to a single point
or an (ndim, npoints) list of coordinates for many points in space.
Examples
--------
>>> a = [0.1, 0.1, 0.1]
>>> b = [0.9, 0,9, 0.9]
>>> period = 1.
>>> dist = euclidean_dist(a, b)
>>> dist
1.38564064606
"""
a = np.array(a)
b = np.array(b)
if a.shape != b.shape: RuntimeError("Arrays must be the same shape.")
c = a.copy()
np.subtract(c, b, c)
np.power(c, 2, c)
c = c.sum(axis = 0)
if isinstance(c, np.ndarray):
np.sqrt(c, c)
else:
# This happens if a and b only have one entry.
c = math.sqrt(c)
return c
[docs]def rotate_vector_3D(a, dim, angle):
r"""Rotates the elements of an array around an axis by some angle.
Given an array of 3D vectors a, this rotates them around a coordinate axis
by a clockwise angle. An alternative way to think about it is the
coordinate axes are rotated counterclockwise, which changes the directions
of the vectors accordingly.
Parameters
----------
a : array
An array of 3D vectors with dimension Nx3.
dim : integer
A integer giving the axis around which the vectors will be rotated.
(x, y, z) = (0, 1, 2).
angle : float
The angle in radians through which the vectors will be rotated
clockwise.
Examples
--------
>>> a = np.array([[1, 1, 0], [1, 0, 1], [0, 1, 1], [1, 1, 1], [3, 4, 5]])
>>> b = rotate_vector_3D(a, 2, np.pi/2)
>>> print b
[[ 1.00000000e+00 -1.00000000e+00 0.00000000e+00]
[ 6.12323400e-17 -1.00000000e+00 1.00000000e+00]
[ 1.00000000e+00 6.12323400e-17 1.00000000e+00]
[ 1.00000000e+00 -1.00000000e+00 1.00000000e+00]
[ 4.00000000e+00 -3.00000000e+00 5.00000000e+00]]
"""
mod = False
if len(a.shape) == 1:
mod = True
a = np.array([a])
if a.shape[1] !=3:
raise SyntaxError("The second dimension of the array a must be == 3!")
if dim == 0:
R = np.array([[1, 0,0],
[0, np.cos(angle), np.sin(angle)],
[0, -np.sin(angle), np.cos(angle)]])
elif dim == 1:
R = np.array([[np.cos(angle), 0, -np.sin(angle)],
[0, 1, 0],
[np.sin(angle), 0, np.cos(angle)]])
elif dim == 2:
R = np.array([[np.cos(angle), np.sin(angle), 0],
[-np.sin(angle), np.cos(angle), 0],
[0, 0, 1]])
else:
raise SyntaxError("dim must be 0, 1, or 2!")
if mod:
return np.dot(R, a.T).T[0]
else:
return np.dot(R, a.T).T
[docs]def modify_reference_frame(CoM, L, P=None, V=None):
r"""Rotates and translates data into a new reference frame to make
calculations easier.
This is primarily useful for calculations of halo data.
The data is translated into the center of mass frame.
Next, it is rotated such that the angular momentum vector for the data
is aligned with the z-axis. Put another way, if one calculates the angular
momentum vector on the data that comes out of this function, it will
always be along the positive z-axis.
If the center of mass is re-calculated, it will be at the origin.
Parameters
----------
CoM : array
The center of mass in 3D.
L : array
The angular momentum vector.
Optional
--------
P : array
The positions of the data to be modified (i.e. particle or grid cell
postions). The array should be Nx3.
V : array
The velocities of the data to be modified (i.e. particle or grid cell
velocities). The array should be Nx3.
Returns
-------
L : array
The angular momentum vector equal to [0, 0, 1] modulo machine error.
P : array
The modified positional data. Only returned if P is not None
V : array
The modified velocity data. Only returned if V is not None
Examples
--------
>>> CoM = np.array([0.5, 0.5, 0.5])
>>> L = np.array([1, 0, 0])
>>> P = np.array([[1, 0.5, 0.5], [0, 0.5, 0.5], [0.5, 0.5, 0.5], [0, 0, 0]])
>>> V = p.copy()
>>> LL, PP, VV = modify_reference_frame(CoM, L, P, V)
>>> LL
array([ 6.12323400e-17, 0.00000000e+00, 1.00000000e+00])
>>> PP
array([[ 3.06161700e-17, 0.00000000e+00, 5.00000000e-01],
[ -3.06161700e-17, 0.00000000e+00, -5.00000000e-01],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 5.00000000e-01, -5.00000000e-01, -5.00000000e-01]])
>>> VV
array([[ -5.00000000e-01, 5.00000000e-01, 1.00000000e+00],
[ -5.00000000e-01, 5.00000000e-01, 3.06161700e-17],
[ -5.00000000e-01, 5.00000000e-01, 5.00000000e-01],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00]])
"""
# First translate the positions to center of mass reference frame.
if P is not None:
P = P - CoM
# is the L vector pointing in the Z direction?
if np.inner(L[:2], L[:2]) == 0.0:
# the reason why we need to explicitly check for the above
# is that formula is used in denominator
# this just checks if we need to flip the z axis or not
if L[2] < 0.0:
# this is just a simple flip in direction of the z axis
if P is not None:
P = -P
if V is not None:
V = -V
# return the values
if V is None and P is not None:
return L, P
elif P is None and V is not None:
return L, V
else:
return L, P, V
# Normal vector is not aligned with simulation Z axis
# Therefore we are going to have to apply a rotation
# Now find the angle between modified L and the x-axis.
LL = L.copy()
LL[2] = 0.0
theta = np.arccos(np.inner(LL, [1.0, 0.0, 0.0]) / np.inner(LL, LL) ** 0.5)
if L[1] < 0.0:
theta = -theta
# Now rotate all the position, velocity, and L vectors by this much around
# the z axis.
if P is not None:
P = rotate_vector_3D(P, 2, theta)
if V is not None:
V = rotate_vector_3D(V, 2, theta)
L = rotate_vector_3D(L, 2, theta)
# Now find the angle between L and the z-axis.
theta = np.arccos(np.inner(L, [0.0, 0.0, 1.0]) / np.inner(L, L) ** 0.5)
# This time we rotate around the y axis.
if P is not None:
P = rotate_vector_3D(P, 1, theta)
if V is not None:
V = rotate_vector_3D(V, 1, theta)
L = rotate_vector_3D(L, 1, theta)
# return the values
if V is None and P is not None:
return L, P
elif P is None and V is not None:
return L, V
else:
return L, P, V
[docs]def compute_rotational_velocity(CoM, L, P, V):
r"""Computes the rotational velocity for some data around an axis.
This is primarily for halo computations.
Given some data, this computes the circular rotational velocity of each
point (particle) in reference to the axis defined by the angular momentum
vector.
This is accomplished by converting the reference frame of the center of
mass of the halo.
Parameters
----------
CoM : array
The center of mass in 3D.
L : array
The angular momentum vector.
P : array
The positions of the data to be modified (i.e. particle or grid cell
postions). The array should be Nx3.
V : array
The velocities of the data to be modified (i.e. particle or grid cell
velocities). The array should be Nx3.
Returns
-------
v : array
An array N elements long that gives the circular rotational velocity
for each datum (particle).
Examples
--------
>>> CoM = np.array([0, 0, 0])
>>> L = np.array([0, 0, 1])
>>> P = np.array([[1, 0, 0], [1, 1, 1], [0, 0, 1], [1, 1, 0]])
>>> V = np.array([[0, 1, 10], [-1, -1, -1], [1, 1, 1], [1, -1, -1]])
>>> circV = compute_rotational_velocity(CoM, L, P, V)
>>> circV
array([ 1. , 0. , 0. , 1.41421356])
"""
# First we translate into the simple coordinates.
L, P, V = modify_reference_frame(CoM, L, P, V)
# Find the vector in the plane of the galaxy for each position point
# that is perpendicular to the radial vector.
radperp = np.cross([0, 0, 1], P)
# Find the component of the velocity along the radperp vector.
# Unf., I don't think there's a better way to do this.
res = np.empty(V.shape[0], dtype='float64')
for i, rp in enumerate(radperp):
temp = np.dot(rp, V[i]) / np.dot(rp, rp) * rp
res[i] = np.dot(temp, temp)**0.5
return res
[docs]def compute_parallel_velocity(CoM, L, P, V):
r"""Computes the parallel velocity for some data around an axis.
This is primarily for halo computations.
Given some data, this computes the velocity component along the angular
momentum vector.
This is accomplished by converting the reference frame of the center of
mass of the halo.
Parameters
----------
CoM : array
The center of mass in 3D.
L : array
The angular momentum vector.
P : array
The positions of the data to be modified (i.e. particle or grid cell
postions). The array should be Nx3.
V : array
The velocities of the data to be modified (i.e. particle or grid cell
velocities). The array should be Nx3.
Returns
-------
v : array
An array N elements long that gives the parallel velocity for
each datum (particle).
Examples
--------
>>> CoM = np.array([0, 0, 0])
>>> L = np.array([0, 0, 1])
>>> P = np.array([[1, 0, 0], [1, 1, 1], [0, 0, 1], [1, 1, 0]])
>>> V = np.array([[0, 1, 10], [-1, -1, -1], [1, 1, 1], [1, -1, -1]])
>>> paraV = compute_parallel_velocity(CoM, L, P, V)
>>> paraV
array([10, -1, 1, -1])
"""
# First we translate into the simple coordinates.
L, P, V = modify_reference_frame(CoM, L, P, V)
# And return just the z-axis velocities.
return V[:,2]
[docs]def compute_radial_velocity(CoM, L, P, V):
r"""Computes the radial velocity for some data around an axis.
This is primarily for halo computations.
Given some data, this computes the radial velocity component for the data.
This is accomplished by converting the reference frame of the center of
mass of the halo.
Parameters
----------
CoM : array
The center of mass in 3D.
L : array
The angular momentum vector.
P : array
The positions of the data to be modified (i.e. particle or grid cell
postions). The array should be Nx3.
V : array
The velocities of the data to be modified (i.e. particle or grid cell
velocities). The array should be Nx3.
Returns
-------
v : array
An array N elements long that gives the radial velocity for
each datum (particle).
Examples
--------
>>> CoM = np.array([0, 0, 0])
>>> L = np.array([0, 0, 1])
>>> P = np.array([[1, 0, 0], [1, 1, 1], [0, 0, 1], [1, 1, 0]])
>>> V = np.array([[0, 1, 10], [-1, -1, -1], [1, 1, 1], [1, -1, -1]])
>>> radV = compute_radial_velocity(CoM, L, P, V)
>>> radV
array([ 1. , 1.41421356 , 0. , 0.])
"""
# First we translate into the simple coordinates.
L, P, V = modify_reference_frame(CoM, L, P, V)
# We find the tangential velocity by dotting the velocity vector
# with the cylindrical radial vector for this point.
# Unf., I don't think there's a better way to do this.
P[:,2] = 0
res = np.empty(V.shape[0], dtype='float64')
for i, rad in enumerate(P):
temp = np.dot(rad, V[i]) / np.dot(rad, rad) * rad
res[i] = np.dot(temp, temp)**0.5
return res
[docs]def compute_cylindrical_radius(CoM, L, P, V):
r"""Compute the radius for some data around an axis in cylindrical
coordinates.
This is primarily for halo computations.
Given some data, this computes the cylindrical radius for each point.
This is accomplished by converting the reference frame of the center of
mass of the halo.
Parameters
----------
CoM : array
The center of mass in 3D.
L : array
The angular momentum vector.
P : array
The positions of the data to be modified (i.e. particle or grid cell
postions). The array should be Nx3.
V : array
The velocities of the data to be modified (i.e. particle or grid cell
velocities). The array should be Nx3.
Returns
-------
cyl_r : array
An array N elements long that gives the radial velocity for
each datum (particle).
Examples
--------
>>> CoM = np.array([0, 0, 0])
>>> L = np.array([0, 0, 1])
>>> P = np.array([[1, 0, 0], [1, 1, 1], [0, 0, 1], [1, 1, 0]])
>>> V = np.array([[0, 1, 10], [-1, -1, -1], [1, 1, 1], [1, -1, -1]])
>>> cyl_r = compute_cylindrical_radius(CoM, L, P, V)
>>> cyl_r
array([ 1. , 1.41421356, 0. , 1.41421356])
"""
# First we translate into the simple coordinates.
L, P, V = modify_reference_frame(CoM, L, P, V)
# Demote all the positions to the z=0 plane, which makes the distance
# calculation very easy.
P[:,2] = 0
return np.sqrt((P * P).sum(axis=1))
[docs]def ortho_find(vec1):
r"""Find two complementary orthonormal vectors to a given vector.
For any given non-zero vector, there are infinite pairs of vectors
orthonormal to it. This function gives you one arbitrary pair from
that set along with the normalized version of the original vector.
Parameters
----------
vec1 : array_like
An array or list to represent a 3-vector.
Returns
-------
vec1 : array
The original 3-vector array after having been normalized.
vec2 : array
A 3-vector array which is orthonormal to vec1.
vec3 : array
A 3-vector array which is orthonormal to vec1 and vec2.
Raises
------
ValueError
If input vector is the zero vector.
Notes
-----
Our initial vector is `vec1` which consists of 3 components: `x1`, `y1`,
and `z1`. ortho_find determines a vector, `vec2`, which is orthonormal
to `vec1` by finding a vector which has a zero-value dot-product with
`vec1`.
.. math::
vec1 \cdot vec2 = x_1 x_2 + y_1 y_2 + z_1 z_2 = 0
As a starting point, we arbitrarily choose `vec2` to have `x2` = 1,
`y2` = 0:
.. math::
vec1 \cdot vec2 = x_1 + (z_1 z_2) = 0
\rightarrow z_2 = -(x_1 / z_1)
Of course, this will fail if `z1` = 0, in which case, let's say use
`z2` = 1 and `x2` = 0:
.. math::
\rightarrow y_2 = -(z_1 / y_1)
Similarly, if `y1` = 0, this case will fail, in which case we use
`y2` = 1 and `z2` = 0:
.. math::
\rightarrow x_2 = -(y_1 / x_1)
Since we don't allow `vec1` to be zero, all cases are accounted for.
Producing `vec3`, the complementary orthonormal vector to `vec1` and `vec2`
is accomplished by simply taking the cross product of `vec1` and `vec2`.
Examples
--------
>>> a = [1.0, 2.0, 3.0]
>>> a, b, c = ortho_find(a)
>>> a
array([ 0.26726124, 0.53452248, 0.80178373])
>>> b
array([ 0.9486833 , 0. , -0.31622777])
>>> c
array([-0.16903085, 0.84515425, -0.50709255])
"""
vec1 = np.array(vec1, dtype=np.float64)
# Normalize
norm = np.sqrt(np.vdot(vec1, vec1))
if norm == 0:
raise ValueError("Zero vector used as input.")
vec1 /= norm
x1 = vec1[0]
y1 = vec1[1]
z1 = vec1[2]
if z1 != 0:
x2 = 1.0
y2 = 0.0
z2 = -(x1 / z1)
norm2 = (1.0 + z2 ** 2.0) ** (0.5)
elif y1 != 0:
x2 = 0.0
z2 = 1.0
y2 = -(z1 / y1)
norm2 = (1.0 + y2 ** 2.0) ** (0.5)
else:
y2 = 1.0
z2 = 0.0
x2 = -(y1 / x1)
norm2 = (1.0 + z2 ** 2.0) ** (0.5)
vec2 = np.array([x2,y2,z2])
vec2 /= norm2
vec3 = np.cross(vec1, vec2)
return vec1, vec2, vec3
[docs]def quartiles(a, axis=None, out=None, overwrite_input=False):
"""
Compute the quartile values (25% and 75%) along the specified axis
in the same way that the numpy.median calculates the median (50%) value
alone a specified axis. Check numpy.median for details, as it is
virtually the same algorithm.
Returns an array of the quartiles of the array elements [lower quartile,
upper quartile].
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
axis : {None, int}, optional
Axis along which the quartiles are computed. The default (axis=None)
is to compute the quartiles along a flattened version of the array.
out : ndarray, optional
Alternative output array in which to place the result. It must
have the same shape and buffer length as the expected output,
but the type (of the output) will be cast if necessary.
overwrite_input : {False, True}, optional
If True, then allow use of memory of input array (a) for
calculations. The input array will be modified by the call to
quartiles. This will save memory when you do not need to preserve
the contents of the input array. Treat the input as undefined,
but it will probably be fully or partially sorted. Default is
False. Note that, if `overwrite_input` is True and the input
is not already an ndarray, an error will be raised.
Returns
-------
quartiles : ndarray
A new 2D array holding the result (unless `out` is specified, in
which case that array is returned instead). If the input contains
integers, or floats of smaller precision than 64, then the output
data-type is float64. Otherwise, the output data-type is the same
as that of the input.
See Also
--------
numpy.median, numpy.mean, numpy.percentile
Notes
-----
Given a vector V of length N, the quartiles of V are the 25% and 75% values
of a sorted copy of V, ``V_sorted`` - i.e., ``V_sorted[(N-1)/4]`` and
``3*V_sorted[(N-1)/4]``, when N is odd. When N is even, it is the average
of the two values bounding these values of ``V_sorted``.
Examples
--------
>>> a = np.arange(100).reshape(10,10)
>>> a
array([[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19],
[20, 21, 22, 23, 24, 25, 26, 27, 28, 29],
[30, 31, 32, 33, 34, 35, 36, 37, 38, 39],
[40, 41, 42, 43, 44, 45, 46, 47, 48, 49],
[50, 51, 52, 53, 54, 55, 56, 57, 58, 59],
[60, 61, 62, 63, 64, 65, 66, 67, 68, 69],
[70, 71, 72, 73, 74, 75, 76, 77, 78, 79],
[80, 81, 82, 83, 84, 85, 86, 87, 88, 89],
[90, 91, 92, 93, 94, 95, 96, 97, 98, 99]])
>>> mu.quartiles(a)
array([ 24.5, 74.5])
>>> mu.quartiles(a,axis=0)
array([[ 15., 16., 17., 18., 19., 20., 21., 22., 23., 24.],
[ 65., 66., 67., 68., 69., 70., 71., 72., 73., 74.]])
>>> mu.quartiles(a,axis=1)
array([[ 1.5, 11.5, 21.5, 31.5, 41.5, 51.5, 61.5, 71.5, 81.5,
91.5],
[ 6.5, 16.5, 26.5, 36.5, 46.5, 56.5, 66.5, 76.5, 86.5,
96.5]])
"""
if overwrite_input:
if axis is None:
sorted = a.ravel()
sorted.sort()
else:
a.sort(axis=axis)
sorted = a
else:
sorted = np.sort(a, axis=axis)
if axis is None:
axis = 0
indexer = [slice(None)] * sorted.ndim
indices = [int(sorted.shape[axis]/4), int(sorted.shape[axis]*.75)]
result = []
for index in indices:
if sorted.shape[axis] % 2 == 1:
# index with slice to allow mean (below) to work
indexer[axis] = slice(index, index+1)
else:
indexer[axis] = slice(index-1, index+1)
# special cases for small arrays
if sorted.shape[axis] == 2:
# index with slice to allow mean (below) to work
indexer[axis] = slice(index, index+1)
# Use mean in odd and even case to coerce data type
# and check, use out array.
result.append(np.mean(sorted[indexer], axis=axis, out=out))
return np.array(result)
[docs]def get_rotation_matrix(theta, rot_vector):
"""
Given an angle theta and a 3D vector rot_vector, this routine
computes the rotation matrix corresponding to rotating theta
radians about rot_vector.
Parameters
----------
theta : scalar
The angle in radians.
rot_vector : array_like
The axis of rotation. Must be 3D.
Returns
-------
rot_matrix : ndarray
A new 3x3 2D array. This is the representation of a
rotation of theta radians about rot_vector in the simulation
box coordinate frame
See Also
--------
ortho_find
Examples
--------
>>> a = [0,1,0]
>>> theta = 0.785398163 # pi/4
>>> rot = mu.get_rotation_matrix(theta,a)
>>> rot
array([[ 0.70710678, 0. , 0.70710678],
[ 0. , 1. , 0. ],
[-0.70710678, 0. , 0.70710678]])
>>> np.dot(rot,a)
array([ 0., 1., 0.])
# since a is an eigenvector by construction
>>> np.dot(rot,[1,0,0])
array([ 0.70710678, 0. , -0.70710678])
"""
ux = rot_vector[0]
uy = rot_vector[1]
uz = rot_vector[2]
cost = np.cos(theta)
sint = np.sin(theta)
R = np.array([[cost+ux**2*(1-cost), ux*uy*(1-cost)-uz*sint, ux*uz*(1-cost)+uy*sint],
[uy*ux*(1-cost)+uz*sint, cost+uy**2*(1-cost), uy*uz*(1-cost)-ux*sint],
[uz*ux*(1-cost)-uy*sint, uz*uy*(1-cost)+ux*sint, cost+uz**2*(1-cost)]])
return R
[docs]def get_ortho_basis(normal):
xprime = np.cross([0.0,1.0,0.0],normal)
if np.sum(xprime) == 0: xprime = np.array([0.0, 0.0, 1.0])
yprime = np.cross(normal,xprime)
zprime = normal
return (xprime, yprime, zprime)
[docs]def get_sph_r(coords):
# The spherical coordinates radius is simply the magnitude of the
# coordinate vector.
return np.sqrt(np.sum(coords**2, axis=0))
[docs]def resize_vector(vector,vector_array):
if len(vector_array.shape) == 4:
res_vector = np.resize(vector,(3,1,1,1))
else:
res_vector = np.resize(vector,(3,1))
return res_vector
[docs]def get_sph_theta(coords, normal):
# The angle (theta) with respect to the normal (J), is the arccos
# of the dot product of the normal with the normalized coordinate
# vector.
res_normal = resize_vector(normal, coords)
tile_shape = [1] + list(coords.shape)[1:]
J = np.tile(res_normal,tile_shape)
JdotCoords = np.sum(J*coords,axis=0)
return np.arccos( JdotCoords / np.sqrt(np.sum(coords**2,axis=0)) )
[docs]def get_sph_phi(coords, normal):
# We have freedom with respect to what axis (xprime) to define
# the disk angle. Here I've chosen to use the axis that is
# perpendicular to the normal and the y-axis. When normal ==
# y-hat, then set xprime = z-hat. With this definition, when
# normal == z-hat (as is typical), then xprime == x-hat.
#
# The angle is then given by the arctan of the ratio of the
# yprime-component and the xprime-component of the coordinate
# vector.
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, coords)
res_yprime = resize_vector(yprime, coords)
tile_shape = [1] + list(coords.shape)[1:]
Jx = np.tile(res_xprime,tile_shape)
Jy = np.tile(res_yprime,tile_shape)
Px = np.sum(Jx*coords,axis=0)
Py = np.sum(Jy*coords,axis=0)
return np.arctan2(Py,Px)
[docs]def get_cyl_r(coords, normal):
# The cross product of the normal (J) with a coordinate vector
# gives a vector of magnitude equal to the cylindrical radius.
res_normal = resize_vector(normal, coords)
tile_shape = [1] + list(coords.shape)[1:]
J = np.tile(res_normal, tile_shape)
JcrossCoords = np.cross(J, coords, axisa=0, axisb=0, axisc=0)
return np.sqrt(np.sum(JcrossCoords**2, axis=0))
[docs]def get_cyl_z(coords, normal):
# The dot product of the normal (J) with the coordinate vector
# gives the cylindrical height.
res_normal = resize_vector(normal, coords)
tile_shape = [1] + list(coords.shape)[1:]
J = np.tile(res_normal, tile_shape)
return np.sum(J*coords, axis=0)
[docs]def get_cyl_theta(coords, normal):
# This is identical to the spherical phi component
return get_sph_phi(coords, normal)
[docs]def get_cyl_r_component(vectors, theta, normal):
# The r of a vector is the vector dotted with rhat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, vectors)
res_yprime = resize_vector(yprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
Jx = np.tile(res_xprime,tile_shape)
Jy = np.tile(res_yprime,tile_shape)
rhat = Jx*np.cos(theta) + Jy*np.sin(theta)
return np.sum(vectors*rhat,axis=0)
[docs]def get_cyl_theta_component(vectors, theta, normal):
# The theta component of a vector is the vector dotted with thetahat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, vectors)
res_yprime = resize_vector(yprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
Jx = np.tile(res_xprime,tile_shape)
Jy = np.tile(res_yprime,tile_shape)
thetahat = -Jx*np.sin(theta) + Jy*np.cos(theta)
return np.sum(vectors*thetahat, axis=0)
[docs]def get_cyl_z_component(vectors, normal):
# The z component of a vector is the vector dotted with zhat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_zprime = resize_vector(zprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
zhat = np.tile(res_zprime, tile_shape)
return np.sum(vectors*zhat, axis=0)
[docs]def get_sph_r_component(vectors, theta, phi, normal):
# The r component of a vector is the vector dotted with rhat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, vectors)
res_yprime = resize_vector(yprime, vectors)
res_zprime = resize_vector(zprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
Jx, Jy, Jz = (
YTArray(np.tile(rprime, tile_shape), "")
for rprime in (res_xprime, res_yprime, res_zprime))
rhat = Jx*np.sin(theta)*np.cos(phi) + \
Jy*np.sin(theta)*np.sin(phi) + \
Jz*np.cos(theta)
return np.sum(vectors*rhat, axis=0)
[docs]def get_sph_phi_component(vectors, phi, normal):
# The phi component of a vector is the vector dotted with phihat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, vectors)
res_yprime = resize_vector(yprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
Jx = YTArray(np.tile(res_xprime,tile_shape), "")
Jy = YTArray(np.tile(res_yprime,tile_shape), "")
phihat = -Jx*np.sin(phi) + Jy*np.cos(phi)
return np.sum(vectors*phihat, axis=0)
[docs]def get_sph_theta_component(vectors, theta, phi, normal):
# The theta component of a vector is the vector dotted with thetahat
(xprime, yprime, zprime) = get_ortho_basis(normal)
res_xprime = resize_vector(xprime, vectors)
res_yprime = resize_vector(yprime, vectors)
res_zprime = resize_vector(zprime, vectors)
tile_shape = [1] + list(vectors.shape)[1:]
Jx, Jy, Jz = (
YTArray(np.tile(rprime, tile_shape), "")
for rprime in (res_xprime, res_yprime, res_zprime))
thetahat = Jx*np.cos(theta)*np.cos(phi) + \
Jy*np.cos(theta)*np.sin(phi) - \
Jz*np.sin(theta)
return np.sum(vectors*thetahat, axis=0)