# Source code for yt.utilities.math_utils

"""
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 : ~yt.data_objects.static_output.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 is True] np_directions = [i for i,p in enumerate(periodicity) if p is 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 positions). 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 positions). 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 positions). 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 positions). 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 positions). 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_perspective_matrix(fovy, aspect, z_near, z_far): """ Given a field of view in radians, an aspect ratio, and a near and far plane distance, this routine computes the transformation matrix corresponding to perspective projection using homogenous coordinates. Parameters ---------- fovy : scalar The angle in degrees of the field of view. aspect : scalar The aspect ratio of width / height for the projection. z_near : scalar The distance of the near plane from the camera. z_far : scalar The distance of the far plane from the camera. Returns ------- persp_matrix : ndarray A new 4x4 2D array. Represents a perspective transformation in homogeneous coordinates. Note that this matrix does not actually perform the projection. After multiplying a 4D vector of the form (x_0, y_0, z_0, 1.0), the point will be transformed to some (x_1, y_1, z_1, w). The final projection is applied by performing a divide by w, that is (x_1/w, y_1/w, z_1/w, w/w). The matrix uses a row-major ordering, rather than the column major ordering typically used by OpenGL. Notes ----- The usage of 4D homogeneous coordinates is for OpenGL and GPU hardware that automatically performs the divide by w operation. See the following for more details about the OpenGL perspective matrices. http://www.tomdalling.com/blog/modern-opengl/explaining-homogenous-coordinates-and-projective-geometry/ http://www.songho.ca/opengl/gl_projectionmatrix.html """ tan_half_fovy = np.tan(np.radians(fovy) / 2) result = np.zeros( (4, 4), dtype = 'float32', order = 'C') #result[0][0] = 1 / (aspect * tan_half_fovy) #result[1][1] = 1 / tan_half_fovy #result[2][2] = - (z_far + z_near) / (z_far - z_near) #result[3][2] = -1 #result[2][3] = -(2 * z_far * z_near) / (z_far - z_near) f = z_far n = z_near t = tan_half_fovy * n b = -t * aspect r = t * aspect l = - t * aspect result[0][0] = (2 * n) / (r - l) result[2][0] = (r + l) / (r - l) result[1][1] = (2 * n) / (t - b) result[1][2] = (t + b) / (t - b) result[2][2] = -(f + n) / (f - n) result[2][3] = -2*f*n/(f - n) result[3][2] = -1 return result
[docs]def get_orthographic_matrix(maxr, aspect, z_near, z_far): """ Given a field of view in radians, an aspect ratio, and a near and far plane distance, this routine computes the transformation matrix corresponding to perspective projection using homogenous coordinates. Parameters ---------- maxr : scalar should be max(|x|, |y|) aspect : scalar The aspect ratio of width / height for the projection. z_near : scalar The distance of the near plane from the camera. z_far : scalar The distance of the far plane from the camera. Returns ------- persp_matrix : ndarray A new 4x4 2D array. Represents a perspective transformation in homogeneous coordinates. Note that this matrix does not actually perform the projection. After multiplying a 4D vector of the form (x_0, y_0, z_0, 1.0), the point will be transformed to some (x_1, y_1, z_1, w). The final projection is applied by performing a divide by w, that is (x_1/w, y_1/w, z_1/w, w/w). The matrix uses a row-major ordering, rather than the column major ordering typically used by OpenGL. Notes ----- The usage of 4D homogeneous coordinates is for OpenGL and GPU hardware that automatically performs the divide by w operation. See the following for more details about the OpenGL perspective matrices. http://www.scratchapixel.com/lessons/3d-basic-rendering/perspective-and-orthographic-projection-matrix/orthographic-projection-matrix http://www.tomdalling.com/blog/modern-opengl/explaining-homogenous-coordinates-and-projective-geometry/ http://www.songho.ca/opengl/gl_projectionmatrix.html """ r = maxr * aspect t = maxr l = -r b = -t result = np.zeros( (4, 4), dtype = 'float32', order = 'C') result[0][0] = 2.0 / (r - l) result[1][1] = 2.0 / (t - b) result[2][2] = -2.0 / (z_far - z_near) result[3][3] = 1 result[3][0] = - (r+l)/(r-l) result[3][1] = -(t+b)/(t-b) result[3][2] = -(z_far + z_near) / (z_far - z_near) return result
[docs]def get_lookat_matrix(eye, center, up): """ Given the position of a camera, the point it is looking at, and an up-direction. Computes the lookat matrix that moves all vectors such that the camera is at the origin of the coordinate system, looking down the z-axis. Parameters ---------- eye : array_like The position of the camera. Must be 3D. center : array_like The location that the camera is looking at. Must be 3D. up : array_like The direction that is considered up for the camera. Must be 3D. Returns ------- lookat_matrix : ndarray A new 4x4 2D array in homogeneous coordinates. This matrix moves all vectors in the same way required to move the camera to the origin of the coordinate system, with it pointing down the negative z-axis. """ eye = np.array(eye) center = np.array(center) up = np.array(up) f = (center - eye) / np.linalg.norm(center - eye) s = np.cross(f, up) / np.linalg.norm(np.cross(f, up)) u = np.cross(s, f) result = np.zeros ( (4, 4), dtype = 'float32', order = 'C') result[0][0] = s[0] result[0][1] = s[1] result[0][2] = s[2] result[1][0] = u[0] result[1][1] = u[1] result[1][2] = u[2] result[2][0] =-f[0] result[2][1] =-f[1] result[2][2] =-f[2] result[0][3] =-np.dot(s, eye) result[1][3] =-np.dot(u, eye) result[2][3] = np.dot(f, eye) result[3][3] = 1.0 return result
[docs]def get_translate_matrix(dx, dy, dz): """ Given a movement amount for each coordinate, creates a translation matrix that moves the vector by each amount. Parameters ---------- dx : scalar A translation amount for the x-coordinate dy : scalar A translation amount for the y-coordinate dz : scalar A translation amount for the z-coordinate Returns ------- trans_matrix : ndarray A new 4x4 2D array. Represents a translation by dx, dy and dz in each coordinate respectively. """ result = np.zeros( (4, 4), dtype = 'float32', order = 'C') result[0][0] = 1.0 result[1][1] = 1.0 result[2][2] = 1.0 result[3][3] = 1.0 result[0][3] = dx result[1][3] = dy result[2][3] = dz return result
[docs]def get_scale_matrix(dx, dy, dz): """ Given a scaling factor for each coordinate, returns a matrix that corresponds to the given scaling amounts. Parameters ---------- dx : scalar A scaling factor for the x-coordinate. dy : scalar A scaling factor for the y-coordinate. dz : scalar A scaling factor for the z-coordinate. Returns ------- scale_matrix : ndarray A new 4x4 2D array. Represents a scaling by dx, dy, and dz in each coordinate respectively. """ result = np.zeros( (4, 4), dtype = 'float32', order = 'C') result[0][0] = dx result[1][1] = dy result[2][2] = dz result[3][3] = 1 return 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 quaternion_mult(q1, q2): ''' Multiply two quaternions. The inputs are 4-component numpy arrays in the order [w, x, y, z]. ''' w = q1[0]*q2[0] - q1[1]*q2[1] - q1[2]*q2[2] - q1[3]*q2[3] x = q1[0]*q2[1] + q1[1]*q2[0] + q1[2]*q2[3] - q1[3]*q2[2] y = q1[0]*q2[2] + q1[2]*q2[0] + q1[3]*q2[1] - q1[1]*q2[3] z = q1[0]*q2[3] + q1[3]*q2[0] + q1[1]*q2[2] - q1[2]*q2[1] return np.array([w, x, y, z])
[docs]def quaternion_to_rotation_matrix(quaternion): """ This converts a quaternion representation of on orientation to a rotation matrix. The input is a 4-component numpy array in the order [w, x, y, z], and the output is a 3x3 matrix stored as a 2D numpy array. We follow the approach in "3D Math Primer for Graphics and Game Development" by Dunn and Parberry. """ w = quaternion[0] x = quaternion[1] y = quaternion[2] z = quaternion[3] R = np.empty((3, 3), dtype=np.float64) R[0][0] = 1.0 - 2.0*y**2 - 2.0*z**2 R[0][1] = 2.0*x*y + 2.0*w*z R[0][2] = 2.0*x*z - 2.0*w*y R[1][0] = 2.0*x*y - 2.0*w*z R[1][1] = 1.0 - 2.0*x**2 - 2.0*z**2 R[1][2] = 2.0*y*z + 2.0*w*x R[2][0] = 2.0*x*z + 2.0*w*y R[2][1] = 2.0*y*z - 2.0*w*x R[2][2] = 1.0 - 2.0*x**2 - 2.0*y**2 return R
[docs]def rotation_matrix_to_quaternion(rot_matrix): ''' Convert a rotation matrix-based representation of an orientation to a quaternion. The input should be a 3x3 rotation matrix, while the output will be a 4-component numpy array. We follow the approach in "3D Math Primer for Graphics and Game Development" by Dunn and Parberry. ''' m11 = rot_matrix[0][0] m12 = rot_matrix[0][1] m13 = rot_matrix[0][2] m21 = rot_matrix[1][0] m22 = rot_matrix[1][1] m23 = rot_matrix[1][2] m31 = rot_matrix[2][0] m32 = rot_matrix[2][1] m33 = rot_matrix[2][2] four_w_squared_minus_1 = m11 + m22 + m33 four_x_squared_minus_1 = m11 - m22 - m33 four_y_squared_minus_1 = m22 - m11 - m33 four_z_squared_minus_1 = m33 - m11 - m22 max_index = 0 four_max_squared_minus_1 = four_w_squared_minus_1 if (four_x_squared_minus_1 > four_max_squared_minus_1): four_max_squared_minus_1 = four_x_squared_minus_1 max_index = 1 if (four_y_squared_minus_1 > four_max_squared_minus_1): four_max_squared_minus_1 = four_y_squared_minus_1 max_index = 2 if (four_z_squared_minus_1 > four_max_squared_minus_1): four_max_squared_minus_1 = four_z_squared_minus_1 max_index = 3 max_val = 0.5*np.sqrt(four_max_squared_minus_1 + 1.0) mult = 0.25 / max_val if (max_index == 0): w = max_val x = (m23 - m32) * mult y = (m31 - m13) * mult z = (m12 - m21) * mult elif (max_index == 1): x = max_val w = (m23 - m32) * mult y = (m12 + m21) * mult z = (m31 + m13) * mult elif (max_index == 2): y = max_val w = (m31 - m13) * mult x = (m12 + m21) * mult z = (m23 + m32) * mult elif (max_index == 3): z = max_val w = (m12 - m21) * mult x = (m31 + m13) * mult y = (m23 + m32) * mult return np.array([w, x, y, z])
[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 normalize_vector(vector): # this function normalizes # an input vector L2 = np.atleast_1d(np.linalg.norm(vector)) L2[L2==0] = 1.0 vector = vector / L2 return 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) # check if the normal vector is normalized # since arccos requires the vector to be normalised res_normal = normalize_vector(res_normal) tile_shape = [1] + list(coords.shape)[1:] J = np.tile(res_normal,tile_shape) JdotCoords = np.sum(J*coords,axis=0) with np.errstate(invalid='ignore'): ret = np.arccos( JdotCoords / np.sqrt(np.sum(coords**2,axis=0))) ret[np.isnan(ret)] = 0 return ret
[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. normal = normalize_vector(normal) (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) res_normal = normalize_vector(res_normal) 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) res_normal = normalize_vector(res_normal) 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 normal = normalize_vector(normal) (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 normal = normalize_vector(normal) (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 normal = normalize_vector(normal) (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 normal = normalize_vector(normal) (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 normal = normalize_vector(normal) (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 normal = normalize_vector(normal) (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)