Source code for halotools.mock_observables.tensor_calculations.inertia_tensor

from __future__ import absolute_import, division, print_function, unicode_literals

import numpy as np
import multiprocessing
from functools import partial

from .engines import inertia_tensor_per_object_engine

from ..mock_observables_helpers import get_num_threads, get_period, enforce_sample_respects_pbcs
from ..pair_counters.mesh_helpers import (_set_approximate_cell_sizes,
    _cell1_parallelization_indices, _enclose_in_box, _enforce_maximum_search_length)
from ..pair_counters.rectangular_mesh import RectangularDoubleMesh

__all__ = ('inertia_tensor_per_object', )

[docs] def inertia_tensor_per_object(sample1, sample2, weights2, smoothing_scale, period=None, num_threads=1, approx_cell1_size=None, approx_cell2_size=None): r""" For each point in `sample1`, identify all `sample2` points within the input `smoothing_scale`; using those points together with the input `weights2`, the `inertia_tensor_per_object` function calculates the inertia tensor of the mass distribution surrounding each point in `sample1`. For every pair of points, :math:`i, j` in `sample1`, `sample2`, the contribution to the inertia tensor is: .. math:: \mathcal{I}_{\rm ij} = m_{\rm j}\begin{bmatrix} \delta x_{\rm ij}*\delta x_{\rm ij} & \delta x_{\rm ij}*\delta y_{\rm ij} & \delta x_{\rm ij}*\delta z_{\rm ij} \\ \delta y_{\rm ij}*\delta x_{\rm ij} & \delta y_{\rm ij}*\delta y_{\rm ij} & \delta y_{\rm ij}*\delta z_{\rm ij} \\ \delta z_{\rm ij}*\delta x_{\rm ij} & \delta z_{\rm ij}*\delta y_{\rm ij} & \delta z_{\rm ij}*\delta z_{\rm ij} \end{bmatrix} The :math:`\delta x_{\rm ij}`, :math:`\delta y_{\rm ij}`, and :math:`\delta z_{\rm ij}` terms store the coordinate distances between the pair of points (optionally accounting for periodic boundary conditions), and :math:`m_{\rm j}` stores the mass of the `sample2` point. To calculate the inertia tensor :math:`\mathcal{I}_{\rm i}` for the :math:`i^{\rm th}` point in `sample1`, the `inertia_tensor_per_object` function sums up the contributions :math:`\mathcal{I}_{\rm ij}` for all :math:`j` such that the distance between the two points :math:`D_{\rm ij}` is less than the smoothing scale :math:`D_{\rm smooth}`: .. math:: \mathcal{I}_{\rm i} = \sum_{j}^{r_{\rm ij} < D_{\rm smooth}} \mathcal{I}_{\rm ij} Parameters ---------- sample1 : array_like Numpy array of shape (npts1, 3) storing 3-D positions of points. See the :ref:`mock_obs_pos_formatting` documentation page, or the Examples section below, for instructions on how to transform your coordinate position arrays into the format accepted by the ``sample1`` and ``sample2`` arguments. Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools. sample2 : array_like Numpy array of shape (npts2, 3) storing 3-D positions of the point masses used to calculate the inertia tensor of every `sample1` point. weights2 : array_like Numpy array of shape (npts2,) storing the mass of each `sample2` point used to calculate the inertia tensor of every `sample1` point. smoothing_scale : float Three-dimensional distance from each `sample1` point defining which points in `sample2` are used to compute the inertia tensor period : array_like, optional Length-3 sequence defining the periodic boundary conditions in each dimension. If you instead provide a single scalar, Lbox, period is assumed to be the same in all Cartesian directions. Default is None, in which case no PBCs will be applied. num_threads : int, optional Number of threads to use in calculation, where parallelization is performed using the python ``multiprocessing`` module. Default is 1 for a purely serial calculation, in which case a multiprocessing Pool object will never be instantiated. A string 'max' may be used to indicate that the pair counters should use all available cores on the machine. approx_cell1_size : array_like, optional Length-3 array serving as a guess for the optimal manner by how points will be apportioned into subvolumes of the simulation box. The optimum choice unavoidably depends on the specs of your machine. Default choice is to use Lbox/10 in each dimension, which will return reasonable result performance for most use-cases. Performance can vary sensitively with this parameter, so it is highly recommended that you experiment with this parameter when carrying out performance-critical calculations. approx_cell2_size : array_like, optional Analogous to ``approx_cell1_size``, but for sample2. See comments for ``approx_cell1_size`` for details. Returns ------- inertia_tensors : ndarray Numpy array of shape (npts1, 3, 3) storing the inertia tensor for every object in `sample1`. sum_of_masses : ndarray Numpy array of shape (npts1, ) storing the sum of the masses of the `sample2` points that fall within `smoothing_scale` of each `sample1` point Examples -------- >>> npts1, npts2 = 50, 75 >>> sample1 = np.random.random((npts1, 3)) >>> sample2 = np.random.random((npts2, 3)) >>> weights2 = np.random.random(npts2) >>> smoothing_scale = 0.1 >>> result = inertia_tensor_per_object(sample1, sample2, weights2, smoothing_scale) Notes ----- There are several convenience functions available to derive quantities from the returned inertia tensors: * `~halotools.mock_observables.principal_axes_from_inertia_tensors` * `~halotools.mock_observables.sphericity_from_inertia_tensors` * `~halotools.mock_observables.triaxility_from_inertia_tensors` """ num_threads = get_num_threads(num_threads, enforce_max_cores=False) period, PBCs = get_period(period) # At this point, period may still be set to None, # in which case we must remap our points inside the smallest enclosing cube # and set ``period`` equal to this cube size. if period is None: x1in, y1in, z1in, x2in, y2in, z2in, period = ( _enclose_in_box( sample1[:, 0], sample1[:, 1], sample1[:, 2], sample2[:, 0], sample2[:, 1], sample2[:, 2], min_size=[smoothing_scale*3.0, smoothing_scale*3.0, smoothing_scale*3.0])) else: x1in = sample1[:, 0] y1in = sample1[:, 1] z1in = sample1[:, 2] x2in = sample2[:, 0] y2in = sample2[:, 1] z2in = sample2[:, 2] msg = "np.shape(weights2) = {0} should be ({1}, )" assert np.shape(weights2) == (sample2.shape[0], ), msg.format(np.shape(weights2), sample2.shape[0]) xperiod, yperiod, zperiod = period _enforce_maximum_search_length(smoothing_scale, period) enforce_sample_respects_pbcs(x1in, y1in, z1in, period) enforce_sample_respects_pbcs(x2in, y2in, z2in, period) search_xlength = smoothing_scale search_ylength = smoothing_scale search_zlength = smoothing_scale # Compute the estimates for the cell sizes approx_cell1_size, approx_cell2_size = ( _set_approximate_cell_sizes(approx_cell1_size, approx_cell2_size, period) ) approx_x1cell_size, approx_y1cell_size, approx_z1cell_size = approx_cell1_size approx_x2cell_size, approx_y2cell_size, approx_z2cell_size = approx_cell2_size # Build the rectangular mesh double_mesh = RectangularDoubleMesh(x1in, y1in, z1in, x2in, y2in, z2in, approx_x1cell_size, approx_y1cell_size, approx_z1cell_size, approx_x2cell_size, approx_y2cell_size, approx_z2cell_size, search_xlength, search_ylength, search_zlength, xperiod, yperiod, zperiod, PBCs) # Create a function object that has a single argument, for parallelization purposes engine = partial(inertia_tensor_per_object_engine, double_mesh, x1in, y1in, z1in, x2in, y2in, z2in, weights2, smoothing_scale) # Calculate the cell1 indices that will be looped over by the engine num_threads, cell1_tuples = _cell1_parallelization_indices( double_mesh.mesh1.ncells, num_threads) if num_threads > 1: pool = multiprocessing.Pool(num_threads) result =, cell1_tuples) tensors = np.array([r[0] for r in result]) sum_of_masses = np.array([r[1] for r in result]) tensors = np.sum(tensors, axis=0) sum_of_masses = np.sum(sum_of_masses, axis=0) pool.close() else: result = engine(cell1_tuples[0]) tensors, sum_of_masses = result return np.array(tensors), np.array(sum_of_masses)