source: sasmodels/sasmodels/jitter.py @ 0d5a655

#!/usr/bin/env python
"""
Jitter Explorer
===============

Application to explore orientation angle and angular dispersity.
"""
from __future__ import division, print_function

import sys, os
sys.path.insert(0, os.path.dirname(os.path.dirname(os.path.realpath(__file__))))

import argparse

import mpl_toolkits.mplot3d   # Adds projection='3d' option to subplot
import matplotlib.pyplot as plt
from matplotlib.widgets import Slider, CheckButtons
from matplotlib import cm
import numpy as np
from numpy import pi, cos, sin, sqrt, exp, degrees, radians

def draw_beam(ax, view=(0, 0)):
    """
    Draw the beam going from source at (0, 0, 1) to detector at (0, 0, -1)
    """
    #ax.plot([0,0],[0,0],[1,-1])
    #ax.scatter([0]*100,[0]*100,np.linspace(1, -1, 100), alpha=0.8)

    steps = 25
    u = np.linspace(0, 2 * np.pi, steps)
    v = np.linspace(-1, 1, steps)

    r = 0.02
    x = r*np.outer(np.cos(u), np.ones_like(v))
    y = r*np.outer(np.sin(u), np.ones_like(v))
    z = 1.3*np.outer(np.ones_like(u), v)

    theta, phi = view
    shape = x.shape
    points = np.matrix([x.flatten(), y.flatten(), z.flatten()])
    points = Rz(phi)*Ry(theta)*points
    x, y, z = [v.reshape(shape) for v in points]

    ax.plot_surface(x, y, z, rstride=4, cstride=4, color='y', alpha=0.5)

def draw_jitter(ax, view, jitter, dist='gaussian', size=(0.1, 0.4, 1.0)):
    """
    Represent jitter as a set of shapes at different orientations.
    """
    # set max diagonal to 0.95
    scale = 0.95/sqrt(sum(v**2 for v in size))
    size = tuple(scale*v for v in size)
    draw_shape = draw_parallelepiped
    #draw_shape = draw_ellipsoid

    #np.random.seed(10)
    #cloud = np.random.randn(10,3)
    cloud = [
        [-1, -1, -1],
        [-1, -1,  0],
        [-1, -1,  1],
        [-1,  0, -1],
        [-1,  0,  0],
        [-1,  0,  1],
        [-1,  1, -1],
        [-1,  1,  0],
        [-1,  1,  1],
        [ 0, -1, -1],
        [ 0, -1,  0],
        [ 0, -1,  1],
        [ 0,  0, -1],
        [ 0,  0,  0],
        [ 0,  0,  1],
        [ 0,  1, -1],
        [ 0,  1,  0],
        [ 0,  1,  1],
        [ 1, -1, -1],
        [ 1, -1,  0],
        [ 1, -1,  1],
        [ 1,  0, -1],
        [ 1,  0,  0],
        [ 1,  0,  1],
        [ 1,  1, -1],
        [ 1,  1,  0],
        [ 1,  1,  1],
    ]
    dtheta, dphi, dpsi = jitter
    if dtheta == 0:
        cloud = [v for v in cloud if v[0] == 0]
    if dphi == 0:
        cloud = [v for v in cloud if v[1] == 0]
    if dpsi == 0:
        cloud = [v for v in cloud if v[2] == 0]
    draw_shape(ax, size, view, [0, 0, 0], steps=100, alpha=0.8)
    scale = {'gaussian':1, 'rectangle':1/sqrt(3), 'uniform':1/3}[dist]
    for point in cloud:
        delta = [scale*dtheta*point[0], scale*dphi*point[1], scale*dpsi*point[2]]
        draw_shape(ax, size, view, delta, alpha=0.8)
    for v in 'xyz':
        a, b, c = size
        lim = np.sqrt(a**2+b**2+c**2)
        getattr(ax, 'set_'+v+'lim')([-lim, lim])
        getattr(ax, v+'axis').label.set_text(v)

def draw_ellipsoid(ax, size, view, jitter, steps=25, alpha=1):
    """Draw an ellipsoid."""
    a,b,c = size
    u = np.linspace(0, 2 * np.pi, steps)
    v = np.linspace(0, np.pi, steps)
    x = a*np.outer(np.cos(u), np.sin(v))
    y = b*np.outer(np.sin(u), np.sin(v))
    z = c*np.outer(np.ones_like(u), np.cos(v))
    x, y, z = transform_xyz(view, jitter, x, y, z)

    ax.plot_surface(x, y, z, rstride=4, cstride=4, color='w', alpha=alpha)

    draw_labels(ax, view, jitter, [
         ('c+', [ 0, 0, c], [ 1, 0, 0]),
         ('c-', [ 0, 0,-c], [ 0, 0,-1]),
         ('a+', [ a, 0, 0], [ 0, 0, 1]),
         ('a-', [-a, 0, 0], [ 0, 0,-1]),
         ('b+', [ 0, b, 0], [-1, 0, 0]),
         ('b-', [ 0,-b, 0], [-1, 0, 0]),
    ])

def draw_parallelepiped(ax, size, view, jitter, steps=None, alpha=1):
    """Draw a parallelepiped."""
    a, b, c = size
    x = a*np.array([ 1,-1, 1,-1, 1,-1, 1,-1])
    y = b*np.array([ 1, 1,-1,-1, 1, 1,-1,-1])
    z = c*np.array([ 1, 1, 1, 1,-1,-1,-1,-1])
    tri = np.array([
        # counter clockwise triangles
        # z: up/down, x: right/left, y: front/back
        [0,1,2], [3,2,1], # top face
        [6,5,4], [5,6,7], # bottom face
        [0,2,6], [6,4,0], # right face
        [1,5,7], [7,3,1], # left face
        [2,3,6], [7,6,3], # front face
        [4,1,0], [5,1,4], # back face
    ])

    x, y, z = transform_xyz(view, jitter, x, y, z)
    ax.plot_trisurf(x, y, triangles=tri, Z=z, color='w', alpha=alpha)

    # Draw pink face on box.
    # Since I can't control face color, instead draw a thin box situated just
    # in front of the "a+" face.
    if 1:
        x = a*np.array([ 1,-1, 1,-1, 1,-1, 1,-1])
        y = b*np.array([ 1, 1,-1,-1, 1, 1,-1,-1])
        z = c*np.array([ 1, 1, 1, 1,-1,-1,-1,-1])
        x, y, z = transform_xyz(view, jitter, abs(x)*1.05, y, z)
        ax.plot_trisurf(x, y, triangles=tri, Z=z, color=[1,0.6,0.6], alpha=alpha)

    draw_labels(ax, view, jitter, [
         ('c+', [ 0, 0, c], [ 1, 0, 0]),
         ('c-', [ 0, 0,-c], [ 0, 0,-1]),
         ('a+', [ a, 0, 0], [ 0, 0, 1]),
         ('a-', [-a, 0, 0], [ 0, 0,-1]),
         ('b+', [ 0, b, 0], [-1, 0, 0]),
         ('b-', [ 0,-b, 0], [-1, 0, 0]),
    ])

def draw_sphere(ax, radius=10., steps=100):
    """Draw a sphere"""
    u = np.linspace(0, 2 * np.pi, steps)
    v = np.linspace(0, np.pi, steps)

    x = radius * np.outer(np.cos(u), np.sin(v))
    y = radius * np.outer(np.sin(u), np.sin(v))
    z = radius * np.outer(np.ones(np.size(u)), np.cos(v))
    ax.plot_surface(x, y, z, rstride=4, cstride=4, color='w')

PROJECTIONS = [
    # in order of PROJECTION number; do not change without updating the
    # constants in kernel_iq.c
    'equirectangular', 'sinusoidal', 'guyou', 'azimuthal_equidistance',
    'azimuthal_equal_area',
]
def draw_mesh(ax, view, jitter, radius=1.2, n=11, dist='gaussian',
              projection='equirectangular'):
    """
    Draw the dispersion mesh showing the theta-phi orientations at which
    the model will be evaluated.

    jitter projections
    <https://en.wikipedia.org/wiki/List_of_map_projections>

    equirectangular (standard latitude-longitude mesh)
        <https://en.wikipedia.org/wiki/Equirectangular_projection>
        Allows free movement in phi (around the equator), but theta is
        limited to +/- 90, and points are cos-weighted. Jitter in phi is
        uniform in weight along a line of latitude.  With small theta and
        phi ranging over +/- 180 this forms a wobbling disk.  With small
        phi and theta ranging over +/- 90 this forms a wedge like a slice
        of an orange.
    azimuthal_equidistance (Postel)
        <https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection>
        Preserves distance from center, and so is an excellent map for
        representing a bivariate gaussian on the surface.  Theta and phi
        operate identically, cutting wegdes from the antipode of the viewing
        angle.  This unfortunately does not allow free movement in either
        theta or phi since the orthogonal wobble decreases to 0 as the body
        rotates through 180 degrees.
    sinusoidal (Sanson-Flamsteed, Mercator equal-area)
        <https://en.wikipedia.org/wiki/Sinusoidal_projection>
        Preserves arc length with latitude, giving bad behaviour at
        theta near +/- 90.  Theta and phi operate somewhat differently,
        so a system with a-b-c dtheta-dphi-dpsi will not give the same
        value as one with b-a-c dphi-dtheta-dpsi, as would be the case
        for azimuthal equidistance.  Free movement using theta or phi
        uniform over +/- 180 will work, but not as well as equirectangular
        phi, with theta being slightly worse.  Computationally it is much
        cheaper for wide theta-phi meshes since it excludes points which
        lie outside the sinusoid near theta +/- 90 rather than packing
        them close together as in equirectangle.  Note that the poles
        will be slightly overweighted for theta > 90 with the circle
        from theta at 90+dt winding backwards around the pole, overlapping
        the circle from theta at 90-dt.
    Guyou (hemisphere-in-a-square) **not weighted**
        <https://en.wikipedia.org/wiki/Guyou_hemisphere-in-a-square_projection>
        With tiling, allows rotation in phi or theta through +/- 180, with
        uniform spacing.  Both theta and phi allow free rotation, with wobble
        in the orthogonal direction reasonably well behaved (though not as
        good as equirectangular phi). The forward/reverse transformations
        relies on elliptic integrals that are somewhat expensive, so the
        behaviour has to be very good to justify the cost and complexity.
        The weighting function for each point has not yet been computed.
        Note: run the module *guyou.py* directly and it will show the forward
        and reverse mappings.
    azimuthal_equal_area  **incomplete**
        <https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection>
        Preserves the relative density of the surface patches.  Not that
        useful and not completely implemented
    Gauss-Kreuger **not implemented**
        <https://en.wikipedia.org/wiki/Transverse_Mercator_projection#Ellipsoidal_transverse_Mercator>
        Should allow free movement in theta, but phi is distorted.
    """
    theta, phi, psi = view
    dtheta, dphi, dpsi = jitter

    t = np.linspace(-1, 1, n)
    weights = np.ones_like(t)
    if dist == 'gaussian':
        t *= 3
        weights = exp(-0.5*t**2)
    elif dist == 'rectangle':
        # Note: uses sasmodels ridiculous definition of rectangle width
        t *= sqrt(3)
    elif dist == 'uniform':
        pass
    else:
        raise ValueError("expected dist to be gaussian, rectangle or uniform")

    if projection == 'equirectangular':  #define PROJECTION 1
        def rotate(theta_i, phi_j):
            return Rx(phi_j)*Ry(theta_i)
        def weight(theta_i, phi_j, wi, wj):
            return wi*wj*abs(cos(radians(theta_i)))
    elif projection == 'sinusoidal':  #define PROJECTION 2
        def rotate(theta_i, phi_j):
            latitude = theta_i
            scale = cos(radians(latitude))
            longitude = phi_j/scale if abs(phi_j) < abs(scale)*180 else 0
            #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude))
            return Rx(longitude)*Ry(latitude)
        def weight(theta_i, phi_j, wi, wj):
            latitude = theta_i
            scale = cos(radians(latitude))
            w = 1 if abs(phi_j) < abs(scale)*180 else 0
            return w*wi*wj
    elif projection == 'guyou':  #define PROJECTION 3  (eventually?)
        def rotate(theta_i, phi_j):
            from guyou import guyou_invert
            #latitude, longitude = guyou_invert([theta_i], [phi_j])
            longitude, latitude = guyou_invert([phi_j], [theta_i])
            return Rx(longitude[0])*Ry(latitude[0])
        def weight(theta_i, phi_j, wi, wj):
            return wi*wj
    elif projection == 'azimuthal_equidistance':  # Note: Rz Ry, not Rx Ry
        def rotate(theta_i, phi_j):
            latitude = sqrt(theta_i**2 + phi_j**2)
            longitude = degrees(np.arctan2(phi_j, theta_i))
            #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude))
            return Rz(longitude)*Ry(latitude)
        def weight(theta_i, phi_j, wi, wj):
            # Weighting for each point comes from the integral:
            #     \int\int I(q, lat, log) sin(lat) dlat dlog
            # We are doing a conformal mapping from disk to sphere, so we need
            # a change of variables g(theta, phi) -> (lat, long):
            #     lat, long = sqrt(theta^2 + phi^2), arctan(phi/theta)
            # giving:
            #     dtheta dphi = det(J) dlat dlong
            # where J is the jacobian from the partials of g. Using
            #     R = sqrt(theta^2 + phi^2),
            # then
            #     J = [[x/R, Y/R], -y/R^2, x/R^2]]
            # and
            #     det(J) = 1/R
            # with the final integral being:
            #    \int\int I(q, theta, phi) sin(R)/R dtheta dphi
            #
            # This does approximately the right thing, decreasing the weight
            # of each point as you go farther out on the disk, but it hasn't
            # yet been checked against the 1D integral results. Prior
            # to declaring this "good enough" and checking that integrals
            # work in practice, we will examine alternative mappings.
            #
            # The issue is that the mapping does not support the case of free
            # rotation about a single axis correctly, with a small deviation
            # in the orthogonal axis independent of the first axis.  Like the
            # usual polar coordiates integration, the integrated sections
            # form wedges, though at least in this case the wedge cuts through
            # the entire sphere, and treats theta and phi identically.
            latitude = sqrt(theta_i**2 + phi_j**2)
            w = sin(radians(latitude))/latitude if latitude != 0 else 1
            return w*wi*wj if latitude < 180 else 0
    elif projection == 'azimuthal_equal_area':
        def rotate(theta_i, phi_j):
            R = min(1, sqrt(theta_i**2 + phi_j**2)/180)
            latitude = 180-degrees(2*np.arccos(R))
            longitude = degrees(np.arctan2(phi_j, theta_i))
            #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude))
            return Rz(longitude)*Ry(latitude)
        def weight(theta_i, phi_j, wi, wj):
            latitude = sqrt(theta_i**2 + phi_j**2)
            w = sin(radians(latitude))/latitude if latitude != 0 else 1
            return w*wi*wj if latitude < 180 else 0
    else:
        raise ValueError("unknown projection %r"%projection)

    # mesh in theta, phi formed by rotating z
    z = np.matrix([[0], [0], [radius]])
    points = np.hstack([rotate(theta_i, phi_j)*z
                        for theta_i in dtheta*t
                        for phi_j in dphi*t])
    # select just the active points (i.e., those with phi < 180
    w = np.array([weight(theta_i, phi_j, wi, wj)
                  for wi, theta_i in zip(weights, dtheta*t)
                  for wj, phi_j in zip(weights, dphi*t)])
    #print(max(w), min(w), min(w[w>0]))
    points = points[:, w>0]
    w = w[w>0]
    w /= max(w)

    if 0: # Kent distribution
        points = np.hstack([Rx(phi_j)*Ry(theta_i)*z for theta_i in 30*t for phi_j in 60*t])
        xp, yp, zp = [np.array(v).flatten() for v in points]
        kappa = max(1e6, radians(dtheta)/(2*pi))
        beta = 1/max(1e-6, radians(dphi)/(2*pi))/kappa
        w = exp(kappa*zp) #+ beta*(xp**2 + yp**2)
        print(kappa, dtheta, radians(dtheta), min(w), max(w), sum(w))
        #w /= abs(cos(radians(
        #w /= sum(w)

    # rotate relative to beam
    points = orient_relative_to_beam(view, points)

    x, y, z = [np.array(v).flatten() for v in points]
    #plt.figure(2); plt.clf(); plt.hist(z, bins=np.linspace(-1, 1, 51))
    ax.scatter(x, y, z, c=w, marker='o', vmin=0., vmax=1.)

def draw_labels(ax, view, jitter, text):
    """
    Draw text at a particular location.
    """
    labels, locations, orientations = zip(*text)
    px, py, pz = zip(*locations)
    dx, dy, dz = zip(*orientations)

    px, py, pz = transform_xyz(view, jitter, px, py, pz)
    dx, dy, dz = transform_xyz(view, jitter, dx, dy, dz)

    # TODO: zdir for labels is broken, and labels aren't appearing.
    for label, p, zdir in zip(labels, zip(px, py, pz), zip(dx, dy, dz)):
        zdir = np.asarray(zdir).flatten()
        ax.text(p[0], p[1], p[2], label, zdir=zdir)

# Definition of rotation matrices comes from wikipedia:
#    https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations
def Rx(angle):
    """Construct a matrix to rotate points about *x* by *angle* degrees."""
    a = radians(angle)
    R = [[1, 0, 0],
         [0, +cos(a), -sin(a)],
         [0, +sin(a), +cos(a)]]
    return np.matrix(R)

def Ry(angle):
    """Construct a matrix to rotate points about *y* by *angle* degrees."""
    a = radians(angle)
    R = [[+cos(a), 0, +sin(a)],
         [0, 1, 0],
         [-sin(a), 0, +cos(a)]]
    return np.matrix(R)

def Rz(angle):
    """Construct a matrix to rotate points about *z* by *angle* degrees."""
    a = radians(angle)
    R = [[+cos(a), -sin(a), 0],
         [+sin(a), +cos(a), 0],
         [0, 0, 1]]
    return np.matrix(R)

def transform_xyz(view, jitter, x, y, z):
    """
    Send a set of (x,y,z) points through the jitter and view transforms.
    """
    x, y, z = [np.asarray(v) for v in (x, y, z)]
    shape = x.shape
    points = np.matrix([x.flatten(),y.flatten(),z.flatten()])
    points = apply_jitter(jitter, points)
    points = orient_relative_to_beam(view, points)
    x, y, z = [np.array(v).reshape(shape) for v in points]
    return x, y, z

def apply_jitter(jitter, points):
    """
    Apply the jitter transform to a set of points.

    Points are stored in a 3 x n numpy matrix, not a numpy array or tuple.
    """
    dtheta, dphi, dpsi = jitter
    points = Rx(dphi)*Ry(dtheta)*Rz(dpsi)*points
    return points

def orient_relative_to_beam(view, points):
    """
    Apply the view transform to a set of points.

    Points are stored in a 3 x n numpy matrix, not a numpy array or tuple.
    """
    theta, phi, psi = view
    points = Rz(phi)*Ry(theta)*Rz(psi)*points
    return points

# translate between number of dimension of dispersity and the number of
# points along each dimension.
PD_N_TABLE = {
    (0, 0, 0): (0, 0, 0),     # 0
    (1, 0, 0): (100, 0, 0),   # 100
    (0, 1, 0): (0, 100, 0),
    (0, 0, 1): (0, 0, 100),
    (1, 1, 0): (30, 30, 0),   # 900
    (1, 0, 1): (30, 0, 30),
    (0, 1, 1): (0, 30, 30),
    (1, 1, 1): (15, 15, 15),  # 3375
}

def clipped_range(data, portion=1.0, mode='central'):
    """
    Determine range from data.

    If *portion* is 1, use full range, otherwise use the center of the range
    or the top of the range, depending on whether *mode* is 'central' or 'top'.
    """
    if portion == 1.0:
        return data.min(), data.max()
    elif mode == 'central':
        data = np.sort(data.flatten())
        offset = int(portion*len(data)/2 + 0.5)
        return data[offset], data[-offset]
    elif mode == 'top':
        data = np.sort(data.flatten())
        offset = int(portion*len(data) + 0.5)
        return data[offset], data[-1]

def draw_scattering(calculator, ax, view, jitter, dist='gaussian'):
    """
    Plot the scattering for the particular view.

    *calculator* is returned from :func:`build_model`.  *ax* are the 3D axes
    on which the data will be plotted.  *view* and *jitter* are the current
    orientation and orientation dispersity.  *dist* is one of the sasmodels
    weight distributions.
    """
    if dist == 'uniform':  # uniform is not yet in this branch
        dist, scale = 'rectangle', 1/sqrt(3)
    else:
        scale = 1

    # add the orientation parameters to the model parameters
    theta, phi, psi = view
    theta_pd, phi_pd, psi_pd = [scale*v for v in jitter]
    theta_pd_n, phi_pd_n, psi_pd_n = PD_N_TABLE[(theta_pd>0, phi_pd>0, psi_pd>0)]
    ## increase pd_n for testing jitter integration rather than simple viz
    #theta_pd_n, phi_pd_n, psi_pd_n = [5*v for v in (theta_pd_n, phi_pd_n, psi_pd_n)]

    pars = dict(
        theta=theta, theta_pd=theta_pd, theta_pd_type=dist, theta_pd_n=theta_pd_n,
        phi=phi, phi_pd=phi_pd, phi_pd_type=dist, phi_pd_n=phi_pd_n,
        psi=psi, psi_pd=psi_pd, psi_pd_type=dist, psi_pd_n=psi_pd_n,
    )
    pars.update(calculator.pars)

    # compute the pattern
    qx, qy = calculator._data.x_bins, calculator._data.y_bins
    Iqxy = calculator(**pars).reshape(len(qx), len(qy))

    # scale it and draw it
    Iqxy = np.log(Iqxy)
    if calculator.limits:
        # use limits from orientation (0,0,0)
        vmin, vmax = calculator.limits
    else:
        vmax = Iqxy.max()
        vmin = vmax*10**-7
        #vmin, vmax = clipped_range(Iqxy, portion=portion, mode='top')
    #print("range",(vmin,vmax))
    #qx, qy = np.meshgrid(qx, qy)
    if 0:
        level = np.asarray(255*(Iqxy - vmin)/(vmax - vmin), 'i')
        level[level<0] = 0
        colors = plt.get_cmap()(level)
        ax.plot_surface(qx, qy, -1.1, rstride=1, cstride=1, facecolors=colors)
    elif 1:
        ax.contourf(qx/qx.max(), qy/qy.max(), Iqxy, zdir='z', offset=-1.1,
                    levels=np.linspace(vmin, vmax, 24))
    else:
        ax.pcolormesh(qx, qy, Iqxy)

def build_model(model_name, n=150, qmax=0.5, **pars):
    """
    Build a calculator for the given shape.

    *model_name* is any sasmodels model.  *n* and *qmax* define an n x n mesh
    on which to evaluate the model.  The remaining parameters are stored in
    the returned calculator as *calculator.pars*.  They are used by
    :func:`draw_scattering` to set the non-orientation parameters in the
    calculation.

    Returns a *calculator* function which takes a dictionary or parameters and
    produces Iqxy.  The Iqxy value needs to be reshaped to an n x n matrix
    for plotting.  See the :class:`sasmodels.direct_model.DirectModel` class
    for details.
    """
    from sasmodels.core import load_model_info, build_model
    from sasmodels.data import empty_data2D
    from sasmodels.direct_model import DirectModel

    model_info = load_model_info(model_name)
    model = build_model(model_info) #, dtype='double!')
    q = np.linspace(-qmax, qmax, n)
    data = empty_data2D(q, q)
    calculator = DirectModel(data, model)

    # stuff the values for non-orientation parameters into the calculator
    calculator.pars = pars.copy()
    calculator.pars.setdefault('backgound', 1e-3)

    # fix the data limits so that we can see if the pattern fades
    # under rotation or angular dispersion
    Iqxy = calculator(theta=0, phi=0, psi=0, **calculator.pars)
    Iqxy = np.log(Iqxy)
    vmin, vmax = clipped_range(Iqxy, 0.95, mode='top')
    calculator.limits = vmin, vmax+1

    return calculator

def select_calculator(model_name, n=150, size=(10,40,100)):
    """
    Create a model calculator for the given shape.

    *model_name* is one of sphere, cylinder, ellipsoid, triaxial_ellipsoid,
    parallelepiped or bcc_paracrystal. *n* is the number of points to use
    in the q range.  *qmax* is chosen based on model parameters for the
    given model to show something intersting.

    Returns *calculator* and tuple *size* (a,b,c) giving minor and major
    equitorial axes and polar axis respectively.  See :func:`build_model`
    for details on the returned calculator.
    """
    a, b, c = size
    if model_name == 'sphere':
        calculator = build_model('sphere', n=n, radius=c)
        a = b = c
    elif model_name == 'bcc_paracrystal':
        calculator = build_model('bcc_paracrystal', n=n, dnn=c,
                                  d_factor=0.06, radius=40)
        a = b = c
    elif model_name == 'cylinder':
        calculator = build_model('cylinder', n=n, qmax=0.3, radius=b, length=c)
        a = b
    elif model_name == 'ellipsoid':
        calculator = build_model('ellipsoid', n=n, qmax=1.0,
                                 radius_polar=c, radius_equatorial=b)
        a = b
    elif model_name == 'triaxial_ellipsoid':
        calculator = build_model('triaxial_ellipsoid', n=n, qmax=0.5,
                                 radius_equat_minor=a,
                                 radius_equat_major=b,
                                 radius_polar=c)
    elif model_name == 'parallelepiped':
        calculator = build_model('parallelepiped', n=n, a=a, b=b, c=c)
    else:
        raise ValueError("unknown model %s"%model_name)

    return calculator, (a, b, c)

SHAPES = [
    'parallelepiped', 'triaxial_ellipsoid', 'bcc_paracrystal',
    'cylinder', 'ellipsoid',
    'sphere',
 ]

DISTRIBUTIONS = [
    'gaussian', 'rectangle', 'uniform',
]
DIST_LIMITS = {
    'gaussian': 30,
    'rectangle': 90/sqrt(3),
    'uniform': 90,
}

def run(model_name='parallelepiped', size=(10, 40, 100),
        dist='gaussian', mesh=30,
        projection='equirectangular'):
    """
    Show an interactive orientation and jitter demo.

    *model_name* is one of: sphere, cylinder, ellipsoid, parallelepiped,
    triaxial_ellipsoid, or bcc_paracrystal.

    *size* gives the dimensions (a, b, c) of the shape.

    *dist* is the type of dispersition: gaussian, rectangle, or uniform.

    *mesh* is the number of points in the dispersion mesh.

    *projection* is the map projection to use for the mesh: equirectangular,
    sinusoidal, guyou, azimuthal_equidistance, or azimuthal_equal_area.
    """
    # projection number according to 1-order position in list, but
    # only 1 and 2 are implemented so far.
    from sasmodels import generate
    generate.PROJECTION = PROJECTIONS.index(projection) + 1
    if generate.PROJECTION > 2:
        print("*** PROJECTION %s not implemented in scattering function ***"%projection)
        generate.PROJECTION = 2

    # set up calculator
    calculator, size = select_calculator(model_name, n=150, size=size)

    ## uncomment to set an independent the colour range for every view
    ## If left commented, the colour range is fixed for all views
    calculator.limits = None

    ## initial view
    #theta, dtheta = 70., 10.
    #phi, dphi = -45., 3.
    #psi, dpsi = -45., 3.
    theta, phi, psi = 0, 0, 0
    dtheta, dphi, dpsi = 0, 0, 0

    ## create the plot window
    #plt.hold(True)
    plt.subplots(num=None, figsize=(5.5, 5.5))
    plt.set_cmap('gist_earth')
    plt.clf()
    plt.gcf().canvas.set_window_title(projection)
    #gs = gridspec.GridSpec(2,1,height_ratios=[4,1])
    #ax = plt.subplot(gs[0], projection='3d')
    ax = plt.axes([0.0, 0.2, 1.0, 0.8], projection='3d')
    try:  # CRUFT: not all versions of matplotlib accept 'square' 3d projection
        ax.axis('square')
    except Exception:
        pass

    axcolor = 'lightgoldenrodyellow'

    ## add control widgets to plot
    axtheta  = plt.axes([0.1, 0.15, 0.45, 0.04], axisbg=axcolor)
    axphi = plt.axes([0.1, 0.1, 0.45, 0.04], axisbg=axcolor)
    axpsi = plt.axes([0.1, 0.05, 0.45, 0.04], axisbg=axcolor)
    stheta = Slider(axtheta, 'Theta', -90, 90, valinit=theta)
    sphi = Slider(axphi, 'Phi', -180, 180, valinit=phi)
    spsi = Slider(axpsi, 'Psi', -180, 180, valinit=psi)

    axdtheta  = plt.axes([0.75, 0.15, 0.15, 0.04], axisbg=axcolor)
    axdphi = plt.axes([0.75, 0.1, 0.15, 0.04], axisbg=axcolor)
    axdpsi= plt.axes([0.75, 0.05, 0.15, 0.04], axisbg=axcolor)
    # Note: using ridiculous definition of rectangle distribution, whose width
    # in sasmodels is sqrt(3) times the given width.  Divide by sqrt(3) to keep
    # the maximum width to 90.
    dlimit = DIST_LIMITS[dist]
    sdtheta = Slider(axdtheta, 'dTheta', 0, 2*dlimit, valinit=dtheta)
    sdphi = Slider(axdphi, 'dPhi', 0, 2*dlimit, valinit=dphi)
    sdpsi = Slider(axdpsi, 'dPsi', 0, 2*dlimit, valinit=dpsi)

    ## callback to draw the new view
    def update(val, axis=None):
        view = stheta.val, sphi.val, spsi.val
        jitter = sdtheta.val, sdphi.val, sdpsi.val
        # set small jitter as 0 if multiple pd dims
        dims = sum(v > 0 for v in jitter)
        limit = [0, 0, 0.5, 5][dims]
        jitter = [0 if v < limit else v for v in jitter]
        ax.cla()
        draw_beam(ax, (0, 0))
        draw_jitter(ax, view, jitter, dist=dist, size=size)
        #draw_jitter(ax, view, (0,0,0))
        draw_mesh(ax, view, jitter, dist=dist, n=mesh, projection=projection)
        draw_scattering(calculator, ax, view, jitter, dist=dist)
        plt.gcf().canvas.draw()

    ## bind control widgets to view updater
    stheta.on_changed(lambda v: update(v,'theta'))
    sphi.on_changed(lambda v: update(v, 'phi'))
    spsi.on_changed(lambda v: update(v, 'psi'))
    sdtheta.on_changed(lambda v: update(v, 'dtheta'))
    sdphi.on_changed(lambda v: update(v, 'dphi'))
    sdpsi.on_changed(lambda v: update(v, 'dpsi'))

    ## initialize view
    update(None, 'phi')

    ## go interactive
    plt.show()

def main():
    parser = argparse.ArgumentParser(
        description="Display jitter",
        formatter_class=argparse.ArgumentDefaultsHelpFormatter,
        )
    parser.add_argument('-p', '--projection', choices=PROJECTIONS, default=PROJECTIONS[0], help='coordinate projection')
    parser.add_argument('-s', '--size', type=str, default='10,40,100', help='a,b,c lengths')
    parser.add_argument('-d', '--distribution', choices=DISTRIBUTIONS, default=DISTRIBUTIONS[0], help='jitter distribution')
    parser.add_argument('-m', '--mesh', type=int, default=30, help='#points in theta-phi mesh')
    parser.add_argument('shape', choices=SHAPES, nargs='?', default=SHAPES[0], help='oriented shape')
    opts = parser.parse_args()
    size = tuple(int(v) for v in opts.size.split(','))
    run(opts.shape, size=size,
        mesh=opts.mesh, dist=opts.distribution,
        projection=opts.projection)

if __name__ == "__main__":
    main()