plotter.py

# Copyright 2011-2013 Kwant authors.
#
# This file is part of Kwant.  It is subject to the license terms in the file
# LICENSE.rst found in the top-level directory of this distribution and at
# http://kwant-project.org/license.  A list of Kwant authors can be found in
# the file AUTHORS.rst at the top-level directory of this distribution and at
# http://kwant-project.org/authors.

"""Plotter module for Kwant.

This module provides iterators useful for any plotter routine, such as a list
of system sites, their coordinates, lead sites at any lead unit cell, etc.  If
`matplotlib` is available, it also provides simple functions for plotting the
system in two or three dimensions.
"""

from collections import defaultdict
import warnings
import numpy as np
import tinyarray as ta
from scipy import spatial, interpolate
from math import cos, sin, pi, sqrt

# All matplotlib imports must be isolated in a try, because even without
# matplotlib iterators remain useful.  Further, mpl_toolkits used for 3D
# plotting are also imported separately, to ensure that 2D plotting works even
# if 3D does not.
try:
    import matplotlib
    from matplotlib.figure import Figure
    from matplotlib import collections
    from matplotlib.backends.backend_agg import FigureCanvasAgg
    mpl_enabled = True
    try:
        from mpl_toolkits import mplot3d
        has3d = True
    except ImportError:
        warnings.warn("3D plotting not available.", RuntimeWarning)
        has3d = False
except ImportError:
    warnings.warn("matplotlib is not available, only iterator-providing "
                  "functions will work.", RuntimeWarning)
    mpl_enabled = False

from . import system, builder, physics

__all__ = ['plot', 'map', 'bands', 'sys_leads_sites', 'sys_leads_hoppings',
           'sys_leads_pos', 'sys_leads_hopping_pos', 'mask_interpolate']


# Collections that allow for symbols and linewiths to be given in data space
# (not for general use, only implement what's needed for plotter)
def isarray(var):
    if hasattr(var, '__getitem__') and not isinstance(var, basestring):
        return True
    else:
        return False


def nparray_if_array(var):
    return np.asarray(var) if isarray(var) else var


if mpl_enabled:
    class LineCollection(collections.LineCollection):
        def __init__(self, segments, reflen=None, **kwargs):
            super(LineCollection, self).__init__(segments, **kwargs)
            self.reflen = reflen

        def set_linewidths(self, linewidths):
            self.linewidths_orig = nparray_if_array(linewidths)

        def draw(self, renderer):
            if self.reflen is not None:
                # Note: only works for aspect ratio 1!
                #       72.0 - there is 72 points in an inch
                factor = (self.axes.transData.frozen().to_values()[0] * 72.0 *
                          self.reflen / self.figure.dpi)
            else:
                factor = 1

            super(LineCollection, self).set_linewidths(self.linewidths_orig *
                                                       factor)
            return super(LineCollection, self).draw(renderer)


    class PathCollection(collections.PathCollection):
        def __init__(self, paths, sizes=None, reflen=None, **kwargs):
            super(PathCollection, self).__init__(paths, sizes=sizes, **kwargs)

            self.reflen = reflen
            self.linewidths_orig = nparray_if_array(self.get_linewidths())

            self.transforms = np.array(
                [matplotlib.transforms.Affine2D().scale(x).get_matrix()
                 for x in sizes])

        def get_transforms(self):
            return self.transforms

        def get_transform(self):
            Affine2D = matplotlib.transforms.Affine2D
            if self.reflen is not None:
                # For the paths, use the data transformation but strip the
                # offset (will be added later with offsets)
                args = self.axes.transData.frozen().to_values()[:4] + (0, 0)
                return Affine2D().from_values(*args).scale(self.reflen)
            else:
                return Affine2D().scale(self.figure.dpi / 72.0)

        def draw(self, renderer):
            if self.reflen:
                # Note: only works for aspect ratio 1!
                factor = (self.axes.transData.frozen().to_values()[0] /
                          self.figure.dpi * 72.0 * self.reflen)
                self.set_linewidths(self.linewidths_orig * factor)

            return collections.Collection.draw(self, renderer)


    if has3d:
        # Sorting is optional.
        sort3d = True

        # Compute the projection of a 3D length into 2D data coordinates
        # for this we use 2 3D half-circles that are projected into 2D.
        # (This gives the same length as projecting the full unit sphere.)

        phi = np.linspace(0, pi, 21)
        xyz = np.c_[np.cos(phi), np.sin(phi), 0 * phi].T.reshape(-1, 1, 21)
        unit_sphere = np.bmat([[xyz[0], xyz[2]], [xyz[1], xyz[0]],
                                [xyz[2], xyz[1]]])
        unit_sphere = np.asarray(unit_sphere)

        def projected_length(ax, length):
            rc = np.array([ax.get_xlim3d(), ax.get_ylim3d(), ax.get_zlim3d()])
            rc = np.apply_along_axis(np.sum, 1, rc) / 2.

            rs = unit_sphere * length + rc.reshape(-1, 1)

            transform = mplot3d.proj3d.proj_transform
            rp = np.asarray(transform(*(list(rs) + [ax.get_proj()]))[:2])
            rc[:2] = transform(*(list(rc) + [ax.get_proj()]))[:2]

            coords = rp - np.repeat(rc[:2].reshape(-1, 1), len(rs[0]), axis=1)
            return sqrt(np.sum(coords**2, axis=0).max())


        # Auxiliary array for calculating corners of a cube.
        corners = np.zeros((3, 8, 6), np.float_)
        corners[0, [0, 1, 2, 3], 0] = corners[0, [4, 5, 6, 7], 1] = \
        corners[0, [0, 1, 4, 5], 2] = corners[0, [2, 3, 6, 7], 3] = \
        corners[0, [0, 2, 4, 6], 4] = corners[0, [1, 3, 5, 7], 5] = 1.0


        class Line3DCollection(mplot3d.art3d.Line3DCollection):
            def __init__(self, segments, reflen=None, zorder=0, **kwargs):
                super(Line3DCollection, self).__init__(segments, **kwargs)
                self.reflen = reflen
                self.zorder3d = zorder

            def set_linewidths(self, linewidths):
                self.linewidths_orig = nparray_if_array(linewidths)

            def do_3d_projection(self, renderer):
                super(Line3DCollection, self).do_3d_projection(renderer)
                # The whole 3D ordering is flawed in mplot3d when several
                # collections are added. We just use normal zorder. Note the
                # "-" due to the different logic in the 3d plotting, we still
                # want larger zorder values to be plotted on top of smaller
                # ones.
                return -self.zorder3d

            def draw(self, renderer):
                if self.reflen:
                    proj_len = projected_length(self.axes, self.reflen)
                    args = self.axes.transData.frozen().to_values()
                    # Note: unlike in the 2D case, where we can enforce equal
                    #       aspect ratio, this (currently) does not work with
                    #       3D plots in matplotlib. As an approximation, we
                    #       thus scale with the average of the x- and y-axis
                    #       transformation.
                    factor = proj_len * (args[0] +
                                         args[3]) * 0.5 * 72.0 / self.figure.dpi
                else:
                    factor = 1

                super(Line3DCollection, self).set_linewidths(
                                                self.linewidths_orig * factor)
                super(Line3DCollection, self).draw(renderer)


        class Path3DCollection(mplot3d.art3d.Patch3DCollection):
            def __init__(self, paths, sizes, reflen=None, zorder=0,
                         offsets=None, **kwargs):
                paths = [matplotlib.patches.PathPatch(path) for path in paths]

                if offsets is not None:
                    kwargs['offsets'] = offsets[:, :2]

                super(Path3DCollection, self).__init__(paths, **kwargs)

                if offsets is not None:
                    self.set_3d_properties(zs=offsets[:, 2], zdir="z")

                self.reflen = reflen
                self.zorder3d = zorder

                self.paths_orig = np.array(paths, dtype='object')
                self.linewidths_orig = nparray_if_array(self.get_linewidths())
                self.linewidths_orig2 = self.linewidths_orig
                self.array_orig = nparray_if_array(self.get_array())
                self.facecolors_orig = nparray_if_array(self.get_facecolors())
                self.edgecolors_orig = nparray_if_array(self.get_edgecolors())

                Affine2D = matplotlib.transforms.Affine2D
                self.orig_transforms = np.array(
                    [Affine2D().scale(x).get_matrix() for x in sizes])
                self.transforms = self.orig_transforms

            def set_array(self, array):
                self.array_orig = nparray_if_array(array)
                super(Path3DCollection, self).set_array(array)

            def set_color(self, colors):
                self.facecolors_orig = nparray_if_array(colors)
                self.edgecolors_orig = self.facecolors_orig
                super(Path3DCollection, self).set_color(colors)

            def set_edgecolors(self, colors):
                colors = matplotlib.colors.colorConverter.to_rgba_array(colors)
                self.edgecolors_orig = nparray_if_array(colors)
                super(Path3DCollection, self).set_edgecolors(colors)

            def get_transforms(self):
                # this is exact only for an isometric projection, for the
                # perspective projection used in mplot3d it's an approximation
                return self.transforms

            def get_transform(self):
                Affine2D = matplotlib.transforms.Affine2D
                if self.reflen:
                    proj_len = projected_length(self.axes, self.reflen)

                    # For the paths, use the data transformation but strip the
                    # offset (will be added later with the offsets).
                    args = self.axes.transData.frozen().to_values()[:4] + (0, 0)
                    return Affine2D().from_values(*args).scale(proj_len)
                else:
                    return Affine2D().scale(self.figure.dpi / 72.0)

            def do_3d_projection(self, renderer):
                xs, ys, zs = self._offsets3d

                # numpy complains about zero-length index arrays
                if len(xs) == 0:
                    return -self.zorder3d

                proj = mplot3d.proj3d.proj_transform_clip
                vs = np.array(proj(xs, ys, zs, renderer.M)[:3])

                if sort3d:
                    indx = vs[2].argsort()[::-1]

                    self.set_offsets(vs[:2, indx].T)

                    if len(self.paths_orig) > 1:
                        paths = np.resize(self.paths_orig, (vs.shape[1],))
                        self.set_paths(paths[indx])

                    if len(self.orig_transforms) > 1:
                        self.transforms = np.resize(self.orig_transforms,
                                                     (vs.shape[1],))
                        self.transforms = self.transforms[indx]

                    lw_orig = self.linewidths_orig
                    if (isinstance(lw_orig, np.ndarray) and len(lw_orig) > 1):
                        self.linewidths_orig2 = np.resize(lw_orig,
                                                           (vs.shape[1],))[indx]

                    # Note: here array, facecolors and edgecolors are
                    #       guaranteed to be 2d numpy arrays or None.  (And
                    #       array is the same length as the coordinates)

                    if self.array_orig is not None:
                        super(Path3DCollection,
                              self).set_array(self.array_orig[indx])

                    if (self.facecolors_orig is not None and
                        self.facecolors_orig.shape[0] > 1):
                        shape = list(self.facecolors_orig.shape)
                        shape[0] = vs.shape[1]
                        super(Path3DCollection, self).set_facecolors(
                            np.resize(self.facecolors_orig, shape)[indx])

                    if (self.edgecolors_orig is not None and
                        self.edgecolors_orig.shape[0] > 1):
                        shape = list(self.edgecolors_orig.shape)
                        shape[0] = vs.shape[1]
                        super(Path3DCollection, self).set_edgecolors(
                                                np.resize(self.edgecolors_orig,
                                                          shape)[indx])
                else:
                    self.set_offsets(vs[:2].T)

                # the whole 3D ordering is flawed in mplot3d when several
                # collections are added. We just use normal zorder, but correct
                # by the projected z-coord of the "center of gravity",
                # normalized by the projected z-coord of the world coordinates.
                # In doing so, several Path3DCollections are plotted probably
                # in the right order (it's not exact) if they have the same
                # zorder. Still, smaller and larger integer zorders are plotted
                # below or on top.

                bbox = np.asarray(self.axes.get_w_lims())

                proj = mplot3d.proj3d.proj_transform_clip
                cz = proj(*(list(np.dot(corners, bbox)) + [renderer.M]))[2]

                return -self.zorder3d + vs[2].mean() / cz.ptp()

            def draw(self, renderer):
                if self.reflen:
                    proj_len = projected_length(self.axes, self.reflen)
                    args = self.axes.transData.frozen().to_values()
                    factor = proj_len * (args[0] +
                                         args[3]) * 0.5 * 72.0 / self.figure.dpi

                    self.set_linewidths(self.linewidths_orig2 * factor)

                super(Path3DCollection, self).draw(renderer)


# matplotlib helper functions.

def set_colors(color, collection, cmap, norm=None):
    """Process a color specification to a format accepted by collections.

    Parameters
    ----------
    color : color specification
    collection : instance of a subclass of `matplotlib.collections.Collection`
        Collection to which the color is added.
    cmap : `matplotlib` color map specification or None
        Color map to be used if colors are specified as floats.
    norm : `matplotlib` color norm
        Norm to be used if colors are specified as floats.
    """

    length = max(len(collection.get_paths()), len(collection.get_offsets()))

    # matplotlib gets confused if dtype='object'
    if (isinstance(color, np.ndarray) and color.dtype == np.dtype('object')):
        color = tuple(color)

    if isinstance(collection, mplot3d.art3d.Line3DCollection):
        length = len(collection._segments3d)  # Once again, matplotlib fault!

    if isarray(color) and len(color) == length:
        try:
            # check if it is an array of floats for color mapping
            color = np.asarray(color, dtype=float)
            if color.ndim == 1:
                collection.set_array(color)
                collection.set_cmap(cmap)
                collection.set_norm(norm)
                collection.set_color(None)
                return
        except (TypeError, ValueError):
            pass

    colors = matplotlib.colors.colorConverter.to_rgba_array(color)
    collection.set_color(colors)


symbol_dict = {'O': 'o', 's': ('p', 4, 45), 'S': ('P', 4, 45)}

def get_symbol(symbols):
    """Return the path corresponding to the description in `symbol`"""
    # Figure out if list of symbols or single symbol.
    if not hasattr(symbols, '__getitem__'):
        symbols = [symbols]
    elif len(symbols) == 3 and symbols[0] in ('p', 'P'):
        # Most likely a polygon specification (at least not a valid other
        # symbol).
        symbols = [symbols]

    symbols = [symbol_dict[symbol] if symbol in symbol_dict else symbol for
               symbol in symbols]

    paths = []
    for symbol in symbols:
        if isinstance(symbol, matplotlib.path.Path):
            return symbol
        elif hasattr(symbol, '__getitem__') and len(symbol) == 3:
            kind, n, angle = symbol

            if kind in ['p', 'P']:
                if kind == 'p':
                    radius = 1. / cos(pi / n)
                else:
                    # make the polygon such that it has area equal
                    # to a unit circle
                    radius = sqrt(2 * pi / (n * sin(2 * pi / n)))

                angle = pi * angle / 180
                patch = matplotlib.patches.RegularPolygon((0, 0), n,
                                                          radius=radius,
                                                          orientation=angle)
            else:
                raise ValueError("Unknown symbol definition " + str(symbol))
        elif symbol == 'o':
            patch = matplotlib.patches.Circle((0, 0), 1)

        paths.append(patch.get_path().transformed(patch.get_transform()))

    return paths


def symbols(axes, pos, symbol='o', size=1, reflen=None, facecolor='k',
            edgecolor='k', linewidth=None, cmap=None, norm=None, zorder=0,
            **kwargs):
    """Add a collection of symbols (2D or 3D) to an axes instance.

    Parameters
    ----------
    axes : matplotlib.axes.Axes instance
        Axes to which the lines have to be added.
    pos0 : 2d or 3d array_like
        Coordinates of each symbol.
    symbol: symbol definition.
        TODO To be written.
    size: float or 1d array
        Size(s) of the symbols. Defaults to 1.
    reflen: float or None, optional
        If `reflen` is `None`, the symbol sizes and linewidths are
        given in points (absolute size in the figure space). If
        `reflen` is a number, the symbol sizes and linewidths are
        given in units of `reflen` in data space (i.e. scales with the
        scale of the plot). Defaults to `None`.
    facecolor: color definition, optional
    edgecolor: color definition, optional
        Defines the fill and edge color of the symbol, repsectively.
        Either a single object that is a proper matplotlib color
        definition or a sequence of such objects of appropriate
        length.  Defaults to all black.
    cmap : `matplotlib` color map specification or None
        Color map to be used if colors are specified as floats.
    norm : `matplotlib` color norm
        Norm to be used if colors are specified as floats.
    zorder: int
        Order in which different collections are drawn: larger
        `zorder` means the collection is drawn over collections with
        smaller `zorder` values.
    **kwargs : dict keyword arguments to
        pass to `PathCollection` or `Path3DCollection`, respectively.

    Returns
    -------
    `PathCollection` or `Path3DCollection` instance containing all the
    symbols that were added.
    """

    dim = pos.shape[1]
    assert dim == 2 or dim == 3

    #internally, size must be array_like
    try:
        size[0]
    except TypeError:
        size = (size, )

    if dim == 2:
        Collection = PathCollection
    else:
        Collection = Path3DCollection

    if len(pos) == 0 or np.all(symbol == 'no symbol') or np.all(size == 0):
        paths = []
        pos = np.empty((0, dim))
    else:
        paths = get_symbol(symbol)

    coll = Collection(paths, sizes=size, reflen=reflen, linewidths=linewidth,
                      offsets=pos, transOffset=axes.transData, zorder=zorder)

    set_colors(facecolor, coll, cmap, norm)
    coll.set_edgecolors(edgecolor)

    coll.update(kwargs)

    if dim == 2:
        axes.add_collection(coll)
    else:
        axes.add_collection3d(coll)

    return coll


def lines(axes, pos0, pos1, reflen=None, colors='k', linestyles='solid',
          cmap=None, norm=None, zorder=0, **kwargs):
    """Add a collection of line segments (2D or 3D) to an axes instance.

    Parameters
    ----------
    axes : matplotlib.axes.Axes instance
        Axes to which the lines have to be added.
    pos0 : 2d or 3d array_like
        Starting coordinates of each line segment
    pos1 : 2d or 3d array_like
        Ending coordinates of each line segment
    reflen: float or None, optional
        If `reflen` is `None`, the linewidths are given in points (absolute
        size in the figure space). If `reflen` is a number, the linewidths
        are given in units of `reflen` in data space (i.e. scales with
        the scale of the plot). Defaults to `None`.
    colors : color definition, optional
        Either a single object that is a proper matplotlib color definition
        or a sequence of such objects of appropriate length.  Defaults to all
        segments black.
    linestyles :linestyle definition, optional
        Either a single object that is a proper matplotlib line style
        definition or a sequence of such objects of appropriate length.
        Defaults to all segments solid.
    cmap : `matplotlib` color map specification or None
        Color map to be used if colors are specified as floats.
    norm : `matplotlib` color norm
        Norm to be used if colors are specified as floats.
    zorder: int
        Order in which different collections are drawn: larger
        `zorder` means the collection is drawn over collections with
        smaller `zorder` values.
    **kwargs : dict keyword arguments to
        pass to `LineCollection` or `Line3DCollection`, respectively.

    Returns
    -------
    `LineCollection` or `Line3DCollection` instance containing all the
    segments that were added.
    """

    if not pos0.shape == pos1.shape:
        raise ValueError('Incompatible lengths of coordinate arrays.')

    dim = pos0.shape[1]
    assert dim == 2 or dim == 3
    if dim == 2:
        Collection = LineCollection
    else:
        Collection = Line3DCollection

    if (len(pos0) == 0 or
        ('linewidths' in kwargs and kwargs['linewidths'] == 0)):
        coll = Collection([], reflen=reflen, linestyles=linestyles,
                          zorder=zorder)
        coll.update(kwargs)
        if dim == 2:
            axes.add_collection(coll)
        else:
            axes.add_collection3d(coll)
        return coll

    segments = np.c_[pos0, pos1].reshape(pos0.shape[0], 2, dim)

    coll = Collection(segments, reflen=reflen, linestyles=linestyles,
                      zorder=zorder)
    set_colors(colors, coll, cmap, norm)
    coll.update(kwargs)

    if dim == 2:
        axes.add_collection(coll)
    else:
        axes.add_collection3d(coll)

    return coll


def output_fig(fig, output_mode='auto', file=None, savefile_opts=None,
               show=True):
    """Output a matplotlib figure using a given output mode.

    Parameters
    ----------
    fig : matplotlib.figure.Figure instance
        The figure to be output.
    output_mode : string
        The output mode to be used.  Can be one of the following:
        'pyplot' : attach the figure to pyplot, with the same behavior as if
        pyplot.plot was called to create this figure.
        'ipython' : attach a `FigureCanvasAgg` to the figure and return it.
        'return' : return the figure.
        'file' : same as 'ipython', but also save the figure into a file.
        'auto' : if fname is given, save to a file, else if pyplot
        is imported, attach to pyplot, otherwise just return.  See also the
        notes below.
    file : string or a file object
        The name of the target file or the target file itself
        (opened for writing).
    savefile_opts : (list, dict) or None
        args and kwargs passed to `print_figure` of `matplotlib`
    show : bool
        Whether to call `matplotlib.pyplot.show()`.  Only has an effect if the
        output uses pyplot.

    Notes
    -----
    The behavior of this function producing a file is different from that of
    matplotlib in that the `dpi` attribute of the figure is used by defaul
    instead of the matplotlib config setting.
    """
    if not mpl_enabled:
        raise RuntimeError('matplotlib is not installed.')
    if output_mode == 'auto':
        if file is not None:
            output_mode = 'file'
        else:
            try:
                matplotlib.pyplot.get_backend()
                output_mode = 'pyplot'
            except AttributeError:
                output_mode = 'pyplot'
    if output_mode == 'pyplot':
        try:
            fake_fig = matplotlib.pyplot.figure()
        except AttributeError:
            msg = ('matplotlib.pyplot is unavailable.  Execute `import '
                   'matplotlib.pyplot` or use a different output mode.')
            raise RuntimeError(msg)
        fake_fig.canvas.figure = fig
        fig.canvas = fake_fig.canvas
        for ax in fig.axes:
            try:
                ax.mouse_init()  # Make 3D interface interactive.
            except AttributeError:
                pass
        if show:
            matplotlib.pyplot.show()
        return fig
    elif output_mode == 'return':
        canvas = FigureCanvasAgg(fig)
        fig.canvas = canvas
        return fig
    elif output_mode == 'file':
        canvas = FigureCanvasAgg(fig)
        if savefile_opts is None:
            savefile_opts = ([], {})
        if 'dpi' not in savefile_opts[1]:
            savefile_opts[1]['dpi'] = fig.dpi
        canvas.print_figure(file, *savefile_opts[0], **savefile_opts[1])
        return fig
    else:
        assert False, 'Unknown output_mode'


# Extracting necessary data from the system.

def sys_leads_sites(sys, num_lead_cells=2):
    """Return all the sites of the system and of the leads as a list.

    Parameters
    ----------
    sys : kwant.builder.Builder or kwant.system.System instance
        The system, sites of which should be returned.
    num_lead_cells : integer
        The number of times lead sites from each lead should be returned.
        This is useful for showing several unit cells of the lead next to the
        system.

    Returns
    -------
    sites : list of (site, lead_number, copy_number) tuples
        A site is a `builder.Site` instance if the system is not finalized,
        and an integer otherwise.  For system sites `lead_number` is `None` and
        `copy_number` is `0`, for leads both are integers.
    lead_cells : list of slices
        `lead_cells[i]` gives the position of all the coordinates of lead
        `i` within `sites`.

    Notes
    -----
    Leads are only supported if they are of the same type as the original
    system, i.e.  sites of `builder.BuilderLead` leads are returned with an
    unfinalized system, and sites of `system.InfiniteSystem` leads are
    returned with a finalized system.
    """
    lead_cells = []
    if isinstance(sys, builder.Builder):
        sites = [(site, None, 0) for site in sys.sites()]
        for leadnr, lead in enumerate(sys.leads):
            start = len(sites)
            if hasattr(lead, 'builder') and len(lead.interface):
                sites.extend(((site, leadnr, i) for site in
                              lead.builder.sites() for i in
                              xrange(num_lead_cells)))
            lead_cells.append(slice(start, len(sites)))
    elif isinstance(sys, system.FiniteSystem):
        sites = [(i, None, 0) for i in xrange(sys.graph.num_nodes)]
        for leadnr, lead in enumerate(sys.leads):
            start = len(sites)
            # We will only plot leads with a graph and with a symmetry.
            if (hasattr(lead, 'graph') and hasattr(lead, 'symmetry') and
                len(sys.lead_interfaces[leadnr])):
                sites.extend(((site, leadnr, i) for site in
                              xrange(lead.cell_size) for i in
                              xrange(num_lead_cells)))
            lead_cells.append(slice(start, len(sites)))
    else:
        raise TypeError('Unrecognized system type.')
    return sites, lead_cells


def sys_leads_pos(sys, site_lead_nr):
    """Return an array of positions of sites in a system.

    Parameters
    ----------
    sys : `kwant.builder.Builder` or `kwant.system.System` instance
        The system, coordinates of sites of which should be returned.
    site_lead_nr : list of `(site, leadnr, copynr)` tuples
        Output of `sys_leads_sites` applied to the system.

    Returns
    -------
    coords : numpy.ndarray of floats
        Array of coordinates of the sites.

    Notes
    -----
    This function uses `site.pos` property to get the position of a builder
    site and `sys.pos(sitenr)` for finalized systems.  This function requires
    that all the positions of all the sites have the same dimensionality.
    """

    # Note about efficiency (also applies to sys_leads_hoppings_pos)
    # NumPy is really slow when making a NumPy array from a tinyarray
    # (buffer interface seems very slow). It's much faster to first
    # convert to a tuple and then to convert to numpy array ...

    is_builder = isinstance(sys, builder.Builder)
    num_lead_cells = site_lead_nr[-1][2] + 1
    if is_builder:
        pos = np.array(ta.array([i[0].pos for i in site_lead_nr]))
    else:
        sys_from_lead = lambda lead: (sys if (lead is None)
                                      else sys.leads[lead])
        pos = np.array(ta.array([sys_from_lead(i[1]).pos(i[0])
                                 for i in site_lead_nr]))
    if pos.dtype == object:  # Happens if not all the pos are same length.
        raise ValueError("pos attribute of the sites does not have consistent"
                         " values.")
    dim = pos.shape[1]

    def get_vec_domain(lead_nr):
        if lead_nr is None:
            return np.zeros((dim,)), 0
        if is_builder:
            sym = sys.leads[lead_nr].builder.symmetry
            try:
                site = sys.leads[lead_nr].interface[0]
            except IndexError:
                return (0, 0)
        else:
            try:
                sym = sys.leads[lead_nr].symmetry
                site = sys.sites[sys.lead_interfaces[lead_nr][0]]
            except (AttributeError, IndexError):
                # empty leads, or leads without symmetry aren't drawn anyways
                return (0, 0)
        dom = sym.which(site)[0] + 1
        # Conversion to numpy array here useful for efficiency
        vec = np.array(sym.periods)[0]
        return vec, dom
    vecs_doms = dict((i, get_vec_domain(i)) for i in xrange(len(sys.leads)))
    vecs_doms[None] = np.zeros((dim,)), 0
    for k, v in vecs_doms.iteritems():
        vecs_doms[k] = [v[0] * i for i in xrange(v[1], v[1] + num_lead_cells)]
    pos += [vecs_doms[i[1]][i[2]] for i in site_lead_nr]
    return pos


def sys_leads_hoppings(sys, num_lead_cells=2):
    """Return all the hoppings of the system and of the leads as an iterator.

    Parameters
    ----------
    sys : kwant.builder.Builder or kwant.system.System instance
        The system, sites of which should be returned.
    num_lead_cells : integer
        The number of times lead sites from each lead should be returned.
        This is useful for showing several unit cells of the lead next to the
        system.

    Returns
    -------
    hoppings : list of (hopping, lead_number, copy_number) tuples
        A site is a `builder.Site` instance if the system is not finalized,
        and an integer otherwise.  For system sites `lead_number` is `None` and
        `copy_number` is `0`, for leads both are integers.
    lead_cells : list of slices
        `lead_cells[i]` gives the position of all the coordinates of lead
        `i` within `hoppings`.

    Notes
    -----
    Leads are only supported if they are of the same type as the original
    system, i.e.  hoppings of `builder.BuilderLead` leads are returned with an
    unfinalized system, and hoppings of `system.InfiniteSystem` leads are
    returned with a finalized system.
    """
    hoppings = []
    lead_cells = []
    if isinstance(sys, builder.Builder):
        hoppings.extend(((hop, None, 0) for hop in sys.hoppings()))

        def lead_hoppings(lead):
            sym = lead.symmetry
            for site2, site1 in lead.hoppings():
                shift1 = sym.which(site1)[0]
                shift2 = sym.which(site2)[0]
                # We need to make sure that the hopping is between a site in a
                # fundamental domain and a site with a negative domain.  The
                # direction of the hopping is chosen arbitrarily
                # NOTE(Anton): This may need to be revisited with the future
                # builder format changes.
                shift = max(shift1, shift2)
                yield sym.act([-shift], site2), sym.act([-shift], site1)

        for leadnr, lead in enumerate(sys.leads):
            start = len(hoppings)
            if hasattr(lead, 'builder') and len(lead.interface):
                hoppings.extend(((hop, leadnr, i) for hop in
                                 lead_hoppings(lead.builder) for i in
                                 xrange(num_lead_cells)))
            lead_cells.append(slice(start, len(hoppings)))
    elif isinstance(sys, system.System):
        def ll_hoppings(sys):
            for i in xrange(sys.graph.num_nodes):
                for j in sys.graph.out_neighbors(i):
                    if i < j:
                        yield i, j

        hoppings.extend(((hop, None, 0) for hop in ll_hoppings(sys)))
        for leadnr, lead in enumerate(sys.leads):
            start = len(hoppings)
            # We will only plot leads with a graph and with a symmetry.
            if (hasattr(lead, 'graph') and hasattr(lead, 'symmetry') and
                len(sys.lead_interfaces[leadnr])):
                hoppings.extend(((hop, leadnr, i) for hop in ll_hoppings(lead)
                                 for i in xrange(num_lead_cells)))
            lead_cells.append(slice(start, len(hoppings)))
    else:
        raise TypeError('Unrecognized system type.')
    return hoppings, lead_cells


def sys_leads_hopping_pos(sys, hop_lead_nr):
    """Return arrays of coordinates of all hoppings in a system.

    Parameters
    ----------
    sys : `kwant.builder.Builder` or `kwant.system.System` instance
        The system, coordinates of sites of which should be returned.
    hoppings : list of `(hopping, leadnr, copynr)` tuples
        Output of `sys_leads_hoppings` applied to the system.

    Returns
    -------
    coords : (end_site, start_site): tuple of NumPy arrays of floats
        Array of coordinates of the hoppings.  The first half of coordinates
        in each array entry are those of the first site in the hopping, the
        last half are those of the second site.

    Notes
    -----
    This function uses `site.pos` property to get the position of a builder
    site and `sys.pos(sitenr)` for finalized systems.  This function requires
    that all the positions of all the sites have the same dimensionality.
    """
    is_builder = isinstance(sys, builder.Builder)
    if len(hop_lead_nr) == 0:
        return np.empty((0, 3)), np.empty((0, 3))
    num_lead_cells = hop_lead_nr[-1][2] + 1
    if is_builder:
        pos = np.array(ta.array([ta.array(tuple(i[0][0].pos) +
                                          tuple(i[0][1].pos)) for i in
                                 hop_lead_nr]))
    else:
        sys_from_lead = lambda lead: (sys if (lead is None) else
                                      sys.leads[lead])
        pos = ta.array([ta.array(tuple(sys_from_lead(i[1]).pos(i[0][0])) +
                                 tuple(sys_from_lead(i[1]).pos(i[0][1]))) for i
                        in hop_lead_nr])
        pos = np.array(pos)
    if pos.dtype == object:  # Happens if not all the pos are same length.
        raise ValueError("pos attribute of the sites does not have consistent"
                         " values.")
    dim = pos.shape[1]

    def get_vec_domain(lead_nr):
        if lead_nr is None:
            return np.zeros((dim,)), 0
        if is_builder:
            sym = sys.leads[lead_nr].builder.symmetry
            try:
                site = sys.leads[lead_nr].interface[0]
            except IndexError:
                return (0, 0)
        else:
            try:
                sym = sys.leads[lead_nr].symmetry
                site = sys.sites[sys.lead_interfaces[lead_nr][0]]
            except (AttributeError, IndexError):
                # empyt leads or leads without symmetry are not drawn anyways
                return (0, 0)
        dom = sym.which(site)[0] + 1
        vec = np.array(sym.periods)[0]
        return np.r_[vec, vec], dom

    vecs_doms = dict((i, get_vec_domain(i)) for i in xrange(len(sys.leads)))
    vecs_doms[None] = np.zeros((dim,)), 0
    for k, v in vecs_doms.iteritems():
        vecs_doms[k] = [v[0] * i for i in xrange(v[1], v[1] + num_lead_cells)]
    pos += [vecs_doms[i[1]][i[2]] for i in hop_lead_nr]
    return np.copy(pos[:, : dim / 2]), np.copy(pos[:, dim / 2:])


# Useful plot functions (to be extended).

defaults = {'site_symbol': {2: 'o', 3: 'o'},
            'site_size': {2: 0.25, 3: 0.5},
            'site_color': {2: 'black', 3: 'white'},
            'site_edgecolor': {2: 'black', 3: 'black'},
            'site_lw': {2: 0, 3: 0.1},
            'hop_color': {2: 'black', 3: 'black'},
            'hop_lw': {2: 0.1, 3: 0},
            'lead_color': {2: 'red', 3: 'red'}}


def plot(sys, num_lead_cells=2, unit='nn',
         site_symbol=None, site_size=None,
         site_color=None, site_edgecolor=None, site_lw=None,
         hop_color=None, hop_lw=None,
         lead_site_symbol=None, lead_site_size=None, lead_color=None,
         lead_site_edgecolor=None, lead_site_lw=None,
         lead_hop_lw=None, pos_transform=None,
         cmap='gray', colorbar=True, file=None,
         show=True, dpi=None, fig_size=None, ax=None):
    """Plot a system in 2 or 3 dimensions.

    Parameters
    ----------
    sys : kwant.builder.Builder or kwant.system.FiniteSystem
        A system to be plotted.
    num_lead_cells : int
        Number of lead copies to be shown with the system.
    unit : 'nn', 'pt', or float
        The unit used to specify symbol sizes and linewidths.
        Possible choices are:

        - 'nn': unit is the shortest hopping or a typical nearst neighbor
          distance in the system if there are no hoppings.  This means that
          symbol sizes/linewidths will scale as the zoom level of the figure is
          changed.  Very short distances are discarded before searching for the
          shortest.  This choice means that the symbols will scale if the
          figure is zoomed.
        - 'pt': unit is points (point = 1/72 inch) in figure space.  This means
          that symbols and linewidths will always be drawn with the same size
          independent of zoom level of the plot.
        - float: sizes are given in units of this value in real (system) space,
          and will accordingly scale as the plot is zoomed.

        The default value is 'nn', which allows to ensure that the images
        neighboring sites do not overlap.

    site_symbol : symbol specification, function, array, or `None`
        Symbol used for representing a site in the plot. Can be specified as

        - 'o': circle with radius of 1 unit.
        - 's': square with inner circle radius of 1 unit.
        - `('p', nvert, angle)`: regular polygon with `nvert` vertices,
          rotated by `angle`. `angle` is given in degrees, and ``angle=0``
          corresponds to one edge of the polygon pointing upward. The
          radius of the inner circle is 1 unit.
        - 'no symbol': no symbol is plotted.
        - 'S', `('P', nvert, angle)`: as the lower-case variants described
          above, but with an area equal to a circle of radius 1. (Makes
          the visual size of the symbol equal to the size of a circle with
          radius 1).
        - matplotlib.path.Path instance.

        Instead of a single symbol, different symbols can be specified
        for different sites by passing a function that returns a valid
        symbol specification for each site, or by passing an array of
        symbols specifications (only for kwant.system.FiniteSystem).
    site_size : number, function, array, or `None`
        Relative (linear) size of the site symbol.
    site_color : `matplotlib` color description, function, array, or `None`
        A color used for plotting a site in the system. If a colormap is used,
        it should be a function returning single floats or a one-dimensional
        array of floats.