text: """
  Streamline plotting for 2D vector fields.
  
  """
  from __future__ import division
  import numpy as np
  import matplotlib
  import matplotlib.cm as cm
  import matplotlib.colors as mcolors
  import matplotlib.collections as mcollections
  import matplotlib.patches as patches
  
  
  __all__ = ['streamplot']
  
  
  def streamplot(axes, x, y, u, v, density=1, linewidth=None, color=None,
                 cmap=None, norm=None, arrowsize=1, arrowstyle='-|>',
                 minlength=0.1, transform=None):
      """Draws streamlines of a vector flow.
  
      *x*, *y* : 1d arrays
          an *evenly spaced* grid.
      *u*, *v* : 2d arrays
          x and y-velocities. Number of rows should match length of y, and
          the number of columns should match x.
      *density* : float or 2-tuple
          Controls the closeness of streamlines. When `density = 1`, the domain
          is divided into a 25x25 grid---*density* linearly scales this grid.
          Each cell in the grid can have, at most, one traversing streamline.
          For different densities in each direction, use [density_x, density_y].
      *linewidth* : numeric or 2d array
          vary linewidth when given a 2d array with the same shape as velocities.
      *color* : matplotlib color code, or 2d array
          Streamline color. When given an array with the same shape as
          velocities, *color* values are converted to colors using *cmap*.
      *cmap* : :class:`~matplotlib.colors.Colormap`
          Colormap used to plot streamlines and arrows. Only necessary when using
          an array input for *color*.
      *norm* : :class:`~matplotlib.colors.Normalize`
          Normalize object used to scale luminance data to 0, 1. If None, stretch
          (min, max) to (0, 1). Only necessary when *color* is an array.
      *arrowsize* : float
          Factor scale arrow size.
      *arrowstyle* : str
          Arrow style specification.
          See :class:`~matplotlib.patches.FancyArrowPatch`.
      *minlength* : float
          Minimum length of streamline in axes coordinates.
  
      Returns:
      
          *stream_container* : StreamplotSet
              Container object with attributes
                  lines : `matplotlib.collections.LineCollection` of streamlines
                  arrows : collection of `matplotlib.patches.FancyArrowPatch` objects
                      repesenting arrows half-way along stream lines.
              This container will probably change in the future to allow changes to
              the colormap, alpha, etc. for both lines and arrows, but these changes
              should be backward compatible.
      
      """
      grid = Grid(x, y)
      mask = StreamMask(density)
      dmap = DomainMap(grid, mask)
  
      if color is None:
          color = axes._get_lines.color_cycle.next()
  
      if linewidth is None:
          linewidth = matplotlib.rcParams['lines.linewidth']
  
      line_kw = {}
      arrow_kw = dict(arrowstyle=arrowstyle, mutation_scale=10*arrowsize)
  
      use_multicolor_lines = isinstance(color, np.ndarray)
      if use_multicolor_lines:
          assert color.shape == grid.shape
          line_colors = []
      else:
          line_kw['color'] = color
          arrow_kw['color'] = color
  
      if isinstance(linewidth, np.ndarray):
          assert linewidth.shape == grid.shape
          line_kw['linewidth'] = []
      else:
          line_kw['linewidth'] = linewidth
          arrow_kw['linewidth'] = linewidth
  
      ## Sanity checks.
      assert u.shape == grid.shape
      assert v.shape == grid.shape
  
      if np.any(np.isnan(u)):
          u = np.ma.array(u, mask=np.isnan(u))
      if np.any(np.isnan(v)):
          v = np.ma.array(v, mask=np.isnan(v))
  
      integrate = get_integrator(u, v, dmap, minlength)
  
      trajectories = []
      for xm, ym in _gen_starting_points(mask.shape):
          if mask[ym, xm] == 0:
              xg, yg = dmap.mask2grid(xm, ym)
              t = integrate(xg, yg)
              if t != None:
                  trajectories.append(t)
  
      if use_multicolor_lines:
          if norm is None:
              norm = mcolors.normalize(color.min(), color.max())
          if cmap is None:
              cmap = cm.get_cmap(matplotlib.rcParams['image.cmap'])
          else:
              cmap = cm.get_cmap(cmap)
  
      streamlines = []
      arrows = []
      for t in trajectories:
          tgx = np.array(t[0])
          tgy = np.array(t[1])
          # Rescale from grid-coordinates to data-coordinates.
          tx = np.array(t[0]) * grid.dx + grid.x_origin
          ty = np.array(t[1]) * grid.dy + grid.y_origin
  
          points = np.transpose([tx, ty]).reshape(-1, 1, 2)
          streamlines.extend(np.hstack([points[:-1], points[1:]]))
  
          # Add arrows half way along each trajectory.
          s = np.cumsum(np.sqrt(np.diff(tx)**2 + np.diff(ty)**2))
          n = np.searchsorted(s, s[-1] / 2.)
          arrow_tail = (tx[n], ty[n])
          arrow_head = (np.mean(tx[n:n+2]), np.mean(ty[n:n+2]))
  
          if isinstance(linewidth, np.ndarray):
              line_widths = interpgrid(linewidth, tgx, tgy)[:-1]
              line_kw['linewidth'].extend(line_widths)
              arrow_kw['linewidth'] = line_widths[n]
  
          if use_multicolor_lines:
              color_values = interpgrid(color, tgx, tgy)[:-1]
              line_colors.extend(color_values)
              arrow_kw['color'] = cmap(norm(color_values[n]))
  
          p = patches.FancyArrowPatch(arrow_tail, 
                                      arrow_head, 
                                      transform=transform, 
                                      **arrow_kw)
          axes.add_patch(p)
          arrows.append(p)
  
      lc = mcollections.LineCollection(streamlines, 
                                       transform=transform, 
                                       **line_kw)
      if use_multicolor_lines:
          lc.set_array(np.asarray(line_colors))
          lc.set_cmap(cmap)
          lc.set_norm(norm)
      axes.add_collection(lc)
  
      axes.update_datalim(((x.min(), y.min()), (x.max(), y.max())))
      axes.autoscale_view(tight=True)
  
      ac = matplotlib.collections.PatchCollection(arrows)
      stream_container = StreamplotSet(lc, ac)
      return stream_container
  
  
  class StreamplotSet(object):
  
      def __init__(self, lines, arrows, **kwargs):
          self.lines = lines
          self.arrows = arrows
  
  
  # Coordinate definitions
  #========================
  
  class DomainMap(object):
      """Map representing different coordinate systems.
  
      Coordinate definitions:
  
      * axes-coordinates goes from 0 to 1 in the domain.
      * data-coordinates are specified by the input x-y coordinates.
      * grid-coordinates goes from 0 to N and 0 to M for an N x M grid,
        where N and M match the shape of the input data.
      * mask-coordinates goes from 0 to N and 0 to M for an N x M mask,
        where N and M are user-specified to control the density of streamlines.
  
      This class also has methods for adding trajectories to the StreamMask.
      Before adding a trajectory, run `start_trajectory` to keep track of regions
      crossed by a given trajectory. Later, if you decide the trajectory is bad
      (e.g. if the trajectory is very short) just call `undo_trajectory`.
      """
  
      def __init__(self, grid, mask):
          self.grid = grid
          self.mask = mask
          ## Constants for conversion between grid- and mask-coordinates
          self.x_grid2mask = float(mask.nx - 1) / grid.nx
          self.y_grid2mask = float(mask.ny - 1) / grid.ny
  
          self.x_mask2grid = 1. / self.x_grid2mask
          self.y_mask2grid = 1. / self.y_grid2mask
  
          self.x_data2grid = grid.nx / grid.width
          self.y_data2grid = grid.ny / grid.height
  
      def grid2mask(self, xi, yi):
          """Return nearest space in mask-coords from given grid-coords."""
          return int((xi * self.x_grid2mask) + 0.5), \
                 int((yi * self.y_grid2mask) + 0.5)
  
      def mask2grid(self, xm, ym):
          return xm * self.x_mask2grid, ym * self.y_mask2grid
  
      def data2grid(self, xd, yd):
          return xd * self.x_data2grid, yd * self.y_data2grid
  
      def start_trajectory(self, xg, yg):
          xm, ym = self.grid2mask(xg, yg)
          self.mask._start_trajectory(xm, ym)
  
      def reset_start_point(self, xg, yg):
          xm, ym = self.grid2mask(xg, yg)
          self.mask._current_xy = (xm, ym)
  
      def update_trajectory(self, xg, yg):
          if not self.grid.within_grid(xg, yg):
              raise InvalidIndexError
          xm, ym = self.grid2mask(xg, yg)
          self.mask._update_trajectory(xm, ym)
  
      def undo_trajectory(self):
          self.mask._undo_trajectory()
  
  
  class Grid(object):
      """Grid of data."""
      def __init__(self, x, y):
  
          if len(x.shape) == 2:
              x_row = x[0]
              assert np.allclose(x_row, x)
              x = x_row
          else:
              assert len(x.shape) == 1
  
          if len(y.shape) == 2:
              y_col = y[:, 0]
              assert np.allclose(y_col, y.T)
              y = y_col
          else:
              assert len(y.shape) == 1
  
          self.nx = len(x)
          self.ny = len(y)
  
          self.dx = x[1] - x[0]
          self.dy = y[1] - y[0]
  
          self.x_origin = x[0]
          self.y_origin = y[0]
  
          self.width = x[-1] - x[0]
          self.height = y[-1] - y[0]
  
      @property
      def shape(self):
          return self.ny, self.nx
  
      def within_grid(self, xi, yi):
          """Return True if point is a valid index of grid."""
          # Note that xi/yi can be floats; so, for example, we can't simply check
          # `xi < self.nx` since `xi` can be `self.nx - 1 < xi < self.nx`
          return xi >= 0 and xi <= self.nx-1 and yi >= 0 and yi <= self.ny-1
  
  
  class StreamMask(object):
      """Mask to keep track of discrete regions crossed by streamlines.
  
      The resolution of this grid determines the approximate spacing between
      trajectories. Streamlines are only allowed to pass through zeroed cells:
      When a streamline enters a cell, that cell is set to 1, and no new
      streamlines are allowed to enter.
      """
  
      def __init__(self, density):
          if np.isscalar(density):
              assert density > 0
              self.nx = self.ny = int(30 * density)
          else:
              assert len(density) == 2
              self.nx = int(25 * density[0])
              self.ny = int(25 * density[1])
          self._mask = np.zeros((self.ny, self.nx))
          self.shape = self._mask.shape
  
          self._current_xy = None
  
      def __getitem__(self, *args):
          return self._mask.__getitem__(*args)
  
      def _start_trajectory(self, xm, ym):
          """Start recording streamline trajectory"""
          self._traj = []
          self._update_trajectory(xm, ym)
  
      def _undo_trajectory(self):
          """Remove current trajectory from mask"""
          for t in self._traj:
              self._mask.__setitem__(t, 0)
  
      def _update_trajectory(self, xm, ym):
          """Update current trajectory position in mask.
  
          If the new position has already been filled, raise `InvalidIndexError`.
          """
          if self._current_xy != (xm, ym):
              if self[ym, xm] == 0:
                  self._traj.append((ym, xm))
                  self._mask[ym, xm] = 1
                  self._current_xy = (xm, ym)
              else:
                  raise InvalidIndexError
  
  
  class InvalidIndexError(Exception):
      pass
  
  class TerminateTrajectory(Exception):
      pass
  
  
  # Integrator definitions
  #========================
  
  def get_integrator(u, v, dmap, minlength):
  
      # rescale velocity onto grid-coordinates for integrations.
      u, v = dmap.data2grid(u, v)
  
      # speed (path length) will be in axes-coordinates
      u_ax = u / dmap.grid.nx
      v_ax = v / dmap.grid.ny
      speed = np.ma.sqrt(u_ax**2 + v_ax**2)
  
      def forward_time(xi, yi):
          ds_dt = interpgrid(speed, xi, yi)
          if ds_dt == 0:
              raise TerminateTrajectory()
          dt_ds = 1. / ds_dt
          ui = interpgrid(u, xi, yi)
          vi = interpgrid(v, xi, yi)
          return ui * dt_ds, vi * dt_ds
  
      def backward_time(xi, yi):
          dxi, dyi = forward_time(xi, yi)
          return -dxi, -dyi
  
      def integrate(x0, y0):
          """Return x, y grid-coordinates of trajectory based on starting point.
  
          Integrate both forward and backward in time from starting point in
          grid coordinates.
  
          Integration is terminated when a trajectory reaches a domain boundary
          or when it crosses into an already occupied cell in the StreamMask. The
          resulting trajectory is None if it is shorter than `minlength`.
          """
  
          dmap.start_trajectory(x0, y0)
          sf, xf_traj, yf_traj = _integrate_rk12(x0, y0, dmap, forward_time)
          dmap.reset_start_point(x0, y0)
          sb, xb_traj, yb_traj = _integrate_rk12(x0, y0, dmap, backward_time)
          # combine forward and backward trajectories
          stotal = sf + sb
          x_traj = xb_traj[::-1] + xf_traj[1:]
          y_traj = yb_traj[::-1] + yf_traj[1:]
  
          if stotal > minlength:
              return x_traj, y_traj
          else: # reject short trajectories
              dmap.undo_trajectory()
              return None
  
      return integrate
  
  
  def _integrate_rk12(x0, y0, dmap, f):
      """2nd-order Runge-Kutta algorithm with adaptive step size.
  
      This method is also referred to as the improved Euler's method, or Heun's
      method. This method is favored over higher-order methods because:
  
      1. To get decent looking trajectories and to sample every mask cell
         on the trajectory we need a small timestep, so a lower order
         solver doesn't hurt us unless the data is *very* high resolution.
         In fact, for cases where the user inputs
         data smaller or of similar grid size to the mask grid, the higher
         order corrections are negligible because of the very fast linear
         interpolation used in `interpgrid`.
  
      2. For high resolution input data (i.e. beyond the mask
         resolution), we must reduce the timestep. Therefore, an adaptive
         timestep is more suited to the problem as this would be very hard
         to judge automatically otherwise.
  
      This integrator is about 1.5 - 2x as fast as both the RK4 and RK45
      solvers in most setups on my machine. I would recommend removing the
      other two to keep things simple.
      """
      ## This error is below that needed to match the RK4 integrator. It
      ## is set for visual reasons -- too low and corners start
      ## appearing ugly and jagged. Can be tuned.
      maxerror = 0.003
  
      ## This limit is important (for all integrators) to avoid the
      ## trajectory skipping some mask cells. We could relax this
      ## condition if we use the code which is commented out below to
      ## increment the location gradually. However, due to the efficient
      ## nature of the interpolation, this doesn't boost speed by much
      ## for quite a bit of complexity.
      maxds = min(1./dmap.mask.nx, 1./dmap.mask.ny, 0.1)
  
      ds = maxds
      stotal = 0
      xi = x0
      yi = y0
      xf_traj = []
      yf_traj = []
  
      while dmap.grid.within_grid(xi, yi):
          xf_traj.append(xi)
          yf_traj.append(yi)
          try:
              k1x, k1y = f(xi, yi)
              k2x, k2y = f(xi + ds * k1x,
                           yi + ds * k1y)
          except IndexError:
              # Out of the domain on one of the intermediate integration steps.
              # Take an Euler step to the boundary to improve neatness.
              ds, xf_traj, yf_traj = _euler_step(xf_traj, yf_traj, dmap, f)
              stotal += ds
              break
          except TerminateTrajectory:
              break
  
          dx1 = ds * k1x
          dy1 = ds * k1y
          dx2 = ds * 0.5 * (k1x + k2x)
          dy2 = ds * 0.5 * (k1y + k2y)
  
          nx, ny = dmap.grid.shape
          # Error is normalized to the axes coordinates
          error = np.sqrt(((dx2-dx1)/nx)**2 + ((dy2-dy1)/ny)**2)
  
          # Only save step if within error tolerance
          if error < maxerror:
              xi += dx2
              yi += dy2
              try:
                  dmap.update_trajectory(xi, yi)
              except InvalidIndexError:
                  break
              if (stotal + ds) > 2:
                  break
              stotal += ds
  
          # recalculate stepsize based on step error
          if error == 0:
              ds = maxds
          else:
              ds = min(maxds, 0.85 * ds * (maxerror/error)**0.5)
  
      return stotal, xf_traj, yf_traj
  
  
  def _euler_step(xf_traj, yf_traj, dmap, f):
      """Simple Euler integration step that extends streamline to boundary."""
      ny, nx = dmap.grid.shape
      xi = xf_traj[-1]
      yi = yf_traj[-1]
      cx, cy = f(xi, yi)
      if cx == 0:
          dsx = np.inf
      elif cx < 0:
          dsx = xi / -cx
      else:
          dsx = (nx - 1 - xi) / cx
      if cy == 0:
          dsy = np.inf
      elif cy < 0:
          dsy = yi / -cy
      else:
          dsy = (ny - 1 - yi) / cy
      ds = min(dsx, dsy)
      xf_traj.append(xi + cx*ds)
      yf_traj.append(yi + cy*ds)
      return ds, xf_traj, yf_traj
  
  
  # Utility functions
  #========================
  
  def interpgrid(a, xi, yi):
      """Fast 2D, linear interpolation on an integer grid"""
  
      Ny, Nx = np.shape(a)
      if isinstance(xi, np.ndarray):
          x = xi.astype(np.int)
          y = yi.astype(np.int)
          # Check that xn, yn don't exceed max index
          xn = np.clip(x + 1, 0, Nx - 1)
          yn = np.clip(y + 1, 0, Ny - 1)
      else:
          x = np.int(xi)
          y = np.int(yi)
          # conditional is faster than clipping for integers
          if x == (Nx - 2): xn = x
          else: xn = x + 1
          if y == (Ny - 2): yn = y
          else: yn = y + 1
  
      a00 = a[y, x]
      a01 = a[y, xn]
      a10 = a[yn, x]
      a11 = a[yn, xn]
      xt = xi - x
      yt = yi - y
      a0 = a00 * (1 - xt) + a01 * xt
      a1 = a10 * (1 - xt) + a11 * xt
      ai = a0 * (1 - yt) + a1 * yt
  
      if not isinstance(xi, np.ndarray):
          if np.ma.is_masked(ai):
              raise TerminateTrajectory
  
      return ai
  
  
  def _gen_starting_points(shape):
      """Yield starting points for streamlines.
  
      Trying points on the boundary first gives higher quality streamlines.
      This algorithm starts with a point on the mask corner and spirals inward.
      This algorithm is inefficient, but fast compared to rest of streamplot.
      """
      ny, nx = shape
      xfirst = 0
      yfirst = 1
      xlast = nx - 1
      ylast = ny - 1
      x, y = 0, 0
      i = 0
      direction = 'right'
      for i in xrange(nx * ny):
  
          yield x, y
  
          if direction == 'right':
              x += 1
              if x >= xlast:
                  xlast -=1
                  direction = 'up'
          elif direction == 'up':
              y += 1
              if y >= ylast:
                  ylast -=1
                  direction = 'left'
          elif direction == 'left':
              x -= 1
              if x <= xfirst:
                  xfirst +=1
                  direction = 'down'
          elif direction == 'down':
              y -= 1
              if y <= yfirst:
                  yfirst +=1
                  direction = 'right'
  