text: """
  Streamline plotting for 2D vector fields.
  
  """
  
  import numpy as np
  
  import matplotlib
  from matplotlib import _api, cm
  import matplotlib.colors as mcolors
  import matplotlib.collections as mcollections
  import matplotlib.lines as mlines
  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, zorder=None, start_points=None,
                 maxlength=4.0, integration_direction='both'):
      """
      Draw streamlines of a vector flow.
  
      Parameters
      ----------
      x, y : 1D/2D arrays
          Evenly spaced strictly increasing arrays to make a grid.  If 2D, all
          rows of *x* must be equal and all columns of *y* must be equal; i.e.,
          they must be as if generated by ``np.meshgrid(x_1d, y_1d)``.
      u, v : 2D arrays
          *x* and *y*-velocities. The number of rows and columns must match
          the length of *y* and *x*, respectively.
      density : float or (float, float)
          Controls the closeness of streamlines. When ``density = 1``, the domain
          is divided into a 30x30 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 a tuple
          (density_x, density_y).
      linewidth : float or 2D array
          The width of the stream lines. With a 2D array the line width can be
          varied across the grid. The array must have the same shape as *u*
          and *v*.
      color : color or 2D array
          The streamline color. If given an array, its values are converted to
          colors using *cmap* and *norm*.  The array must have the same shape
          as *u* and *v*.
      cmap : `~matplotlib.colors.Colormap`
          Colormap used to plot streamlines and arrows. This is only used if
          *color* is an array.
      norm : `~matplotlib.colors.Normalize`
          Normalize object used to scale luminance data to 0, 1. If ``None``,
          stretch (min, max) to (0, 1). This is only used if *color* is an array.
      arrowsize : float
          Scaling factor for the arrow size.
      arrowstyle : str
          Arrow style specification.
          See `~matplotlib.patches.FancyArrowPatch`.
      minlength : float
          Minimum length of streamline in axes coordinates.
      start_points : Nx2 array
          Coordinates of starting points for the streamlines in data coordinates
          (the same coordinates as the *x* and *y* arrays).
      zorder : int
          The zorder of the stream lines and arrows.
          Artists with lower zorder values are drawn first.
      maxlength : float
          Maximum length of streamline in axes coordinates.
      integration_direction : {'forward', 'backward', 'both'}, default: 'both'
          Integrate the streamline in forward, backward or both directions.
      data : indexable object, optional
          DATA_PARAMETER_PLACEHOLDER
  
      Returns
      -------
      StreamplotSet
          Container object with attributes
  
          - ``lines``: `.LineCollection` of streamlines
  
          - ``arrows``: `.PatchCollection` containing `.FancyArrowPatch`
            objects representing the 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 zorder is None:
          zorder = mlines.Line2D.zorder
  
      # default to data coordinates
      if transform is None:
          transform = axes.transData
  
      if color is None:
          color = axes._get_lines.get_next_color()
  
      if linewidth is None:
          linewidth = matplotlib.rcParams['lines.linewidth']
  
      line_kw = {}
      arrow_kw = dict(arrowstyle=arrowstyle, mutation_scale=10 * arrowsize)
  
      _api.check_in_list(['both', 'forward', 'backward'],
                         integration_direction=integration_direction)
  
      if integration_direction == 'both':
          maxlength /= 2.
  
      use_multicolor_lines = isinstance(color, np.ndarray)
      if use_multicolor_lines:
          if color.shape != grid.shape:
              raise ValueError("If 'color' is given, it must match the shape of "
                               "'Grid(x, y)'")
          line_colors = []
          color = np.ma.masked_invalid(color)
      else:
          line_kw['color'] = color
          arrow_kw['color'] = color
  
      if isinstance(linewidth, np.ndarray):
          if linewidth.shape != grid.shape:
              raise ValueError("If 'linewidth' is given, it must match the "
                               "shape of 'Grid(x, y)'")
          line_kw['linewidth'] = []
      else:
          line_kw['linewidth'] = linewidth
          arrow_kw['linewidth'] = linewidth
  
      line_kw['zorder'] = zorder
      arrow_kw['zorder'] = zorder
  
      # Sanity checks.
      if u.shape != grid.shape or v.shape != grid.shape:
          raise ValueError("'u' and 'v' must match the shape of 'Grid(x, y)'")
  
      u = np.ma.masked_invalid(u)
      v = np.ma.masked_invalid(v)
  
      integrate = _get_integrator(u, v, dmap, minlength, maxlength,
                                  integration_direction)
  
      trajectories = []
      if start_points is None:
          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 is not None:
                      trajectories.append(t)
      else:
          sp2 = np.asanyarray(start_points, dtype=float).copy()
  
          # Check if start_points are outside the data boundaries
          for xs, ys in sp2:
              if not (grid.x_origin <= xs <= grid.x_origin + grid.width and
                      grid.y_origin <= ys <= grid.y_origin + grid.height):
                  raise ValueError("Starting point ({}, {}) outside of data "
                                   "boundaries".format(xs, ys))
  
          # Convert start_points from data to array coords
          # Shift the seed points from the bottom left of the data so that
          # data2grid works properly.
          sp2[:, 0] -= grid.x_origin
          sp2[:, 1] -= grid.y_origin
  
          for xs, ys in sp2:
              xg, yg = dmap.data2grid(xs, ys)
              t = integrate(xg, yg)
              if t is not None:
                  trajectories.append(t)
  
      if use_multicolor_lines:
          if norm is None:
              norm = mcolors.Normalize(color.min(), color.max())
          cmap = cm.get_cmap(cmap)
  
      streamlines = []
      arrows = []
      for t in trajectories:
          tgx, tgy = t.T
          # Rescale from grid-coordinates to data-coordinates.
          tx, ty = dmap.grid2data(tgx, tgy)
          tx += grid.x_origin
          ty += 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.hypot(np.diff(tx), np.diff(ty)))
          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.append(color_values)
              arrow_kw['color'] = cmap(norm(color_values[n]))
  
          p = patches.FancyArrowPatch(
              arrow_tail, arrow_head, transform=transform, **arrow_kw)
          arrows.append(p)
  
      lc = mcollections.LineCollection(
          streamlines, transform=transform, **line_kw)
      lc.sticky_edges.x[:] = [grid.x_origin, grid.x_origin + grid.width]
      lc.sticky_edges.y[:] = [grid.y_origin, grid.y_origin + grid.height]
      if use_multicolor_lines:
          lc.set_array(np.ma.hstack(line_colors))
          lc.set_cmap(cmap)
          lc.set_norm(norm)
      axes.add_collection(lc)
  
      ac = matplotlib.collections.PatchCollection(arrows)
      # Adding the collection itself is broken; see #2341.
      for p in arrows:
          axes.add_patch(p)
  
      axes.autoscale_view()
      stream_container = StreamplotSet(lc, ac)
      return stream_container
  
  
  class StreamplotSet:
  
      def __init__(self, lines, arrows):
          self.lines = lines
          self.arrows = arrows
  
  
  # Coordinate definitions
  # ========================
  
  class DomainMap:
      """
      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 = (mask.nx - 1) / (grid.nx - 1)
          self.y_grid2mask = (mask.ny - 1) / (grid.ny - 1)
  
          self.x_mask2grid = 1. / self.x_grid2mask
          self.y_mask2grid = 1. / self.y_grid2mask
  
          self.x_data2grid = 1. / grid.dx
          self.y_data2grid = 1. / grid.dy
  
      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 grid2data(self, xg, yg):
          return xg / self.x_data2grid, yg / 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:
      """Grid of data."""
      def __init__(self, x, y):
  
          if x.ndim == 1:
              pass
          elif x.ndim == 2:
              x_row = x[0, :]
              if not np.allclose(x_row, x):
                  raise ValueError("The rows of 'x' must be equal")
              x = x_row
          else:
              raise ValueError("'x' can have at maximum 2 dimensions")
  
          if y.ndim == 1:
              pass
          elif y.ndim == 2:
              y_col = y[:, 0]
              if not np.allclose(y_col, y.T):
                  raise ValueError("The columns of 'y' must be equal")
              y = y_col
          else:
              raise ValueError("'y' can have at maximum 2 dimensions")
  
          if not (np.diff(x) > 0).all():
              raise ValueError("'x' must be strictly increasing")
          if not (np.diff(y) > 0).all():
              raise ValueError("'y' must be strictly increasing")
  
          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]
  
          if not np.allclose(np.diff(x), self.width / (self.nx - 1)):
              raise ValueError("'x' values must be equally spaced")
          if not np.allclose(np.diff(y), self.height / (self.ny - 1)):
              raise ValueError("'y' values must be equally spaced")
  
      @property
      def shape(self):
          return self.ny, self.nx
  
      def within_grid(self, xi, yi):
          """Return whether (*xi*, *yi*) is a valid index of the 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 0 <= xi <= self.nx - 1 and 0 <= yi <= self.ny - 1
  
  
  class StreamMask:
      """
      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):
          try:
              self.nx, self.ny = (30 * np.broadcast_to(density, 2)).astype(int)
          except ValueError as err:
              raise ValueError("'density' must be a scalar or be of length "
                               "2") from err
          if self.nx < 0 or self.ny < 0:
              raise ValueError("'density' must be positive")
          self._mask = np.zeros((self.ny, self.nx))
          self.shape = self._mask.shape
  
          self._current_xy = None
  
      def __getitem__(self, args):
          return self._mask[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[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, maxlength, integration_direction):
  
      # 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 - 1)
      v_ax = v / (dmap.grid.ny - 1)
      speed = np.ma.sqrt(u_ax ** 2 + v_ax ** 2)
  
      def forward_time(xi, yi):
          if not dmap.grid.within_grid(xi, yi):
              raise OutOfBounds
          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 (N, 2) 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`.
          """
  
          stotal, xy_traj = 0., []
  
          try:
              dmap.start_trajectory(x0, y0)
          except InvalidIndexError:
              return None
          if integration_direction in ['both', 'backward']:
              s, xyt = _integrate_rk12(x0, y0, dmap, backward_time, maxlength)
              stotal += s
              xy_traj += xyt[::-1]
  
          if integration_direction in ['both', 'forward']:
              dmap.reset_start_point(x0, y0)
              s, xyt = _integrate_rk12(x0, y0, dmap, forward_time, maxlength)
              stotal += s
              xy_traj += xyt[1:]
  
          if stotal > minlength:
              return np.broadcast_arrays(xy_traj, np.empty((1, 2)))[0]
          else:  # reject short trajectories
              dmap.undo_trajectory()
              return None
  
      return integrate
  
  
  @_api.deprecated("3.5")
  def get_integrator(u, v, dmap, minlength, maxlength, integration_direction):
      xy_traj = _get_integrator(
          u, v, dmap, minlength, maxlength, integration_direction)
      return (None if xy_traj is None
              else ([], []) if not len(xy_traj)
              else [*zip(*xy_traj)])
  
  
  class OutOfBounds(IndexError):
      pass
  
  
  def _integrate_rk12(x0, y0, dmap, f, maxlength):
      """
      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 RK4 and RK45 solvers (using
      similar Python implementations) in most setups.
      """
      # 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
      xyf_traj = []
  
      while True:
          try:
              if dmap.grid.within_grid(xi, yi):
                  xyf_traj.append((xi, yi))
              else:
                  raise OutOfBounds
  
              # Compute the two intermediate gradients.
              # f should raise OutOfBounds if the locations given are
              # outside the grid.
              k1x, k1y = f(xi, yi)
              k2x, k2y = f(xi + ds * k1x, yi + ds * k1y)
  
          except OutOfBounds:
              # Out of the domain during this step.
              # Take an Euler step to the boundary to improve neatness
              # unless the trajectory is currently empty.
              if xyf_traj:
                  ds, xyf_traj = _euler_step(xyf_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)
  
          ny, nx = dmap.grid.shape
          # Error is normalized to the axes coordinates
          error = np.hypot((dx2 - dx1) / (nx - 1), (dy2 - dy1) / (ny - 1))
  
          # 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 > maxlength:
                  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, xyf_traj
  
  
  def _euler_step(xyf_traj, dmap, f):
      """Simple Euler integration step that extends streamline to boundary."""
      ny, nx = dmap.grid.shape
      xi, yi = xyf_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)
      xyf_traj.append((xi + cx * ds, yi + cy * ds))
      return ds, xyf_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(int)
          y = yi.astype(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 = int(xi)
          y = int(yi)
          # conditional is faster than clipping for integers
          if x == (Nx - 1):
              xn = x
          else:
              xn = x + 1
          if y == (Ny - 1):
              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
      direction = 'right'
      for i in range(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'
  