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""" Streamline plotting for 2D vector fields.
"""
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.
*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 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 [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. *start_points*: Nx2 array Coordinates of starting points for the streamlines. In data coordinates, the same as the ``x`` and ``y`` arrays. *zorder* : int any number *maxlength* : float Maximum length of streamline in axes coordinates. *integration_direction* : ['forward', 'backward', 'both'] Integrate the streamline in forward, backward or both directions.
Returns:
*stream_container* : StreamplotSet Container object with attributes
- lines: `matplotlib.collections.LineCollection` of streamlines
- arrows: collection of `matplotlib.patches.FancyArrowPatch` objects representing 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)
if integration_direction not in ['both', 'forward', 'backward']: errstr = ("Integration direction '%s' not recognised. " "Expected 'both', 'forward' or 'backward'." % integration_direction) raise ValueError(errstr)
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, must have 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, must have 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 be of shape '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()) 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, ty = dmap.grid2data(*np.array(t)) 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.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.append(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) 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) axes.autoscale_view()
ac = matplotlib.collections.PatchCollection(arrows) stream_container = StreamplotSet(lc, ac) return stream_container
self.lines = lines self.arrows = arrows
# Coordinate definitions # ========================
"""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`. """
self.grid = grid self.mask = mask # Constants for conversion between grid- and mask-coordinates self.x_grid2mask = (mask.nx - 1) / grid.nx self.y_grid2mask = (mask.ny - 1) / grid.ny
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
"""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))
return xm * self.x_mask2grid, ym * self.y_mask2grid
return xd * self.x_data2grid, yd * self.y_data2grid
return xg / self.x_data2grid, yg / self.y_data2grid
xm, ym = self.grid2mask(xg, yg) self.mask._start_trajectory(xm, ym)
xm, ym = self.grid2mask(xg, yg) self.mask._current_xy = (xm, ym)
if not self.grid.within_grid(xg, yg): raise InvalidIndexError xm, ym = self.grid2mask(xg, yg) self.mask._update_trajectory(xm, ym)
self.mask._undo_trajectory()
"""Grid of data."""
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")
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]
def shape(self): return self.ny, self.nx
"""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
"""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. """
if np.isscalar(density): if density <= 0: raise ValueError("If a scalar, 'density' must be positive") self.nx = self.ny = int(30 * density) else: if len(density) != 2: raise ValueError("'density' can have at maximum 2 dimensions") self.nx = int(30 * density[0]) self.ny = int(30 * density[1]) self._mask = np.zeros((self.ny, self.nx)) self.shape = self._mask.shape
self._current_xy = None
return self._mask.__getitem__(*args)
"""Start recording streamline trajectory""" self._traj = [] self._update_trajectory(xm, ym)
"""Remove current trajectory from mask""" for t in self._traj: self._mask.__setitem__(t, 0)
"""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
# Integrator definitions #========================
# 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`. """
stotal, x_traj, y_traj = 0., [], []
try: dmap.start_trajectory(x0, y0) except InvalidIndexError: return None if integration_direction in ['both', 'backward']: s, xt, yt = _integrate_rk12(x0, y0, dmap, backward_time, maxlength) stotal += s x_traj += xt[::-1] y_traj += yt[::-1]
if integration_direction in ['both', 'forward']: dmap.reset_start_point(x0, y0) s, xt, yt = _integrate_rk12(x0, y0, dmap, forward_time, maxlength) if len(x_traj) > 0: xt = xt[1:] yt = yt[1:] stotal += s x_traj += xt y_traj += yt
if stotal > minlength: return x_traj, y_traj else: # reject short trajectories dmap.undo_trajectory() return None
return integrate
"""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 > 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, xf_traj, yf_traj
"""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 # ========================
"""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
"""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 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' |