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"""
Streamline plotting for 2D vector fields.
"""
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):
"""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 *streamlines* : :class:`~matplotlib.collections.LineCollection`
Line collection with all streamlines as a series of line segments.
Currently, there is no way to differentiate between line segments
on different streamlines (other than manually checking that segments
are connected).
"""
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 = []
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, **arrow_kw)
axes.add_patch(p)
lc = mcollections.LineCollection(streamlines, **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)
return lc
# 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.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
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."""
ny, nx = dmap.grid.shape
xi = xf_traj[-1]
yi = yf_traj[-1]
cx, cy = f(xi, yi)
if cx > 0:
dsx = (nx - 1 - xi) / cx
else:
dsx = xi / -cx
if cy > 0:
dsy = (ny - 1 - yi) / cy
else:
dsy = 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'
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