"""
Streamline plotting for 2D vector fields.
"""
import numpy as np
import matplotlib
import matplotlib.cbook as cbook
import matplotlib.cm as 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 arrays
An evenly spaced grid.
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.
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)
cbook._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())
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.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)
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
class StreamplotSet:
def __init__(self, lines, arrows, **kwargs):
if kwargs:
cbook.warn_deprecated(
"3.3",
message="Passing arbitrary keyword arguments to StreamplotSet "
"is deprecated since %(since) and will become an "
"error %(removal)s.")
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")
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 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 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 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
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 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 True:
try:
if dmap.grid.within_grid(xi, yi):
xf_traj.append(xi)
yf_traj.append(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 xf_traj:
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.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, 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(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'
