"""

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)

# Floating point issues can cause xg, yg to be slightly out of

# bounds for xs, ys on the upper boundaries. Because we have

# already checked that the starting points are within the original

# grid, clip the xg, yg to the grid to work around this issue

xg = np.clip(xg, 0, grid.nx - 1)

yg = np.clip(yg, 0, grid.ny - 1)

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'