matplotlib/lib/matplotlib/streamplot.py at master · thomasgit/matplotlib · GitHub

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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.

Parameters

----------

*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'

matplotlib/lib/matplotlib/streamplot.py at master · thomasgit/matplotlib · GitHub

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