I am working with a custom projection of Matplotlib and don't understand how to do vector transformations within the projection (Note: The custom projection is a Lambert azimuthal equal-area projection with equatorial aspect).
In my example I want to transform a point that is dipping 30° to the North (meaning that the point is 60°N of the equator) into a point that dips 30° East (meaning that is lies 60° east of the prime meridian). I want to do this with the help of a vector transformation matrix, in order to do more complicated calculations with the program in the future. But I don't really understand how to get the length of the transformed vector right (or getting the correct longitude and latitude of that point).
I am also studying this example, but it uses a slightly different approach for the transformations:
https://github.com/joferkington/mplstereonet/blob/master/mplstereonet/stereonet_math.py
Testfile:
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
from numpy import pi, sin, cos, sqrt, tan, arctan2, arccos
#Internal imports
import projection
def transformVector(geom, raxis, rot):
"""
Input:
geom: single point geometry (vector)
raxis: rotation axis as a vector (vector)
([0][1][2]) = (x,y,z) = (Longitude, Latitude, Down)
rot: rotation in radian
Returns:
Array: a vector that has been transformed
"""
sr = sin(rot)
cr = cos(rot)
omcr = 1.0 - cr
tf = np.array([
[cr + raxis[0]**2 * omcr,
-raxis[2] * sr + raxis[0] * raxis[1] * omcr,
raxis[1] * sr + raxis[0] * raxis[2] * omcr],
[raxis[2] * sr + raxis[1] * raxis[0] * omcr,
cr + raxis[1]**2 * omcr,
-raxis[0] * sr + raxis[1] * raxis[2] * omcr],
[-raxis[1] * sr + raxis[2] * raxis[0] * omcr,
raxis[0] * sr + raxis[2] * raxis[1] * omcr,
cr + raxis[2]**2 * omcr]])
ar = np.dot(geom, tf)
return ar
def sphericalToVector(inp_ar):
"""
Convert a spherical measurement into a vector in cartesian space
[0] = x (+) east (-) west
[1] = y (+) north (-) south
[2] = z (+) down
"""
ar = np.array([0.0, 0.0, 0.0])
ar[0] = sin(inp_ar[0]) * cos(inp_ar[1])
ar[1] = cos(inp_ar[0]) * cos(inp_ar[1])
ar[2] = sin(inp_ar[1])
return ar
def vectorToGeogr(vect):
"""
Returns:
Array with the components [0] longitude, [1] latitude
"""
ar = np.array([0.0, 0.0])
ar[0] = np.arctan2(vect[0], vect[2])
ar[1] = np.arctan2(vect[1], vect[2])
ar = ar * pi/2
return ar
def plotPoint(dip):
"""
Testfunction for converting, transforming and plotting a point
"""
plt.subplot(111, projection="lmbrt_equ_area_equ_aspect")
#Convert to radians
dip_rad = np.radians(dip)
#Set rotation to azimuth and convert dip to latitude on north-south axis
rot = dip_rad[0]
dip_lat = pi/2 - dip_rad[1]
plt.plot(0, dip_lat, "ro")
print(dip_lat, rot)
#Convert the dip into a vector along the north-south axis
#x = 0, y = dip
vect = sphericalToVector([0, dip_lat])
print(vect, np.linalg.norm(vect))
#Transfrom the dip to its proper azimuth
tvect = transformVector(vect, [0,0,1], rot)
print(tvect, np.linalg.norm(tvect))
#Transform the vector back to geographic coordinates
geo = vectorToGeogr(tvect)
print(geo)
plt.plot(geo[0], geo[1], "bo")
plt.grid(True)
plt.show()
datapoint = np.array([090.0,30])
plotPoint(datapoint)
Custom projection:
import matplotlib
from matplotlib.axes import Axes
from matplotlib.patches import Circle
from matplotlib.path import Path
from matplotlib.ticker import NullLocator, Formatter, FixedLocator
from matplotlib.transforms import Affine2D, BboxTransformTo, Transform
from matplotlib.projections import register_projection
import matplotlib.spines as mspines
import matplotlib.axis as maxis
import matplotlib.pyplot as plt
import numpy as np
from numpy import pi, sin, cos, sqrt, arctan2
# This example projection class is rather long, but it is designed to
# illustrate many features, not all of which will be used every time.
# It is also common to factor out a lot of these methods into common
# code used by a number of projections with similar characteristics
# (see geo.py).
class LambertAxes(Axes):
"""
A custom class for the Lambert azimuthal equal-area projection
with equatorial aspect. In geosciences this is also referre to
as a "Schmidt plot". For more information see:
http://pubs.er.usgs.gov/publication/pp1395
"""
# The projection must specify a name. This will be used be the
# user to select the projection, i.e. ``subplot(111,
# projection='lmbrt_equ_area_equ_aspect')``.
name = 'lmbrt_equ_area_equ_aspect'
def __init__(self, *args, **kwargs):
Axes.__init__(self, *args, **kwargs)
self.set_aspect(1, adjustable='box', anchor='C')
self.cla()
def _init_axis(self):
self.xaxis = maxis.XAxis(self)
self.yaxis = maxis.YAxis(self)
# Do not register xaxis or yaxis with spines -- as done in
# Axes._init_axis() -- until LambertAxes.xaxis.cla() works.
# self.spines['hammer'].register_axis(self.yaxis)
self._update_transScale()
def cla(self):
"""
Override to set up some reasonable defaults.
"""
# Don't forget to call the base class
Axes.cla(self)
# Set up a default grid spacing
self.set_longitude_grid(10)
self.set_latitude_grid(10)
self.set_longitude_grid_ends(80)
# Turn off minor ticking altogether
self.xaxis.set_minor_locator(NullLocator())
self.yaxis.set_minor_locator(NullLocator())
# Do not display ticks -- we only want gridlines and text
self.xaxis.set_ticks_position('none')
self.yaxis.set_ticks_position('none')
# The limits on this projection are fixed -- they are not to
# be changed by the user. This makes the math in the
# transformation itself easier, and since this is a toy
# example, the easier, the better.
Axes.set_xlim(self, -pi/2, pi/2)
Axes.set_ylim(self, -pi, pi)
def _set_lim_and_transforms(self):
"""
This is called once when the plot is created to set up all the
transforms for the data, text and grids.
"""
# There are three important coordinate spaces going on here:
#
# 1. Data space: The space of the data itself
#
# 2. Axes space: The unit rectangle (0, 0) to (1, 1)
# covering the entire plot area.
#
# 3. Display space: The coordinates of the resulting image,
# often in pixels or dpi/inch.
# This function makes heavy use of the Transform classes in
# ``lib/matplotlib/transforms.py.`` For more information, see
# the inline documentation there.
# The goal of the first two transformations is to get from the
# data space (in this case longitude and latitude) to axes
# space. It is separated into a non-affine and affine part so
# that the non-affine part does not have to be recomputed when
# a simple affine change to the figure has been made (such as
# resizing the window or changing the dpi).
# 1) The core transformation from data space into
# rectilinear space defined in the LambertEqualAreaTransform class.
self.transProjection = self.LambertEqualAreaTransform()
# 2) The above has an output range that is not in the unit
# rectangle, so scale and translate it so it fits correctly
# within the axes. The peculiar calculations of xscale and
# yscale are specific to a Aitoff-Hammer projection, so don't
# worry about them too much.
xscale = sqrt(2.0) * sin(0.5 * pi)
yscale = sqrt(2.0) * sin(0.5 * pi)
self.transAffine = Affine2D() \
.scale(0.5 / xscale, 0.5 / yscale) \
.translate(0.5, 0.5)
# 3) This is the transformation from axes space to display
# space.
self.transAxes = BboxTransformTo(self.bbox)
# Now put these 3 transforms together -- from data all the way
# to display coordinates. Using the '+' operator, these
# transforms will be applied "in order". The transforms are
# automatically simplified, if possible, by the underlying
# transformation framework.
self.transData = \
self.transProjection + \
self.transAffine + \
self.transAxes
# The main data transformation is set up. Now deal with
# gridlines and tick labels.
# Longitude gridlines and ticklabels. The input to these
# transforms are in display space in x and axes space in y.
# Therefore, the input values will be in range (-xmin, 0),
# (xmax, 1). The goal of these transforms is to go from that
# space to display space. The tick labels will be offset 4
# pixels from the equator.
self._xaxis_pretransform = \
Affine2D() \
.scale(1.0, pi) \
.translate(0.0, -pi)
self._xaxis_transform = \
self._xaxis_pretransform + \
self.transData
self._xaxis_text1_transform = \
Affine2D().scale(1.0, 0.0) + \
self.transData + \
Affine2D().translate(0.0, 4.0)
self._xaxis_text2_transform = \
Affine2D().scale(1.0, 0.0) + \
self.transData + \
Affine2D().translate(0.0, -4.0)
# Now set up the transforms for the latitude ticks. The input to
# these transforms are in axes space in x and display space in
# y. Therefore, the input values will be in range (0, -ymin),
# (1, ymax). The goal of these transforms is to go from that
# space to display space. The tick labels will be offset 4
# pixels from the edge of the axes ellipse.
yaxis_stretch = Affine2D().scale(pi * 2.0, 1.0).translate(-pi, 0.0)
yaxis_space = Affine2D().scale(1.0, 1.0)
self._yaxis_transform = \
yaxis_stretch + \
self.transData
yaxis_text_base = \
yaxis_stretch + \
self.transProjection + \
(yaxis_space + \
self.transAffine + \
self.transAxes)
self._yaxis_text1_transform = \
yaxis_text_base + \
Affine2D().translate(-8.0, 0.0)
self._yaxis_text2_transform = \
yaxis_text_base + \
Affine2D().translate(8.0, 0.0)
def get_xaxis_transform(self,which='grid'):
"""
Override this method to provide a transformation for the
x-axis grid and ticks.
"""
assert which in ['tick1','tick2','grid']
return self._xaxis_transform
def get_xaxis_text1_transform(self, pixelPad):
"""
Override this method to provide a transformation for the
x-axis tick labels.
Returns a tuple of the form (transform, valign, halign)
"""
return self._xaxis_text1_transform, 'bottom', 'center'
def get_xaxis_text2_transform(self, pixelPad):
"""
Override this method to provide a transformation for the
secondary x-axis tick labels.
Returns a tuple of the form (transform, valign, halign)
"""
return self._xaxis_text2_transform, 'top', 'center'
def get_yaxis_transform(self,which='grid'):
"""
Override this method to provide a transformation for the
y-axis grid and ticks.
"""
assert which in ['tick1','tick2','grid']
return self._yaxis_transform
def get_yaxis_text1_transform(self, pixelPad):
"""
Override this method to provide a transformation for the
y-axis tick labels.
Returns a tuple of the form (transform, valign, halign)
"""
return self._yaxis_text1_transform, 'center', 'right'
def get_yaxis_text2_transform(self, pixelPad):
"""
Override this method to provide a transformation for the
secondary y-axis tick labels.
Returns a tuple of the form (transform, valign, halign)
"""
return self._yaxis_text2_transform, 'center', 'left'
def _gen_axes_patch(self):
"""
Override this method to define the shape that is used for the
background of the plot. It should be a subclass of Patch.
In this case, it is a Circle (that may be warped by the axes
transform into an ellipse). Any data and gridlines will be
clipped to this shape.
"""
return Circle((0.5, 0.5), 0.5)
def _gen_axes_spines(self):
return {'lmbrt_equ_area_equ_aspect':mspines.Spine.circular_spine(self,
(0.5, 0.5), 0.5)}
# Prevent the user from applying scales to one or both of the
# axes. In this particular case, scaling the axes wouldn't make
# sense, so we don't allow it.
def set_xscale(self, *args, **kwargs):
if args[0] != 'linear':
raise NotImplementedError
Axes.set_xscale(self, *args, **kwargs)
def set_yscale(self, *args, **kwargs):
if args[0] != 'linear':
raise NotImplementedError
Axes.set_yscale(self, *args, **kwargs)
# Prevent the user from changing the axes limits. In our case, we
# want to display the whole sphere all the time, so we override
# set_xlim and set_ylim to ignore any input. This also applies to
# interactive panning and zooming in the GUI interfaces.
def set_xlim(self, *args, **kwargs):
Axes.set_xlim(self, -pi, pi)
Axes.set_ylim(self, -pi, pi)
set_ylim = set_xlim
def format_coord(self, lon, lat):
"""
Override this method to change how the values are displayed in
the status bar.
In this case, we want them to be displayed in degrees N/S/E/W.
"""
lon = np.degrees(lon)
lat = np.degrees(lat)
#if lat >= 0.0:
# ns = 'N'
#else:
# ns = 'S'
#if lon >= 0.0:
# ew = 'E'
#else:
# ew = 'W'
return "{0} / {1}".format(round(lon,1), round(lat,1))
class DegreeFormatter(Formatter):
"""
This is a custom formatter that converts the native unit of
radians into (truncated) degrees and adds a degree symbol.
"""
def __init__(self, round_to=1.0):
self._round_to = round_to
def __call__(self, x, pos=None):
degrees = (x / pi) * 180.0
degrees = round(degrees / self._round_to) * self._round_to
return "%d\u00b0" % degrees
def set_longitude_grid(self, degrees):
"""
Set the number of degrees between each longitude grid.
This is an example method that is specific to this projection
class -- it provides a more convenient interface to set the
ticking than set_xticks would.
"""
# Set up a FixedLocator at each of the points, evenly spaced
# by degrees.
number = (360.0 / degrees) + 1
self.xaxis.set_major_locator(
plt.FixedLocator(
np.linspace(-pi, pi, number, True)[1:-1]))
# Set the formatter to display the tick labels in degrees,
# rather than radians.
self.xaxis.set_major_formatter(self.DegreeFormatter(degrees))
def set_latitude_grid(self, degrees):
"""
Set the number of degrees between each longitude grid.
This is an example method that is specific to this projection
class -- it provides a more convenient interface than
set_yticks would.
"""
# Set up a FixedLocator at each of the points, evenly spaced
# by degrees.
number = (180.0 / degrees) + 1
self.yaxis.set_major_locator(
FixedLocator(
np.linspace(-pi / 2.0, pi / 2.0, number, True)[1:-1]))
# Set the formatter to display the tick labels in degrees,
# rather than radians.
self.yaxis.set_major_formatter(self.DegreeFormatter(degrees))
def set_longitude_grid_ends(self, degrees):
"""
Set the latitude(s) at which to stop drawing the longitude grids.
Often, in geographic projections, you wouldn't want to draw
longitude gridlines near the poles. This allows the user to
specify the degree at which to stop drawing longitude grids.
This is an example method that is specific to this projection
class -- it provides an interface to something that has no
analogy in the base Axes class.
"""
longitude_cap = degrees * (pi / 180.0)
# Change the xaxis gridlines transform so that it draws from
# -degrees to degrees, rather than -pi to pi.
self._xaxis_pretransform \
.clear() \
.scale(1.0, longitude_cap * 2.0) \
.translate(0.0, -longitude_cap)
def get_data_ratio(self):
"""
Return the aspect ratio of the data itself.
This method should be overridden by any Axes that have a
fixed data ratio.
"""
return 1.0
# Interactive panning and zooming is not supported with this projection,
# so we override all of the following methods to disable it.
def can_zoom(self):
"""
Return True if this axes support the zoom box
"""
return False
def start_pan(self, x, y, button):
pass
def end_pan(self):
pass
def drag_pan(self, button, key, x, y):
pass
class LambertEqualAreaTransform(Transform):
"""
The basic transformation class.
"""
input_dims = 2
output_dims = 2
is_separable = False
def transform_non_affine(self, ll):
"""
Override the transform_non_affine method to implement the custom
transform.
The input and output are Nx2 numpy arrays.
"""
xi = ll[:, 0:1]
yi = ll[:, 1:2]
k = 1 + np.absolute(cos(yi) * cos(xi))
k = 2 / k
if np.isposinf(k[0]) == True:
k[0] = 1e+15
if np.isneginf(k[0]) == True:
k[0] = -1e+15
if k[0] == 0:
k[0] = 1e-15
k = sqrt(k)
x = k * cos(yi) * sin(xi)
y = k * sin(yi)
return np.concatenate((x, y), 1)
# This is where things get interesting. With this projection,
# straight lines in data space become curves in display space.
# This is done by interpolating new values between the input
# values of the data. Since ``transform`` must not return a
# differently-sized array, any transform that requires
# changing the length of the data array must happen within
# ``transform_path``.
def transform_path_non_affine(self, path):
ipath = path.interpolated(path._interpolation_steps)
return Path(self.transform(ipath.vertices), ipath.codes)
transform_path_non_affine.__doc__ = \
Transform.transform_path_non_affine.__doc__
if matplotlib.__version__ < '1.2':
# Note: For compatibility with matplotlib v1.1 and older, you'll
# need to explicitly implement a ``transform`` method as well.
# Otherwise a ``NotImplementedError`` will be raised. This isn't
# necessary for v1.2 and newer, however.
transform = transform_non_affine
# Similarly, we need to explicitly override ``transform_path`` if
# compatibility with older matplotlib versions is needed. With v1.2
# and newer, only overriding the ``transform_path_non_affine``
# method is sufficient.
transform_path = transform_path_non_affine
transform_path.__doc__ = Transform.transform_path.__doc__
def inverted(self):
return LambertAxes.InvertedLambertEqualAreaTransform()
inverted.__doc__ = Transform.inverted.__doc__
class InvertedLambertEqualAreaTransform(Transform):
#This is not working yet !!!
input_dims = 2
output_dims = 2
is_separable = False
def transform_non_affine(self, xy):
x = xy[:, 0:1]
y = xy[:, 1:2]
#quarter_x = 0.25 * x
#half_y = 0.5 * y
#z = sqrt(1.0 - quarter_x*quarter_x - half_y*half_y)
#longitude = 2 * np.arctan((z*x) / (2.0 * (2.0*z*z - 1.0)))
r = sqrt(2)
p = sqrt(x**2 * y**2)
c = 2 * np.arcsin(p / (2 * r))
phi1 = pi/2
lbd0 = 0
#print(x,y)
if y[0] == 0:
lat = 0
else:
lat = np.arcsin(cos(c) * sin(phi1) + (y * sin(c) * cos(phi1 / p)))
#if phi == phi1:
# lon = lbd0 + np.arctan(x / (-y))
#elif phi == -phi1:
# lon = lbd0 + np.arctan(x / y)
#else:
# lon = lbd0 + np.arctan(x * sin(c) / (p * cos(phi1) * cos(c) - y * sin(phi1) * sin(c)))
if x[0] == 0:
lon = 0
else:
lon = lbd0 + np.arctan(x * sin(c) / (p * cos(phi1) * cos(c) - y * sin(phi1) * sin(c)))
return np.concatenate((lon, lat), 1)
transform_non_affine.__doc__ = Transform.transform_non_affine.__doc__
# As before, we need to implement the "transform" method for
# compatibility with matplotlib v1.1 and older.
if matplotlib.__version__ < '1.2':
transform = transform_non_affine
def inverted(self):
# The inverse of the inverse is the original transform... ;)
return LambertAxes.LambertEqualAreaTransform()
inverted.__doc__ = Transform.inverted.__doc__
# Now register the projection with matplotlib so the user can select
# it.
register_projection(LambertAxes)
It appears that the problem is in both your vectorToGeogr and spherical2vector functions. Based on the comments in those and the pole that you were rotating around, it looks (?) like you were intending to have the following relationship:
x : east-west (east-positive)
y : north-south (north-positive)
z : up-down (down-positive)
However, you had mixed in math in places that assumed mathematical coordinates:
x : towards the equator/prime-meridian intersection
y : towards the equator/90 intersection
z : towards the north pole
A quick-but-not-foolproof test is to try "round-tripping" any coordinate conversion functions. It doesn't guarantee that what you're doing is correct, but it guarantees that it's internally consistent. Your current version of things fails this test:
for lat in range(-90, 100, 10):
for lon in range(-180, 190, 10):
point = np.radians([lon, lat])
round_trip = vectorToGeogr(sphericalToVector(point))
assert np.allclose(point, round_trip)
As an aside, I highly reccomend getting at least a few tests up and running and using a test runner of some sort (py.test is my favorite). It will save you a lot of pain in the long run!
Quick side note:
Personally I prefer to separate "real-world" Cartesian space from the Cartesian space used in a stereonet.
It makes the math simpler, and converting between real-world and "stereonet" space is straight-forward (e.g. see the mplstereonet.xyz2stereonet and mplstereonet.stereonet2xyz functions. They're both in the file you linked to.). The examples in stereonet_math.py all use the second set of conventions. When you need to deal with "real" vectors, (e.g. the contour_normal_vectors.py example) they can be converted over with either xyz2stereonet (outputs lon, lat) or one of the various normal2<foo> functions (outputs plunge/bearing, strike/dip, etc).
However, if you do want to use "real-world" Cartesian coordinates internally, you'll need to change your conversion functions.
Your original sphericalToVector function:
def sphericalToVector(inp_ar):
ar = np.array([0.0, 0.0, 0.0])
ar[0] = sin(inp_ar[0]) * cos(inp_ar[1])
ar[1] = cos(inp_ar[0]) * cos(inp_ar[1])
ar[2] = sin(inp_ar[1])
return ar
Should be changed to:
def sphericalToVector(inp_ar):
ar = np.array([0.0, 0.0, 0.0])
ar[0] = -sin(inp_ar[1])
ar[1] = sin(inp_ar[0]) * cos(inp_ar[1])
ar[2] = cos(inp_ar[0]) * cos(inp_ar[1])
return ar
And your original vectorToGeogr function:
def vectorToGeogr(vect):
ar = np.array([0.0, 0.0])
ar[0] = np.arctan2(vect[0], vect[2])
ar[1] = np.arctan2(vect[1], vect[2])
ar = ar * pi/2
return ar
Should be changed to:
def vectorToGeogr(vect):
ar = np.array([0.0, 0.0])
ar[0] = np.arctan2(vect[1], vect[2])
ar[1] = np.arcsin(-vect[0] / np.linalg.norm(vect))
return ar
The modified version of your example is here: https://gist.github.com/joferkington/ddd90715421720033066 The only things changed are the functions above in test.py. As an example of the result:
Related
General aim
I am trying to write some plotting functionality that (at its core)
plots arbitrary paths with a constant width given in data coordinates
(i.e. unlike lines in matplotlib which have widths given in display coordinates).
Previous solutions
This answer achieves
the basic goal. However, this answer converts between display and data
coordinates and then uses a matplotlib line with adjusted
coordinates. The existing functionality in my code that I would like
to replace / extend inherits from matplotlib.patches.Polygon. Since
the rest of the code base makes extensive use of
matplotlib.patches.Polygon attributes and methods, I would like to
continue to inherit from that class.
Problem
My current implementation (code below) seems to come close. However,
the patch created by simple_test seems to be subtly thicker towards
the centre than it is at the start and end point, and I have no
explanation why that may be the case.
I suspect that the problem lies in the computation of the orthogonal vector.
As supporting evidence, I would like to point to the start and end points of the patch in the figure created by complicated_test, which do not seem exactly orthogonal to the path. However, the dot product of the orthonormal vector and the tangent vector is always zero, so I am not sure that what is going on here.
Output of simple_test:
Output of complicated_test:
Code
#!/usr/bin/env python
import numpy as np
import matplotlib.patches
import matplotlib.pyplot as plt
class CurvedPatch(matplotlib.patches.Polygon):
def __init__(self, path, width, *args, **kwargs):
vertices = self.get_vertices(path, width)
matplotlib.patches.Polygon.__init__(self, list(map(tuple, vertices)),
closed=True,
*args, **kwargs)
def get_vertices(self, path, width):
left = _get_parallel_path(path, -width/2)
right = _get_parallel_path(path, width/2)
full = np.concatenate([left, right[::-1]])
return full
def _get_parallel_path(path, delta):
# initialise output
offset = np.zeros_like(path)
# use the previous and the following point to
# determine the tangent at each point in the path;
for ii in range(1, len(path)-1):
offset[ii] += _get_shift(path[ii-1], path[ii+1], delta)
# handle start and end points
offset[0] = _get_shift(path[0], path[1], delta)
offset[-1] = _get_shift(path[-2], path[-1], delta)
return path + offset
def _get_shift(p1, p2, delta):
# unpack coordinates
x1, y1 = p1
x2, y2 = p2
# get orthogonal unit vector;
# adapted from https://stackoverflow.com/a/16890776/2912349
v = np.r_[x2-x1, y2-y1] # vector between points
v = v / np.linalg.norm(v) # unit vector
w = np.r_[-v[1], v[0]] # orthogonal vector
w = w / np.linalg.norm(w) # orthogonal unit vector
# check that vectors are indeed orthogonal
assert np.isclose(np.dot(v, w), 0.)
# rescale unit vector
dx, dy = delta * w
return dx, dy
def simple_test():
x = np.linspace(-1, 1, 1000)
y = np.sqrt(1. - x**2)
path = np.c_[x, y]
curve = CurvedPatch(path, 0.1, facecolor='red', alpha=0.5)
fig, ax = plt.subplots(1,1)
ax.add_artist(curve)
ax.plot(x, y) # plot path for reference
plt.show()
def complicated_test():
random_points = np.random.rand(10, 2)
# Adapted from https://stackoverflow.com/a/35007804/2912349
import scipy.interpolate as si
def scipy_bspline(cv, n=100, degree=3, periodic=False):
""" Calculate n samples on a bspline
cv : Array ov control vertices
n : Number of samples to return
degree: Curve degree
periodic: True - Curve is closed
"""
cv = np.asarray(cv)
count = cv.shape[0]
# Closed curve
if periodic:
kv = np.arange(-degree,count+degree+1)
factor, fraction = divmod(count+degree+1, count)
cv = np.roll(np.concatenate((cv,) * factor + (cv[:fraction],)),-1,axis=0)
degree = np.clip(degree,1,degree)
# Opened curve
else:
degree = np.clip(degree,1,count-1)
kv = np.clip(np.arange(count+degree+1)-degree,0,count-degree)
# Return samples
max_param = count - (degree * (1-periodic))
spl = si.BSpline(kv, cv, degree)
return spl(np.linspace(0,max_param,n))
x, y = scipy_bspline(random_points, n=1000).T
path = np.c_[x, y]
curve = CurvedPatch(path, 0.1, facecolor='red', alpha=0.5)
fig, ax = plt.subplots(1,1)
ax.add_artist(curve)
ax.plot(x, y) # plot path for reference
plt.show()
if __name__ == '__main__':
plt.ion()
simple_test()
complicated_test()
I've been working with matplotlib and basemap to show some information about New York City. Up until now, I've been following this guide, but I've hit an issue. I'm trying to show manhattan island within my visualization, but I can't figure out why basemap isn't showing it as an island.
Here's the visualization that basemap is giving me:
Here's a screenshot of the bounding box I'm using:
And here's the code that is generating the image:
wl = -74.04006
sl = 40.683092
el = -73.834067
nl = 40.88378
m = Basemap(resolution='f', # c, l, i, h, f or None
projection='merc',
area_thresh=50,
lat_0=(wl + sl)/2, lon_0=(el + nl)/2,
llcrnrlon= wl, llcrnrlat= sl, urcrnrlon= el, urcrnrlat= nl)
m.drawmapboundary(fill_color='#46bcec')
m.fillcontinents(color='#f2f2f2',lake_color='#46bcec')
m.drawcoastlines()
m.drawrivers()
I thought that it might consider the water in between a river, but m.drawrivers() didn't appear to fix it. Any help is obviously extremely appreciated.
Thanks in advance!
One approach to get a better quality base map for your plots is building one from web map tiles at an appropriate zoom level. Here I demonstrate how to get them from openstreetmap web map servers. In this case, I use zoom level 10, and get 2 map tiles to combined as single image array. One of the drawbacks, the extent of the combined image is always larger than the values we asked for. Here is the working code:
from mpl_toolkits.basemap import Basemap
import matplotlib.pyplot as plt
import numpy as np
import math
import urllib2
import StringIO
from PIL import Image
# === Begin block1 ===
# Credit: BerndGit, answered Feb 15 '15 at 19:47. And ...
# Source: https://wiki.openstreetmap.org/wiki/Slippy_map_tilenames
def deg2num(lat_deg, lon_deg, zoom):
'''Lon./lat. to tile numbers'''
lat_rad = math.radians(lat_deg)
n = 2.0 ** zoom
xtile = int((lon_deg + 180.0) / 360.0 * n)
ytile = int((1.0 - math.log(math.tan(lat_rad) + (1 / math.cos(lat_rad))) / math.pi) / 2.0 * n)
return (xtile, ytile)
def num2deg(xtile, ytile, zoom):
'''Tile numbers to lon./lat.'''
n = 2.0 ** zoom
lon_deg = xtile / n * 360.0 - 180.0
lat_rad = math.atan(math.sinh(math.pi * (1 - 2 * ytile / n)))
lat_deg = math.degrees(lat_rad)
return (lat_deg, lon_deg) # NW-corner of the tile.
def getImageCluster(lat_deg, lon_deg, delta_lat, delta_long, zoom):
# access map tiles from internet
# no access/key or password is needed
smurl = r"http://a.tile.openstreetmap.org/{0}/{1}/{2}.png"
# useful snippet: smurl.format(zoom, xtile, ytile) -> complete URL
# x increases L-R; y Top-Bottom
xmin, ymax =deg2num(lat_deg, lon_deg, zoom) # get tile numbers (x,y)
xmax, ymin =deg2num(lat_deg+delta_lat, lon_deg+delta_long, zoom)
# PIL is used to build new image from tiles
Cluster = Image.new('RGB',((xmax-xmin+1)*256-1,(ymax-ymin+1)*256-1) )
for xtile in range(xmin, xmax+1):
for ytile in range(ymin, ymax+1):
try:
imgurl = smurl.format(zoom, xtile, ytile)
print("Opening: " + imgurl)
imgstr = urllib2.urlopen(imgurl).read()
# TODO: study, what these do?
tile = Image.open(StringIO.StringIO(imgstr))
Cluster.paste(tile, box=((xtile-xmin)*256 , (ytile-ymin)*255))
except:
print("Couldn't download image")
tile = None
return Cluster
# ===End Block1===
# Credit to myself
def getextents(latmin_deg, lonmin_deg, delta_lat, delta_long, zoom):
'''Return LL and UR, each with (long,lat) of real extent of combined tiles.
latmin_deg: bottom lat of extent
lonmin_deg: left long of extent
delta_lat: extent of lat
delta_long: extent of long, all in degrees
'''
# Tile numbers(x,y): x increases L-R; y Top-Bottom
xtile_LL, ytile_LL = deg2num(latmin_deg, lonmin_deg, zoom) #get tile numbers as specified by (x, y)
xtile_UR, ytile_UR = deg2num(latmin_deg + delta_lat, lonmin_deg + delta_long, zoom)
# from tile numbers, we get NW corners
lat_NW_LL, lon_NW_LL = num2deg(xtile_LL, ytile_LL, zoom)
lat_NW_LLL, lon_NW_LLL = num2deg(xtile_LL, ytile_LL+1, zoom) # next down below
lat_NW_UR, lon_NW_UR = num2deg(xtile_UR, ytile_UR, zoom)
lat_NW_URR, lon_NW_URR = num2deg(xtile_UR+1, ytile_UR, zoom) # next to the right
# get extents
minLat = lat_NW_LLL
minLon = lon_NW_LL
maxLat = lat_NW_UR
maxLon = lon_NW_URR
return (minLon, maxLon, minLat, maxLat) # (left, right, bottom, top) in degrees
# OP's values of extents for target area to plot
# some changes here (with larger zoom level) may lead to better final plot
wl = -74.04006
sl = 40.683092
el = -73.834067
nl = 40.88378
lat_deg = sl
lon_deg = wl
d_lat = nl - sl
d_long = el - wl
zoom = 10 # zoom level
# Acquire images. The combined images will be slightly larger that the extents
timg = getImageCluster(lat_deg, lon_deg, d_lat, d_long, zoom)
# This computes real extents of the combined tile images, and get (left, right, bottom, top)
latmin_deg, lonmin_deg, delta_lat, delta_long = sl, wl, nl-sl, el-wl
(left, right, bottom, top) = getextents(latmin_deg, lonmin_deg, delta_lat, delta_long, zoom) #units: degrees
# Set Basemap with proper parameters
m = Basemap(resolution='h', # h is nice
projection='merc',
area_thresh=50,
lat_0=(bottom + top)/2, lon_0=(left + right)/2,
llcrnrlon=left, llcrnrlat=bottom, urcrnrlon=right, urcrnrlat=top)
fig = plt.figure()
fig.set_size_inches(10, 12)
m.imshow(np.asarray(timg), extent=[left, right, bottom, top], origin='upper' )
m.drawcoastlines(color='gray', linewidth=3.0) # intentionally thick line
#m.fillcontinents(color='#f2f2f2', lake_color='#46bcec', alpha=0.6)
plt.show()
Hope it helps. The resulting plot:
Edit
To crop the image in order to get the exact area to plot is not difficult. The PIL module can handle that. Numpy's array slicing also works.
We have a class that have three functions called(Bdisk, Bhalo,and BX).
all of these functions accept arrays (e.g. shape (1000))not matrices (e.g. shape (2,1000)).
I want to get the total of all these functions( total= Bdisk + Bhalo+BX), total these all functions give the magnetic field in all three components (B_r, B_phi, B_z) for thousand coordinate points (r, phi, z).
the code is here:
import numpy as np
import logging
import warnings
import gmf
signum = lambda x: (x < 0.) * -1. + (x >= 0) * 1.
pi = np.pi
#Class with analytical functions that describe the GMF according to the model of JF12
class GMF(object):
def __init__(self): # self:is automatically set to reference the newly created object that needs to be initialized
self.Rsun = -8.5 # position of the sun along the x axis in kpc
############################################################################
# Disk Parameters
############################################################################
self.bring, self.bring_unc = 0.1,0.1 # floats, field strength in ring at 3 kpc < r < 5 kpc
self.hdisk, self.hdisk_unc = 0.4, 0.03 # float, disk/halo transition height
self.wdisk, self.wdisk_unc = 0.27,0.08 # floats, transition width
self.b = np.array([0.1,3.,-0.9,-0.8,-2.0,-4.2,0.,2.7]) # (8,1)-dim np.arrays, field strength of spiral arms at 5 kpc
self.b_unc = np.array([1.8,0.6,0.8,0.3,0.1,0.5,1.8,1.8]) # uncertainty
self.rx = np.array([5.1,6.3,7.1,8.3,9.8,11.4,12.7,15.5])# (8,1)-dim np.array,dividing lines of spiral lines coordinates of neg. x-axes that intersect with arm
self.idisk = 11.5 * pi/180. # float, spiral arms pitch angle
#############################################################################
# Halo Parameters
#############################################################################
self.Bn, self.Bn_unc = 1.4,0.1 # floats, field strength northern halo
self.Bs, self.Bs_unc = -1.1,0.1 # floats, field strength southern halo
self.rn, self.rn_unc = 9.22,0.08 # floats, transition radius south, lower limit
self.rs, self.rs_unc = 16.7,0. # transition radius south, lower limit
self.whalo, self.whalo_unc = 0.2,0.12 # floats, transition width
self.z0, self.z0_unc = 5.3, 1.6 # floats, vertical scale height
##############################################################################
# Out of plaxe or "X" component Parameters
##############################################################################
self.BX0, self.BX_unc = 4.6,0.3 # floats, field strength at origin
self.ThetaX0, self.ThetaX0_unc = 49. * pi/180., pi/180. # elev. angle at z = 0, r > rXc
self.rXc, self.rXc_unc = 4.8, 0.2 # floats, radius where thetaX = thetaX0
self.rX, self.rX_unc = 2.9, 0.1 # floats, exponential scale length
# striated field
self.gamma, self.gamma_unc = 2.92,0.14 # striation and/or rel. elec. number dens. rescaling
return
##################################################################################
##################################################################################
# Transition function given by logistic function eq.5
##################################################################################
def L(self,z,h,w):
if np.isscalar(z):
z = np.array([z]) # scalar or numpy array with positions (height above disk, z; distance from center, r)
ones = np.ones(z.shape[0])
return 1./(ones + np.exp(-2. *(np.abs(z)- h)/w))
####################################################################################
# return distance from center for angle phi of logarithmic spiral
# r(phi) = rx * exp(b * phi) as np.array
####################################################################################
def r_log_spiral(self,phi):
if np.isscalar(phi): #Returns True if the type of num is a scalar type.
phi = np.array([phi])
ones = np.ones(phi.shape[0])
# self.rx.shape = 8
# phi.shape = p
# then result is given as (8,p)-dim array, each row stands for one rx
# vstack : Take a sequence of arrays and stack them vertically to make a single array
# tensordot(a, b, axes=2):Compute tensor dot product along specified axes for arrays >=1D.
result = np.tensordot(self.rx , np.exp((phi - 3.*pi*ones) / np.tan(pi/2. - self.idisk)),axes = 0)
result = np.vstack((result, np.tensordot(self.rx , np.exp((phi - pi*ones) / np.tan(pi/2. - self.idisk)),axes = 0) ))
result = np.vstack((result, np.tensordot(self.rx , np.exp((phi + pi*ones) / np.tan(pi/2. - self.idisk)),axes = 0) ))
return np.vstack((result, np.tensordot(self.rx , np.exp((phi + 3.*pi*ones) / np.tan(pi/2. - self.idisk)),axes = 0) ))
#############################################################################################
# Disk component in galactocentric cylindrical coordinates (r,phi,z)
#############################################################################################
def Bdisk(self,r,phi,z):
# Bdisk is purely azimuthal (toroidal) with the field strength b_ring
"""
r: N-dim np.array, distance from origin in GC cylindrical coordinates, is in kpc
z: N-dim np.array, height in kpc in GC cylindrical coordinates
phi:N-dim np.array, polar angle in GC cylindircal coordinates, in radian
Bdisk: (3,N)-dim np.array with (r,phi,z) components of disk field for each coordinate tuple
|Bdisk|: N-dim np.array, absolute value of Bdisk for each coordinate tuple
"""
if (not r.shape[0] == phi.shape[0]) and (not z.shape[0] == phi.shape[0]):
warnings.warn("List do not have equal shape! returning -1", RuntimeWarning)
return -1
# Return a new array of given shape and type, filled with zeros.
Bdisk = np.zeros((3,r.shape[0])) # Bdisk vector in r, phi, z
ones = np.ones(r.shape[0])
r_center = (r >= 3.) & (r < 5.1)
r_disk = (r >= 5.1) & (r <= 20.)
Bdisk[1,r_center] = self.bring
# Determine in which arm we are
# this is done for each coordinate individually
if np.sum(r_disk):
rls = self.r_log_spiral(phi[r_disk])
rls = np.abs(rls - r[r_disk])
arms = np.argmin(rls, axis = 0) % 8
# The magnetic spiral defined at r=5 kpc and fulls off as 1/r ,the field direction is given by:
Bdisk[0,r_disk] = np.sin(self.idisk)* self.b[arms] * (5. / r[r_disk])
Bdisk[1,r_disk] = np.cos(self.idisk)* self.b[arms] * (5. / r[r_disk])
Bdisk *= (ones - self.L(z,self.hdisk,self.wdisk)) # multiplied by L
return Bdisk, np.sqrt(np.sum(Bdisk**2.,axis = 0)) # the Bdisk, the normalization
# axis=0 : sum over index 0(row)
# axis=1 : sum over index 1(columns)
##############################################################################################
# Halo component
###############################################################################################
def Bhalo(self,r,z):
# Bhalo is purely azimuthal (toroidal), i.e. has only a phi component
if (not r.shape[0] == z.shape[0]):
warnings.warn("List do not have equal shape! returning -1", RuntimeWarning)
return -1
Bhalo = np.zeros((3,r.shape[0])) # Bhalo vector in r, phi, z rows: r, phi and z component
ones = np.ones(r.shape[0])
m = ( z != 0. )
# SEE equation 6.
Bhalo[1,m] = np.exp(-np.abs(z[m])/self.z0) * self.L(z[m], self.hdisk, self.wdisk) * \
( self.Bn * (ones[m] - self.L(r[m], self.rn, self.whalo)) * (z[m] > 0.) \
+ self.Bs * (ones[m] - self.L(r[m], self.rs, self.whalo)) * (z[m] < 0.) )
return Bhalo , np.sqrt(np.sum(Bhalo**2.,axis = 0))
##############################################################################################
# BX component (OUT OF THE PLANE)
###############################################################################################
def BX(self,r,z):
#BX is purely ASS and poloidal, i.e. phi component = 0
if (not r.shape[0] == z.shape[0]):
warnings.warn("List do not have equal shape! returning -1", RuntimeWarning)
return -1
BX= np.zeros((3,r.shape[0])) # BX vector in r, phi, z rows: r, phi and z component
m = np.sqrt(r**2. + z**2.) >= 1.
bx = lambda r_p: self.BX0 * np.exp(-r_p / self.rX) # eq.7
thetaX = lambda r,z,r_p: np.arctan(np.abs(z)/(r - r_p)) # eq.10
r_p = r[m] *self.rXc/(self.rXc + np.abs(z[m] ) / np.tan(self.ThetaX0)) # eq 9
m_r_b = r_p > self.rXc # region with constant elevation angle
m_r_l = r_p <= self.rXc # region with varying elevation angle
theta = np.zeros(z[m].shape[0])
b = np.zeros(z[m].shape[0])
r_p0 = (r[m])[m_r_b] - np.abs( (z[m])[m_r_b] ) / np.tan(self.ThetaX0) # eq.8
b[m_r_b] = bx(r_p0) * r_p0/ (r[m])[m_r_b] # the field strength in the constant elevation angle (b_x(r_p)r_p/r)
theta[m_r_b] = self.ThetaX0 * np.ones(theta.shape[0])[m_r_b]
b[m_r_l] = bx(r_p[m_r_l]) * (r_p[m_r_l]/(r[m])[m_r_l] )**2. # the field strength with varying elevation angle (b_x(r_p)(r_p/r)**2)
theta[m_r_l] = thetaX((r[m])[m_r_l] ,(z[m])[m_r_l] ,r_p[m_r_l])
mz = (z[m] == 0.)
theta[mz] = np.pi/2.
BX[0,m] = b * (np.cos(theta) * (z[m] >= 0) + np.cos(pi*np.ones(theta.shape[0]) - theta) * (z[m] < 0))
BX[2,m] = b * (np.sin(theta) * (z[m] >= 0) + np.sin(pi*np.ones(theta.shape[0]) - theta) * (z[m] < 0))
return BX, np.sqrt(np.sum(BX**2.,axis=0))
then, I create three arrays, one for r, one for phi, one for z. Each of these arrays has (e.g: thousand elements). like this:
import gmf
gmfm = gmf.GMF()
x = np.linspace(-20.,20.,100)
y = np.linspace(-20.,20.,100)
z = np.linspace(-1.,1.,x.shape[0])
xx,yy = np.meshgrid(x,y)
rr = np.sqrt(xx**2. + yy**2.)
theta = np.arctan2(yy,xx)
for i,r in enumerate(rr[:]):
Bdisk, Babs_d = gmfm.Bdisk(r,theta[i],z)
Bhalo, Babs_h = gmfm.Bhalo(r,z)
BX, Babs_x = gmfm.BX(r,z)
Btotal = Bdisk + Bhalo + BX
but I am getting when I make the addition of the three functions Btotal= Bdisk + Bhalo+BX) in 2d matrix with 3 rows and 100 columns.
My question is how can I add these three functions together to get Btotal in shape (n,) e.g( shape(100,)
because as I said in the beginning the three functions accept accept arrays (e.g. shape (1000) )then when we adding the three functions together we have to get the total also in the same shape (shape (n,)?
I do not know how can I do it, could you please tell me how can I make it.
thank you for your cooperation.
You need to correct the indention, for example in the def Bdisk method.
More importantly in
for i,r in enumerate(rr[:]):
Bdisk, Babs_d = gmfm.Bdisk(r,theta[i],z)
Bhalo, Babs_h = gmfm.Bhalo(r,z)
BX, Babs_x = gmfm.BX(r,z)
Btotal = Bdisk + Bhalo + BX
are you doing this addition for each iteration, or once at the end of the loop? You aren't accumulating any values over iterations. You are just throwing away the old ones, leaving you with the final iteration.
As for adding the array - it appears that all your arrays are initialed like:
Bdisk = np.zeros((3,r.shape[0]))
If that's what the method returns, then
Bdisk + Bhalo + BX
will just sum the corresponding elements of each array, resulting in a Btotal with the same shape. If you don't not like the shape of Btotal then change how Bdisk is calculated, because it has the same shape.
I would like to represent the elliptical orbit of a binary system of two stars. What I aim to, is something like this:
Where I have a grid of the sizes along the axes, an in-scale star at the focus, and the orbit of the secondary star. The decimal numbers along the orbit are the orbital phases. The arrow at the bottom is the Earth direction, and the thick part at the orbit is related to the observation for that specific case - I don't need it. What I want to change from this plot is:
Orbital phase: instead of numbers along the orbit, I would like "dashed rays" from the focus to the orbit, and the orbital phase above them:
I don't want the cross along (0, 0);
I would like to re-orient the orbit, in order that the 0.0 phase is around the top left part of the plot, and the Earth direction is an upward pointing straight arrow (the parameters of my system are different from the one plotted here).
I tried to look for python examples, but the only thing I came out with (from here), is a polar plot:
which is not really representative of what I want, but still is a beginning:
import numpy as np
import matplotlib.pyplot as plt
cos = np.cos
pi = np.pi
a = 10
e = 0.1
theta = np.linspace(0,2*pi, 360)
r = (a*(1-e**2))/(1+e*cos(theta))
fig = plt.figure()
ax = fig.add_subplot(111, polar=True)
ax.set_yticklabels([])
ax.plot(theta,r)
print(np.c_[r,theta])
plt.show()
Here's something that gets you very close. You do not need polar coordinates to plot a decent ellipse. There is a so-called artist you can readily utilize.
You probably have to customize the axis labels and maybe insert an arrow or two if you want:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse
# initializing the figure:
fig = plt.figure()
# the (carthesian) axis:
ax = fig.add_subplot(111,aspect='equal')
ax.grid(True)
# parameters of the ellipse:
a = 5.0
e = 4.0
b = np.sqrt(a**2.0 - e**2.0)
# the center of the ellipse:
x = 6.0
y = 6.0
# the angle by which the ellipse is rotated:
angle = -45.0
#angle = 0.0
# plotting the ellipse, using an artist:
ax.add_artist(Ellipse(xy=[x,y], width=2.0*a, height=2.0*b, \
angle=angle, facecolor='none'))
ax.set_xlim(0,2.0*x)
ax.set_ylim(0,2.0*y)
# marking the focus (actually, both)
# and accounting for the rotation of the ellipse by angle
xf = [x - e*np.cos(angle * np.pi/180.0),
x + e*np.cos(angle * np.pi/180.0)]
yf = [y - e*np.sin(angle * np.pi/180.0),
y + e*np.sin(angle * np.pi/180.0)]
ax.plot(xf,yf,'xr')
# plotting lines from the focus to the ellipse:
# these should be your "rays"
t = np.arange(np.pi,3.0*np.pi,np.pi/5.0)
p = b**2.0 / a
E = e / a
r = [p/(1-E*np.cos(ti)) for ti in t]
# converting the radius based on the focus
# into x,y coordinates on the ellipse:
xr = [ri*np.cos(ti) for ri,ti in zip(r,t)]
yr = [ri*np.sin(ti) for ri,ti in zip(r,t)]
# accounting for the rotation by anlge:
xrp = [xi*np.cos(angle * np.pi/180.0) - \
yi*np.sin(angle * np.pi/180.0) for xi,yi in zip(xr,yr)]
yrp = [xi*np.sin(angle * np.pi/180.0) + \
yi*np.cos(angle * np.pi/180.0) for xi,yi in zip(xr,yr)]
for q in range(0,len(t)):
ax.plot([xf[0], xf[0]+xrp[q]],[yf[0], yf[0]+yrp[q]],'--b')
# put labels outside the "rays"
offset = 0.75
rLabel = [ri+offset for ri in r]
xrl = [ri*np.cos(ti) for ri,ti in zip(rLabel,t)]
yrl = [ri*np.sin(ti) for ri,ti in zip(rLabel,t)]
xrpl = [xi*np.cos(angle * np.pi/180.0) - \
yi*np.sin(angle * np.pi/180.0) for xi,yi in zip(xrl,yrl)]
yrpl = [xi*np.sin(angle * np.pi/180.0) + \
yi*np.cos(angle * np.pi/180.0) for xi,yi in zip(xrl,yrl)]
# for fancy label rotation reduce the range of the angle t:
tlabel = [(ti -np.pi)*180.0/np.pi for ti in t]
for q in range(0,len(tlabel)):
if tlabel[q] >= 180.0:
tlabel[q] -= 180.0
# convert the angle t from radians into degrees:
tl = [(ti-np.pi)*180.0/np.pi for ti in t]
for q in range(0,len(t)):
rotate_label = angle + tlabel[q]
label_text = '%.1f' % tl[q]
ax.text(xf[0]+xrpl[q],yf[0]+yrpl[q],label_text,\
va='center', ha='center',rotation=rotate_label)
plt.show()
The example above will result in this figure:
Explanations:
You can use an artist to plot the ellipse, instead of using polar coordinates
The nomenclature is based on the definitions available on Wikipedia
The angle in the artist setup rotates the ellipse. This angle is later used to rotate coordinates for the rays and labels (this is just math)
The rays are derived from the polar form of the ellipse relative to a focus.
The angle t runs from pi to 3.0*pi because I assumed that this would correspond to your idea of where the rays should start. You get the same rays for 0 to 2.0*pi. I used np.arange instead of linspace because I wanted a defined increment (pi/5.0, or 36 degrees) in this example.
The labels at the end of the rays are placed as text, the variable offset controls the distance between the ellipse and the labels. Adjust this as needed.
For the alignment of the label text orientation with the rays, I reduced the angle, t, to the range 0 to 180 degrees. This makes for better readability compared to the full range of 0 to 360 degrees.
For the label text I just used the angle, t, for simplicity. Replace this with whatever information better suits your purpose.
The angle, t, was converted from radians to degrees before the loop that places the label. Inside the loop, each element of tl is converted to a string. This allows for more formatting control (e.g. %.3f if you needed 3 decimals)
[COMPLETELY REWRITTEN]
I'm looking for a way of interpolating large set of data in dimensions ranging from 3 to 7.
The data is, by nature, on a non rectangular grid and non uniformly spaced.
I looked every option I could think of (griddata, KDTree + magic, linear interpolation, reworked map_coordinates...): the fastest and most usable tool seems to be Scipy's LinearNDInterpolator function. Linear interpolation in such high dimensions space is fine and should be precise enough.
However, there is one big shortcoming with this class: data with gaps or "concave regions" will produce extrapolated results when I want only interpolation.
This is best seen with some pictures (2-D test). In the following I produce some randomly generated data for X, and VALUE, while Y is upper-bounded by a function of X (just so I create gaps).
After rescaling data (mostly done using pieces of code of the LinearNDIterpolator from the master, ie. development, branch), Delaunay triangulation will produce a Convex Hull that includes the gap, and will "extrapolate" in this region. The term "extrapolate" is not really correct here, in a technical sense, but I think would be appropriate given the fact original data is assumed to be sufficiently well sampled so that the big gaps means "no data allowed" (not physical).
To start handling the problem, I "tagged" every Delaunay (hyper-)triangles whose (hyper-)volume is higher than a user-defined threshold (by default the volume equivalent to 5% of the data extent in each dimension).
Generating random data, and evaluating the values using this technique would produce the following figure:
Black dots (with red or white rings) is the randomly generated data to be evaluated. Red rings indicates points that are being rejected (ie. value = NaN) by my custom class based on LinearNDInterpolator, and white rings show accepted points.
For clarity I've plotted triangles that have been rejected from the original Delaunay triangulation.
As you can see, there are still some white rings points that fall in the gap, which I do not want. This is because the simplex to which they belong has a volume less than the authorized maximum volume (some of these triangles even appear as lines on the figure, so it is hard to see)
My question is: how could I improve from here? What could be done?
I was thinking of grabbing all points that fall in a small ball around each evaluated point, and see if there are points in that. But this is not a good solution since it would be resource-consuming and not precise enough (eg. what about points very close to the bottom of the gap, but yet outside the upper envelope?)
Here is my custom interpolation module I used:
#!/usr/bin/env python
"""
Custom N-D linear interpolation class, based on scipy's LinearNDInterpolator.
The main differences are:
- auto-scaling
- interpolation: inside convex hull (normal behavior), and "close enough" to original data.
This rejects points that would normally be interpolated by LinearNDInterpolator.
"""
# ================
# Python modules
# ================
import cPickle
import numpy as np
from scipy.spatial import Delaunay
from scipy.interpolate import LinearNDInterpolator
from scipy.misc import factorial
# =======================
# Convenience functions
# =======================
def _inv_log10(x):
return 10**x
def det(coords): #, n):
"""
Return the determinant of the given coordinates (not the usual determinant, but the one used to compute
the hyper-volume of an hyper-triangle)
From a Delaunay triangulation, the coordinates of one simplex (ie. hyper-triangle) is given by:
coords_i = tri.points[simplex_i]
where
tri = Delaunay(points)
simplex_i = tri.simplices[i]
In an N-dimensional space, the simplex will have N+1 points, each one of them of dimension N.
Eg. in 3D, a points i has coordinates pi = (xi, yi, zi). Therefore p1 = points 1 = (x1, y1, z1)
|x1 x2 x3 x4|
|y1 y2 y3 y4| |(x1-x4) (x2-x4) (x3-x4)|
det = |z1 z2 z3 z4| = |(y1-y4) (y2-y4) (y3-y4)|
|1 1 1 1 | |(z1-z4) (z2-z4) (z3-z4)|
"""
# assert n == len(coords[0]), 'number of dimensions of coordinates (%d) != %d' % (len(coords[0]), n)
q = coords[:-1, :] - coords[-1, None, :]
sign, logdet = np.linalg.slogdet(q)
return sign * np.exp(logdet)
# ==============================
# LinearNDInterpolator wrapper
# ==============================
class Interp(object):
"""
Simple wrapper around LinearNDInterpolator.
"""
def __init__(self, points, values, **kwargs):
"""
:param points: list of coordinates (eg. [(0, 1), (0, 3), (4, 4.5)] for 3 points in 2-D)
:param values: list of associated value(s) for each point (eg. [1, 2, 3] for 3 points of single value)
:keyword rescale: rescale data points so that the final extents is [0, 1] in every dimensions
:keyword transform: transform data points (prior to rescaling). If True, automatically transform dimension coordinates
if extents span more than 2 order of magnitudes. It can also be a list of tuples of
(transformation function, inverse function), that will be applied whenever needed.
:keyword fill_value: outside bounds interpolation values (default: np.nan)
"""
try:
points = np.asanyarray(points, dtype=np.float64)
values = np.asanyarray(values, dtype=np.float64)
except ValueError:
raise ValueError('Cannot convert input points to an array of floats')
# dimensions / number of points and values
self.ndim = points.shape[1]
self.nvalues = values.shape[1]
self.npoints = points.shape[0]
# locals
self._idims = range(self.ndim)
# extents
self.minis = np.min(points, axis=0)
self.maxis = np.max(points, axis=0)
self.ranges = self.maxis - self.minis
self.magnitudes = self.maxis / self.minis
# options
rescale = kwargs.pop('rescale', True)
transform = kwargs.pop('transform', True)
fill_value = kwargs.pop('fill_value', np.nan)
# transformation
if transform:
transforms = []
if transform is True:
# automatic transformation -> if extent >= 2 order of magnitudes: f(x) = log10(x)
for i, e in enumerate(self.magnitudes):
if e >= 100.:
transforms.append((np.log10, _inv_log10))
else:
transforms.append(None)
if not transforms:
transforms = None
else:
err_msg = 'transform: both the transformation function and its inverse must be given in a tuple'
if not isinstance(transform, (tuple, list)):
raise ValueError(err_msg)
if (self.ndim > 1) and (len(transform) != self.ndim):
raise ValueError('transform: None or transformations tuple must be given for every dimension')
for t in transform:
if not isinstance(t, (tuple, list)):
raise ValueError(err_msg)
elif t is None:
transforms.append(None)
else:
transforms.append(t)
self.transforms = transforms
else:
self.transforms = None
points = self._transform(points)
# scaling
self.offset = 0.
self.scale = 1.
self.rescale = rescale
if rescale:
self.offset = np.mean(points, axis=0)
self.scale = (points - self.offset).ptp(axis=0)
self.scale[~(self.scale > 0)] = 1.0 # avoid division by 0
points = self._rescale(points)
# triangulation
self.tri = self._triangulate(points)
# volumes
self.fact = 1. / factorial(self.ndim)
self.volume_max = np.product(self.tri.points.ptp(axis=0) * 0.05) # 5% peak-to-peak in each dimension
self.rej_idx = None
self.rej_vol = None
self.cached_rej = False
# linear interpolation
self.fill_value = fill_value
self.func = LinearNDInterpolator(self.tri, values, fill_value=fill_value)
def _triangulate(self, points, **kwargs):
"""
Delaunay triangulation
"""
return Delaunay(points, **kwargs)
def _get_volume_simplex(self, point):
"""
Compute the simplex volume of the given point
"""
i = self.tri.find_simplex(point)
idx = self.tri.simplices[i]
return np.abs(self.fact * det(self.tri.points[idx]))
def cache_rejected_triangles(self, p=None, check_min=False):
"""
Cache the indexes of rejected triangles.
OPTIONS
p -- peak-to-peak percentage in each dimension for the maximum volume calculation
Default: None (default at __init__: p = 0.05)
Type: float (0 < p <= 1)
Type: list of floats (length = # dimensions)
check_min -- check that the minimum spacing in each dimension is at least equal to p * extent
Default: False
Warning: *p* must be given
"""
self.cached_rej = True
if p is not None:
p = np.array(p)
# update the maximum hyper-triangle volume (p % of the extent in each dimension)
self.volume_max = np.product(self.tri.points.ptp(axis=0) * p)
if check_min:
assert p is not None, 'You must give *p* parameter for checking minimum volume of hyper-triangle'
ptps = self.tri.points.ptp(axis=0)
ps = np.ones(self.ndim) * p
n_up = 0
for i in self._idims:
_x = np.unique(self.tri.points[:, i])
mini = np.min(_x[1:] - _x[:-1])
if mini > (ptps[i] * ps[i]):
n_up += 1
print 'WARNING: changed max. volume axis of dim. %d from %.3g to %.3g' % (i+1, ps[i], mini)
ps[i] = mini
if n_up:
new_vol = np.product(ptps * ps)
print 'CHANGE: old volume was = %.3g, and is now = %.3g' % (self.volume_max, new_vol)
self.volume_max = new_vol
rej_idx = []
rej_vol = []
for i, simplex in enumerate(self.tri.simplices):
vol = np.abs(self.fact * det(self.tri.points[simplex]))
if vol > self.volume_max:
rej_idx.append(i)
rej_vol.append(vol)
self.rej_idx = np.array(rej_idx)
self.rej_vol = np.array(rej_vol)
def _transform(self, points, inverse=False):
"""
Transform point coordinates using functions. Set 'inverse' to True to transform back.
"""
if self.transforms is not None:
j = 1 - int(inverse)
for i in self._idims:
t = self.transforms[i]
if t is None:
continue
points[:, i] = t[j](points[:, i])
return points
def _rescale(self, points, inverse=False):
"""
Rescale point coordinates so that extents in each dimensions span [0, 1]. Set 'inverse' to True to scale back.
"""
if self.rescale:
if inverse:
points = points * self.scale + self.offset
else:
points = (points - self.offset) / self.scale
return points
def _check(self, x, res):
"""
Check that interpolation results are close enough to real data and have not been extrapolated.
"""
points = np.asanyarray(x)
if points.ndim == 1:
# only 1 point
values = np.asanyarray(res).reshape(1, self.ndim)
else:
# more than 1 point
values = np.asanyarray(res).reshape(points.shape[0], self.ndim)
if self.cached_rej:
idx = np.unique(np.where(np.isfinite(values))[0])
ui_tri, uii = np.unique(self.tri.find_simplex(points[idx]), return_inverse=True)
umask = np.lib.arraysetops.in1d(ui_tri, self.rej_idx, assume_unique=True)
mask = umask[uii]
values[idx[mask], :] = self.fill_value
else:
for i, v in enumerate(values):
if not np.isnan(v[0]):
vol = self._get_volume_simplex(points[i])
if vol > self.volume_max:
# reject
values[i][:] = self.fill_value
return values.reshape(res.shape)
def __call__(self, x, check=False):
"""
Interpolate. If 'check' is True, check that interpolated points are close enough to real data.
"""
_x = self._rescale(self._transform(x))
res = self.func(_x)
if check:
res = self._check(_x, res)
return res
def ev(self, x, check=False):
"""
Alias for __call__
"""
return self.__call__(x, check=check)
def get_original_points(self):
"""
Return original points
"""
return self._transform(self._rescale(self.func.points, inverse=True), inverse=True)
def get_original_values(self):
"""
Return original values
"""
return self.func.values
# ===========================
# Save / load interpolation
# ===========================
def save(filename, interp):
"""
Dump the Interp instance to a binary file with cPickle (protocol 2)
"""
with open(filename, 'wb') as f:
cPickle.dump(interp, f, protocol=2)
def load(filename):
"""
Load a previously saved (cPickled with save_interp function) Interp instance
"""
with open(filename, 'rb') as f:
interp = cPickle.load(f)
return interp
And the test script:
#!/usr/bin/env python
"""
Test the custom interpolation class (see interp.py)
"""
import sys
import numpy as np
from interp import Interp
import matplotlib.pyplot as plt
# generate random data
n = 2000 # number of generated points
x = np.random.random(n)
def f(v):
maxi = v ** (1/(v+1e-5)) * (v - 5.) ** 2 - np.exp(v-7) + 1
return np.random.random() * maxi
y = map(f, x * 10)
z = np.random.random(n)
points = np.array((x, y)).T
values = np.random.random(points.shape)
# create interpolation function
func = Interp(points, values, transform=False)
func.cache_rejected_triangles(p=0.05, check_min=True)
# generate random data + evaluate
pts = np.random.random((500, points.shape[1]))
pts *= points.ptp(0)
pts += points.min(0)
res = func(pts, check=True)
# rejected points indexes
idx_rej = np.unique(np.where(np.isnan(res))[0])
n_rej = len(idx_rej)
print '%d points (%.0f%%) have been rejected' % (n_rej, 100.*n_rej/pts.shape[0])
# plot rejected triangles
fig = plt.figure()
ax = plt.gca()
for i in func.rej_idx:
_x = [p for p in points[func.tri.simplices[i], 0]]
_x += [points[func.tri.simplices[i][0], 0]]
_y = [p for p in points[func.tri.simplices[i], 1]]
_y += [points[func.tri.simplices[i][0], 1]]
ax.plot(_x, _y, c='k', ls='-', zorder=100)
# plot original data
ax.scatter(points[:, 0], points[:, 1], c='b', linewidths=0, s=20, zorder=50)
# plot all points (both accepted and rejected): in white
ax.scatter(pts[:, 0], pts[:, 1], c='k', edgecolors='w', linewidths=1, zorder=150, s=30)
# re-plot rejected points: in red
ax.scatter(pts[idx_rej, 0], pts[idx_rej, 1], c='k', edgecolors='r', linewidths=1, zorder=200, s=30)
fig.savefig('img_tri.png', transparent=True, dpi=300)