Coverage for local/lib/python2.7/site-packages/sage/plot/complex_plot.pyx : 94%

""" 

Complex Plots 

""" 

#***************************************************************************** 

# Copyright (C) 2009 Robert Bradshaw <robertwb@math.washington.edu>, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import absolute_import 

# TODO: use NumPy buffers and complex fast_callable (when supported) 

from cysignals.signals cimport sig_on, sig_off 

cimport numpy as cnumpy 

from sage.plot.primitive import GraphicPrimitive 

from sage.misc.decorators import options 

from sage.rings.complex_double cimport ComplexDoubleElement 

from sage.arith.srange import srange 

from libc.math cimport hypot, atan2, atan, log, sqrt 

cdef double PI = 4 * atan(1) 

cdef inline ComplexDoubleElement new_CDF_element(double x, double y): 

cdef ComplexDoubleElement z = ComplexDoubleElement.__new__(ComplexDoubleElement) 

z._complex.dat[0] = x 

z._complex.dat[1] = y 

return z 

cdef inline double mag_to_lightness(double r): 

""" 

Tweak this to adjust how the magnitude affects the color. 

For instance, changing ``sqrt(r)`` to ``r`` will cause 

anything near a zero to be much darker and poles to be 

much lighter, while ``r**(.25)`` would cause the reverse 

effect. 

INPUT: 

- ``r`` - a non-negative real number 

OUTPUT: 

A value between `-1` (black) and `+1` (white), inclusive. 

EXAMPLES: 

This tests it implicitly:: 

sage: from sage.plot.complex_plot import complex_to_rgb 

sage: complex_to_rgb([[0, 1, 10]]) 

array([[[ 0. , 0. , 0. ], 

[ 0.77172568, 0. , 0. ], 

[ 1. , 0.22134776, 0.22134776]]]) 

""" 

return atan(log(sqrt(r)+1)) * (4/PI) - 1 

def complex_to_rgb(z_values): 

""" 

INPUT: 

- ``z_values`` -- A grid of complex numbers, as a list of lists 

OUTPUT: 

An `N \\times M \\times 3` floating point Numpy array ``X``, where 

``X[i,j]`` is an (r,g,b) tuple. 

EXAMPLES:: 

sage: from sage.plot.complex_plot import complex_to_rgb 

sage: complex_to_rgb([[0, 1, 1000]]) 

array([[[ 0. , 0. , 0. ], 

[ 0.77172568, 0. , 0. ], 

[ 1. , 0.64421177, 0.64421177]]]) 

sage: complex_to_rgb([[0, 1j, 1000j]]) 

array([[[ 0. , 0. , 0. ], 

[ 0.38586284, 0.77172568, 0. ], 

[ 0.82210588, 1. , 0.64421177]]]) 

""" 

import numpy 

cdef unsigned int i, j, imax, jmax 

cdef double x, y, mag, arg 

cdef double lightness, hue, top, bot 

cdef double r, g, b 

cdef int ihue 

cdef ComplexDoubleElement z 

from sage.rings.complex_double import CDF 

imax = len(z_values) 

jmax = len(z_values[0]) 

cdef cnumpy.ndarray[cnumpy.float_t, ndim=3, mode='c'] rgb = numpy.empty(dtype=numpy.float, shape=(imax, jmax, 3)) 

sig_on() 

for i from 0 <= i < imax: 

row = z_values[i] 

for j from 0 <= j < jmax: 

zz = row[j] 

if type(zz) is ComplexDoubleElement: 

z = <ComplexDoubleElement>zz 

else: 

z = CDF(zz) 

x, y = z._complex.dat[0], z._complex.dat[1] 

mag = hypot(x, y) 

arg = atan2(y, x) # math module arctan has range from -pi to pi, so cut along negative x-axis 

lightness = mag_to_lightness(mag) 

if lightness < 0: # in hsv, variable value, full saturation (s=1, v=1+lightness) 

bot = 0 

top = (1+lightness) 

else: # in hsv, variable saturation, full value (v=1, s=1-lightness) 

bot = lightness 

top = 1 

hue = 3*arg/PI # Note that does same thing as colorsys module hsv_to_rgb for this setup, but in Cython 

if hue < 0: hue += 6 # usual hsv hue is thus h=arg/(2*pi) for positive, h=arg/(2*PI)+1 for negative 

ihue = <int>hue 

if ihue == 0: 

r = top 

g = bot + hue * (top-bot) 

b = bot 

elif ihue == 1: 

r = bot + (2-hue) * (top-bot) 

g = top 

b = bot 

elif ihue == 2: 

r = bot 

g = top 

b = bot + (hue-2) * (top-bot) 

elif ihue == 3: 

r = bot 

g = bot + (4-hue) * (top-bot) 

b = top 

elif ihue == 4: 

r = bot + (hue-4) * (top-bot) 

g = bot 

b = top 

else: 

r = top 

g = bot 

b = bot + (6-hue) * (top-bot) 

rgb[i, j, 0] = r 

rgb[i, j, 1] = g 

rgb[i, j, 2] = b 

sig_off() 

return rgb 

class ComplexPlot(GraphicPrimitive): 

""" 

The GraphicsPrimitive to display complex functions in using the domain 

coloring method 

INPUT: 

- ``rgb_data`` -- An array of colored points to be plotted. 

- ``xrange`` -- A minimum and maximum x value for the plot. 

- ``yrange`` -- A minimum and maximum y value for the plot. 

TESTS:: 

sage: p = complex_plot(lambda z: z^2-1, (-2, 2), (-2, 2)) 

""" 

def __init__(self, rgb_data, xrange, yrange, options): 

""" 

TESTS:: 

sage: p = complex_plot(lambda z: z^2-1, (-2, 2), (-2, 2)) 

""" 

self.xrange = xrange 

self.yrange = yrange 

self.x_count = len(rgb_data) 

self.y_count = len(rgb_data[0]) 

self.rgb_data = rgb_data 

GraphicPrimitive.__init__(self, options) 

def get_minmax_data(self): 

""" 

Returns a dictionary with the bounding box data. 

EXAMPLES:: 

sage: p = complex_plot(lambda z: z, (-1, 2), (-3, 4)) 

sage: sorted(p.get_minmax_data().items()) 

[('xmax', 2.0), ('xmin', -1.0), ('ymax', 4.0), ('ymin', -3.0)] 

""" 

from sage.plot.plot import minmax_data 

return minmax_data(self.xrange, self.yrange, dict=True) 

def _allowed_options(self): 

""" 

TESTS:: 

sage: isinstance(complex_plot(lambda z: z, (-1,1), (-1,1))[0]._allowed_options(), dict) 

True 

""" 

return {'plot_points':'How many points to use for plotting precision', 

'interpolation':'What interpolation method to use'} 

def _repr_(self): 

""" 

TESTS:: 

sage: isinstance(complex_plot(lambda z: z, (-1,1), (-1,1))[0]._repr_(), str) 

True 

""" 

return "ComplexPlot defined by a %s x %s data grid"%(self.x_count, self.y_count) 

def _render_on_subplot(self, subplot): 

""" 

TESTS:: 

sage: complex_plot(lambda x: x^2, (-5, 5), (-5, 5)) 

Graphics object consisting of 1 graphics primitive 

""" 

options = self.options() 

x0,x1 = float(self.xrange[0]), float(self.xrange[1]) 

y0,y1 = float(self.yrange[0]), float(self.yrange[1]) 

subplot.imshow(self.rgb_data, origin='lower', extent=(x0,x1,y0,y1), interpolation=options['interpolation']) 

@options(plot_points=100, interpolation='catrom') 

def complex_plot(f, xrange, yrange, **options): 

r""" 

``complex_plot`` takes a complex function of one variable, 

`f(z)` and plots output of the function over the specified 

``xrange`` and ``yrange`` as demonstrated below. The magnitude of the 

output is indicated by the brightness (with zero being black and 

infinity being white) while the argument is represented by the 

hue (with red being positive real, and increasing through orange, 

yellow, ... as the argument increases). 

``complex_plot(f, (xmin, xmax), (ymin, ymax), ...)`` 

INPUT: 

- ``f`` -- a function of a single complex value `x + iy` 

- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values 

- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values 

The following inputs must all be passed in as named parameters: 

- ``plot_points`` -- integer (default: 100); number of points to plot 

in each direction of the grid 

- ``interpolation`` -- string (default: ``'catrom'``), the interpolation 

method to use: ``'bilinear'``, ``'bicubic'``, ``'spline16'``, 

``'spline36'``, ``'quadric'``, ``'gaussian'``, ``'sinc'``, 

``'bessel'``, ``'mitchell'``, ``'lanczos'``, ``'catrom'``, 

``'hermite'``, ``'hanning'``, ``'hamming'``, ``'kaiser'`` 

EXAMPLES: 

Here we plot a couple of simple functions:: 

sage: complex_plot(sqrt(x), (-5, 5), (-5, 5)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(sqrt(x), (-5, 5), (-5, 5))) 

:: 

sage: complex_plot(sin(x), (-5, 5), (-5, 5)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(sin(x), (-5, 5), (-5, 5))) 

:: 

sage: complex_plot(log(x), (-10, 10), (-10, 10)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(log(x), (-10, 10), (-10, 10))) 

:: 

sage: complex_plot(exp(x), (-10, 10), (-10, 10)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(exp(x), (-10, 10), (-10, 10))) 

A function with some nice zeros and a pole:: 

sage: f(z) = z^5 + z - 1 + 1/z 

sage: complex_plot(f, (-3, 3), (-3, 3)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

def f(z): return z**5 + z - 1 + 1/z 

sphinx_plot(complex_plot(f, (-3, 3), (-3, 3))) 

Here is the identity, useful for seeing what values map to what colors:: 

sage: complex_plot(lambda z: z, (-3, 3), (-3, 3)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(lambda z: z, (-3, 3), (-3, 3))) 

The Riemann Zeta function:: 

sage: complex_plot(zeta, (-30,30), (-30,30)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

sphinx_plot(complex_plot(zeta, (-30,30), (-30,30))) 

Extra options will get passed on to show(), as long as they are valid:: 

sage: complex_plot(lambda z: z, (-3, 3), (-3, 3), figsize=[1,1]) 

Graphics object consisting of 1 graphics primitive 

:: 

sage: complex_plot(lambda z: z, (-3, 3), (-3, 3)).show(figsize=[1,1]) # These are equivalent 

TESTS: 

Test to make sure that using fast_callable functions works:: 

sage: f(x) = x^2 

sage: g = fast_callable(f, domain=CC, vars='x') 

sage: h = fast_callable(f, domain=CDF, vars='x') 

sage: P = complex_plot(f, (-10, 10), (-10, 10)) 

sage: Q = complex_plot(g, (-10, 10), (-10, 10)) 

sage: R = complex_plot(h, (-10, 10), (-10, 10)) 

sage: S = complex_plot(exp(x)-sin(x), (-10, 10), (-10, 10)) 

sage: P; Q; R; S 

Graphics object consisting of 1 graphics primitive 

Graphics object consisting of 1 graphics primitive 

Graphics object consisting of 1 graphics primitive 

Graphics object consisting of 1 graphics primitive 

Test to make sure symbolic functions still work without declaring 

a variable. (We don't do this in practice because it doesn't use 

fast_callable, so it is much slower.) 

:: 

sage: complex_plot(sqrt, (-5, 5), (-5, 5)) 

Graphics object consisting of 1 graphics primitive 

""" 

from sage.plot.all import Graphics 

from sage.plot.misc import setup_for_eval_on_grid 

from sage.ext.fast_callable import fast_callable 

from sage.rings.complex_double import CDF 

try: 

f = fast_callable(f, domain=CDF, expect_one_var=True) 

except (AttributeError, TypeError, ValueError): 

pass 

cdef double x, y 

ignore, ranges = setup_for_eval_on_grid([], [xrange, yrange], options['plot_points']) 

xrange,yrange=[r[:2] for r in ranges] 

sig_on() 

z_values = [[ f(new_CDF_element(x, y)) for x in srange(*ranges[0], include_endpoint=True)] 

for y in srange(*ranges[1], include_endpoint=True)] 

sig_off() 

g = Graphics() 

g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax'])) 

g.add_primitive(ComplexPlot(complex_to_rgb(z_values), xrange, yrange, options)) 

return g