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""" Riemann Mapping
AUTHORS:
- Ethan Van Andel (2009-2011): initial version and upgrades
- Robert Bradshaw (2009): his "complex_plot" was adapted for plot_colored
Development supported by NSF award No. 0702939. """
#***************************************************************************** # Copyright (C) 2011 Ethan Van Andel <evlutte@gmail.com>, # # 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 print_function, absolute_import
from cysignals.signals cimport sig_on, sig_off
cimport numpy as np
ctypedef np.float64_t FLOAT_T
ctypedef np.complex128_t COMPLEX_T
cdef class Riemann_Map: r""" The ``Riemann_Map`` class computes an interior or exterior Riemann map, or an Ahlfors map of a region given by the supplied boundary curve(s) and center point. The class also provides various methods to evaluate, visualize, or extract data from the map.
A Riemann map conformally maps a simply connected region in the complex plane to the unit disc. The Ahlfors map does the same thing for multiply connected regions.
Note that all the methods are numerical. As a result all answers have some imprecision. Moreover, maps computed with small number of collocation points, or for unusually shaped regions, may be very inaccurate. Error computations for the ellipse can be found in the documentation for :meth:`analytic_boundary` and :meth:`analytic_interior`.
[BSV2010]_ provides an overview of the Riemann map and discusses the research that lead to the creation of this module.
INPUT:
- ``fs`` -- A list of the boundaries of the region, given as complex-valued functions with domain `0` to `2*pi`. Note that the outer boundary must be parameterized counter clockwise (i.e. ``e^(I*t)``) while the inner boundaries must be clockwise (i.e. ``e^(-I*t)``).
- ``fprimes`` -- A list of the derivatives of the boundary functions. Must be in the same order as ``fs``.
- ``a`` -- Complex, the center of the Riemann map. Will be mapped to the origin of the unit disc. Note that ``a`` MUST be within the region in order for the results to be mathematically valid.
The following inputs may be passed in as named parameters:
- ``N`` -- integer (default: ``500``), the number of collocation points used to compute the map. More points will give more accurate results, especially near the boundaries, but will take longer to compute.
- ``exterior`` -- boolean (default: ``False``), if set to ``True``, the exterior map will be computed, mapping the exterior of the region to the exterior of the unit circle.
The following inputs may be passed as named parameters in unusual circumstances:
- ``ncorners`` -- integer (default: ``4``), if mapping a figure with (equally t-spaced) corners -- corners that make a significant change in the direction of the boundary -- better results may be sometimes obtained by accurately giving this parameter. Used to add the proper constant to the theta correspondence function.
- ``opp`` -- boolean (default: ``False``), set to ``True`` in very rare cases where the theta correspondence function is off by ``pi``, that is, if red is mapped left of the origin in the color plot.
EXAMPLES:
The unit circle identity map::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: m = Riemann_Map([f], [fprime], 0) # long time (4 sec) sage: m.plot_colored() + m.plot_spiderweb() # long time Graphics object consisting of 22 graphics primitives
The exterior map for the unit circle::
sage: m = Riemann_Map([f], [fprime], 0, exterior=True) # long time (4 sec) sage: #spiderwebs are not supported for exterior maps sage: m.plot_colored() # long time Graphics object consisting of 1 graphics primitive
The unit circle with a small hole::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: hf(t) = 0.5*e^(-I*t) sage: hfprime(t) = 0.5*-I*e^(-I*t) sage: m = Riemann_Map([f, hf], [fprime, hfprime], 0.5 + 0.5*I) sage: #spiderweb and color plots cannot be added for multiply sage: #connected regions. Instead we do this. sage: m.plot_spiderweb(withcolor = True) # long time Graphics object consisting of 3 graphics primitives
A square::
sage: ps = polygon_spline([(-1, -1), (1, -1), (1, 1), (-1, 1)]) sage: f = lambda t: ps.value(real(t)) sage: fprime = lambda t: ps.derivative(real(t)) sage: m = Riemann_Map([f], [fprime], 0.25, ncorners=4) sage: m.plot_colored() + m.plot_spiderweb() # long time Graphics object consisting of 22 graphics primitives
Compute rough error for this map::
sage: x = 0.75 # long time sage: print("error = {}".format(m.inverse_riemann_map(m.riemann_map(x)) - x)) # long time error = (-0.000...+0.0016...j)
A fun, complex region for demonstration purposes::
sage: f(t) = e^(I*t) sage: fp(t) = I*e^(I*t) sage: ef1(t) = .2*e^(-I*t) +.4+.4*I sage: ef1p(t) = -I*.2*e^(-I*t) sage: ef2(t) = .2*e^(-I*t) -.4+.4*I sage: ef2p(t) = -I*.2*e^(-I*t) sage: pts = [(-.5,-.15-20/1000),(-.6,-.27-10/1000),(-.45,-.45),(0,-.65+10/1000),(.45,-.45),(.6,-.27-10/1000),(.5,-.15-10/1000),(0,-.43+10/1000)] sage: pts.reverse() sage: cs = complex_cubic_spline(pts) sage: mf = lambda x:cs.value(x) sage: mfprime = lambda x: cs.derivative(x) sage: m = Riemann_Map([f,ef1,ef2,mf],[fp,ef1p,ef2p,mfprime],0,N = 400) # long time sage: p = m.plot_colored(plot_points = 400) # long time
ALGORITHM:
This class computes the Riemann Map via the Szego kernel using an adaptation of the method described by [KT1986]_.
""" cdef int N, B, ncorners cdef f cdef opp cdef COMPLEX_T a cdef np.ndarray tk, tk2 cdef np.ndarray cps, dps, szego, p_vector, pre_q_vector cdef np.ndarray p_vector_inverse, sinalpha, cosalpha, theta_array cdef x_range, y_range cdef exterior
def __init__(self, fs, fprimes, COMPLEX_T a, int N=500, int ncorners=4, opp=False, exterior = False):
""" Initializes the ``Riemann_Map`` class. See the class :class:`Riemann_Map` for full documentation on the input of this initialization method.
TESTS::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0, N = 10) """ cdef double xmax, xmin, ymax, ymin, space cdef int i, k raise ValueError( "The number of collocation points must be > 2.") raise ValueError("The exterior map requires a=0") pass raise ValueError( "The exterior map is undefined for multiply connected domains") # Find the points on the boundaries and their derivatives. for k in xrange(self.B): for i in xrange(N): fk = fs[k](self.tk[N-i-1]) cps[k, i] = np.complex(1/fk) dps[k, i] = np.complex(1/fk**2*fprimes[k](self.tk[N-i-1])) else: xmax = (1/cps).real.max() xmin = (1/cps).real.min() ymax = (1/cps).imag.max() ymin = (1/cps).imag.min() else: #The default plotting window for this map. # Now we do some more computation
def _repr_(self): """ Return a string representation of this :class:`Riemann_Map` object.
TESTS::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: isinstance(Riemann_Map([f], [fprime], 0, N = 10)._repr_(), str) # long time True """ return "A Riemann or Ahlfors mapping of a figure to the unit circle."
cdef _generate_theta_array(self): """ Generates the essential data for the Riemann map, primarily the Szego kernel and boundary correspondence. See [KT1986]_ for the algorithm.
TESTS::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0, N = 10) """ cdef int i, k cdef FLOAT_T saa, t0 cdef np.ndarray[FLOAT_T, ndim=1] adp, sadp cdef np.ndarray[COMPLEX_T,ndim =1] h, hconj, g, normalized_dp, C, phi cdef np.ndarray[COMPLEX_T,ndim =2] K cdef np.ndarray[FLOAT_T, ndim=2] theta_array # Setting things up to use the Nystrom method # Nystrom Method for solving 2nd kind integrals # the all-important Szego kernel # Finding the theta correspondence using phase. Misbehaves for some # regions. # Finding the theta correspondence using abs. Well behaved, but # doesn't work on multiply connected domains. else: # Finding the initial value of the theta function. t0 = theta_array[k, tmax] + phase(phimax) else:
def get_szego(self, int boundary=-1, absolute_value=False): """ Returns a discretized version of the Szego kernel for each boundary function.
INPUT:
The following inputs may be passed in as named parameters:
- ``boundary`` -- integer (default: ``-1``) if < 0, :meth:`get_theta_points` will return the points for all boundaries. If >= 0, :meth:`get_theta_points` will return only the points for the boundary specified.
- ``absolute_value`` -- boolean (default: ``False``) if ``True``, will return the absolute value of the (complex valued) Szego kernel instead of the kernel itself. Useful for plotting.
OUTPUT:
A list of points of the form ``[t value, value of the Szego kernel at that t]``.
EXAMPLES:
Generic use::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: sz = m.get_szego(boundary=0) sage: points = m.get_szego(absolute_value=True) sage: list_plot(points) Graphics object consisting of 1 graphics primitive
Extending the points by a spline::
sage: s = spline(points) sage: s(3*pi / 4) 0.0012158... sage: plot(s,0,2*pi) # plot the kernel Graphics object consisting of 1 graphics primitive
The unit circle with a small hole::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: hf(t) = 0.5*e^(-I*t) sage: hfprime(t) = 0.5*-I*e^(-I*t) sage: m = Riemann_Map([f, hf], [fprime, hfprime], 0.5 + 0.5*I)
Getting the szego for a specific boundary::
sage: sz0 = m.get_szego(boundary=0) sage: sz1 = m.get_szego(boundary=1) """ cdef int k, B temptk = np.concatenate([temptk, self.tk]) else: return np.column_stack([temptk, self.szego.flatten()]).tolist() else: return np.column_stack( [self.tk, abs(self.szego[boundary])]).tolist() else:
def get_theta_points(self, int boundary=-1): """ Returns an array of points of the form ``[t value, theta in e^(I*theta)]``, that is, a discretized version of the theta/boundary correspondence function. In other words, a point in this array [t1, t2] represents that the boundary point given by f(t1) is mapped to a point on the boundary of the unit circle given by e^(I*t2).
For multiply connected domains, ``get_theta_points`` will list the points for each boundary in the order that they were supplied.
INPUT:
The following input must all be passed in as named parameters:
- ``boundary`` -- integer (default: ``-1``) if < 0, ``get_theta_points()`` will return the points for all boundaries. If >= 0, ``get_theta_points()`` will return only the points for the boundary specified.
OUTPUT:
A list of points of the form ``[t value, theta in e^(I*theta)]``.
EXAMPLES:
Getting the list of points, extending it via a spline, getting the points for only the outside of a multiply connected domain::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: points = m.get_theta_points() sage: list_plot(points) Graphics object consisting of 1 graphics primitive
Extending the points by a spline::
sage: s = spline(points) sage: s(3*pi / 4) 1.627660...
The unit circle with a small hole::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: hf(t) = 0.5*e^(-I*t) sage: hfprime(t) = 0.5*-I*e^(-I*t) sage: m = Riemann_Map([f, hf], [hf, hfprime], 0.5 + 0.5*I)
Getting the boundary correspondence for a specific boundary::
sage: tp0 = m.get_theta_points(boundary=0) sage: tp1 = m.get_theta_points(boundary=1) """ temptk = np.concatenate([temptk, self.tk2]) else:
cdef _generate_interior_mapper(self): """ Generates the data necessary to use the :meth:`riemann_map` function. As much setup as possible is done here to minimize the computation that must be done in ``riemann_map``.
TESTS::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0, N = 10) # indirect doctest """ cdef int k, i # Lots of setup for Simpson's method of integration. p_vector[k, N] = 1*coeff * dps[k, 0] * exp(I*theta_array[k, 0]) else:
cpdef riemann_map(self, COMPLEX_T pt): """ Returns the Riemann mapping of a point. That is, given ``pt`` on the interior of the mapped region, ``riemann_map`` will return the point on the unit disk that ``pt`` maps to. Note that this method only works for interior points; accuracy breaks down very close to the boundary. To get boundary correspondance, use :meth:`get_theta_points`.
INPUT:
- ``pt`` -- A complex number representing the point to be inverse mapped.
OUTPUT:
A complex number representing the point on the unit circle that the input point maps to.
EXAMPLES:
Can work for different types of complex numbers::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: m.riemann_map(0.25 + sqrt(-0.5)) (0.137514...+0.876696...j) sage: I = CDF.gen() sage: m.riemann_map(1.3*I) (-1.56...e-05+0.989694...j) sage: m.riemann_map(0.4) (0.73324...+3.2...e-06j) sage: import numpy as np sage: m.riemann_map(np.complex(-3, 0.0001)) (1.405757...e-05+8.06...e-10j) """
cdef COMPLEX_T pt1 cdef np.ndarray[COMPLEX_T, ndim=1] q_vector pt1 = 1/pt q_vector = 1 / ( self.pre_q_vector - pt1) return -1/np.dot(self.p_vector, q_vector) else:
cdef _generate_inverse_mapper(self): """ Generates the data necessary to use the :meth:`inverse_riemann_map` function. As much setup as possible is done here to minimize the computation that must be done in ``inverse_riemann_map``.
TESTS::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0, N = 10) # indirect doctest """ cdef float di # Setup for trapezoid integration because integration points are # not equally spaced. di = di - TWOPI
cpdef inverse_riemann_map(self, COMPLEX_T pt): """ Returns the inverse Riemann mapping of a point. That is, given ``pt`` on the interior of the unit disc, ``inverse_riemann_map()`` will return the point on the original region that would be Riemann mapped to ``pt``. Note that this method does not work for multiply connected domains.
INPUT:
- ``pt`` -- A complex number (usually with absolute value <= 1) representing the point to be inverse mapped.
OUTPUT:
The point on the region that Riemann maps to the input point.
EXAMPLES:
Can work for different types of complex numbers::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: m.inverse_riemann_map(0.5 + sqrt(-0.5)) (0.406880...+0.3614702...j) sage: m.inverse_riemann_map(0.95) (0.486319...-4.90019052...j) sage: m.inverse_riemann_map(0.25 - 0.3*I) (0.1653244...-0.180936...j) sage: import numpy as np sage: m.inverse_riemann_map(np.complex(-0.2, 0.5)) (-0.156280...+0.321819...j) """ pt = 1/pt else: return 1/mapped else:
""" Plots the boundaries of the region for the Riemann map. Note that this method DOES work for multiply connected domains.
INPUT:
The following inputs may be passed in as named parameters:
- ``plotjoined`` -- boolean (default: ``True``) If ``False``, discrete points will be drawn; otherwise they will be connected by lines. In this case, if ``plotjoined=False``, the points shown will be the original collocation points used to generate the Riemann map.
- ``rgbcolor`` -- float array (default: ``[0,0,0]``) the red-green-blue color of the boundary.
- ``thickness`` -- positive float (default: ``1``) the thickness of the lines or points in the boundary.
EXAMPLES:
General usage::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0)
Default plot::
sage: m.plot_boundaries() Graphics object consisting of 1 graphics primitive
Big blue collocation points::
sage: m.plot_boundaries(plotjoined=False, rgbcolor=[0,0,1], thickness=6) Graphics object consisting of 1 graphics primitive """ # This conditional should be eliminated when the thickness/pointsize # issue is resolved later. Same for the others in plot_spiderweb(). else:
cpdef compute_on_grid(self, plot_range, int x_points): """ Computes the Riemann map on a grid of points. Note that these points are complex of the form z = x + y*i.
INPUT:
- ``plot_range`` -- a tuple of the form ``[xmin, xmax, ymin, ymax]``. If the value is ``[]``, the default plotting window of the map will be used.
- ``x_points`` -- int, the size of the grid in the x direction The number of points in the y_direction is scaled accordingly
OUTPUT:
- a tuple containing ``[z_values, xmin, xmax, ymin, ymax]`` where ``z_values`` is the evaluation of the map on the specified grid.
EXAMPLES:
General usage::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: data = m.compute_on_grid([],5) sage: data[0][8,1] (-0.0879...+0.9709...j) """ cdef FLOAT_T xmin, xmax, xstep, ymin, ymax, ystep cdef int y_points else: cdef Py_ssize_t i, j cdef COMPLEX_T pt for i in xrange(x_points): for j in xrange(y_points): pt = 1/(xmin + 0.5*xstep + i*xstep + I*(ymin + 0.5*ystep + j*ystep)) z_values[j, i] = 1/(-np.dot(p_vector,1/(pre_q_vector - pt))) else:
plot_points = 200, min_mag = 0.001, **options): """ Generates a traditional "spiderweb plot" of the Riemann map. Shows what concentric circles and radial lines map to. The radial lines may exhibit erratic behavior near the boundary; if this occurs, decreasing ``linescale`` may mitigate the problem.
For multiply connected domains the spiderweb is by necessity generated using the forward mapping. This method is more computationally intensive. In addition, these spiderwebs cannot be ``added`` to color plots. Instead the ``withcolor`` option must be used.
In addition, spiderweb plots are not currently supported for exterior maps.
INPUT:
The following inputs may be passed in as named parameters:
- ``spokes`` -- integer (default: ``16``) the number of equally spaced radial lines to plot.
- ``circles`` -- integer (default: ``4``) the number of equally spaced circles about the center to plot.
- ``pts`` -- integer (default: ``32``) the number of points to plot. Each radial line is made by ``1*pts`` points, each circle has ``2*pts`` points. Note that high values may cause erratic behavior of the radial lines near the boundaries. - only for simply connected domains
- ``linescale`` -- float between 0 and 1. Shrinks the radial lines away from the boundary to reduce erratic behavior. - only for simply connected domains
- ``rgbcolor`` -- float array (default: ``[0,0,0]``) the red-green-blue color of the spiderweb.
- ``thickness`` -- positive float (default: ``1``) the thickness of the lines or points in the spiderweb.
- ``plotjoined`` -- boolean (default: ``True``) If ``False``, discrete points will be drawn; otherwise they will be connected by lines. - only for simply connected domains
- ``withcolor`` -- boolean (default: ``False``) If ``True``, The spiderweb will be overlaid on the basic color plot.
- ``plot_points`` -- integer (default: ``200``) the size of the grid in the x direction The number of points in the y_direction is scaled accordingly. Note that very large values can cause this function to run slowly. - only for multiply connected domains
- ``min_mag`` -- float (default: ``0.001``) The magnitude cutoff below which spiderweb points are not drawn. This only applies to multiply connected domains and is designed to prevent "fuzz" at the edge of the domain. Some complicated multiply connected domains (particularly those with corners) may require a larger value to look clean outside.
EXAMPLES:
General usage::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0)
Default plot::
sage: m.plot_spiderweb() Graphics object consisting of 21 graphics primitives
Simplified plot with many discrete points::
sage: m.plot_spiderweb(spokes=4, circles=1, pts=400, linescale=0.95, plotjoined=False) Graphics object consisting of 6 graphics primitives
Plot with thick, red lines::
sage: m.plot_spiderweb(rgbcolor=[1,0,0], thickness=3) Graphics object consisting of 21 graphics primitives
To generate the unit circle map, it's helpful to see what the original spiderweb looks like::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: m = Riemann_Map([f], [fprime], 0, 1000) sage: m.plot_spiderweb() Graphics object consisting of 21 graphics primitives
A multiply connected region with corners. We set ``min_mag`` higher to remove "fuzz" outside the domain::
sage: ps = polygon_spline([(-4,-2),(4,-2),(4,2),(-4,2)]) sage: z1 = lambda t: ps.value(t); z1p = lambda t: ps.derivative(t) sage: z2(t) = -2+exp(-I*t); z2p(t) = -I*exp(-I*t) sage: z3(t) = 2+exp(-I*t); z3p(t) = -I*exp(-I*t) sage: m = Riemann_Map([z1,z2,z3],[z1p,z2p,z3p],0,ncorners=4) # long time sage: p = m.plot_spiderweb(withcolor=True,plot_points=500, thickness = 2.0, min_mag=0.1) # long time """ cdef int k, i raise ValueError( "Spiderwebs for exterior maps are not currently supported") else: else: return edge + sum(circle_list) + sum(line_list) + \ self.plot_colored(plot_points=plot_points) else: else: # The more difficult multiply connected z_values, xmin, xmax, ymin, ymax = self.compute_on_grid([], plot_points) xstep = (xmax-xmin)/plot_points ystep = (ymax-ymin)/plot_points dr, dtheta= get_derivatives(z_values, xstep, ystep) # clean later
g = Graphics() g.add_primitive(ComplexPlot(complex_to_spiderweb(z_values,dr,dtheta, spokes, circles, rgbcolor,thickness, withcolor, min_mag), (xmin, xmax), (ymin, ymax),options)) return g + self.plot_boundaries(thickness = thickness)
""" Generates a colored plot of the Riemann map. A red point on the colored plot corresponds to a red point on the unit disc.
INPUT:
The following inputs may be passed in as named parameters:
- ``plot_range`` -- (default: ``[]``) list of 4 values ``(xmin, xmax, ymin, ymax)``. Declare if you do not want the plot to use the default range for the figure.
- ``plot_points`` -- integer (default: ``100``), number of points to plot in the x direction. Points in the y direction are scaled accordingly. Note that very large values can cause this function to run slowly.
EXAMPLES:
Given a Riemann map m, general usage::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: m.plot_colored() Graphics object consisting of 1 graphics primitive
Plot zoomed in on a specific spot::
sage: m.plot_colored(plot_range=[0,1,.25,.75]) Graphics object consisting of 1 graphics primitive
High resolution plot::
sage: m.plot_colored(plot_points=1000) # long time (29s on sage.math, 2012) Graphics object consisting of 1 graphics primitive
To generate the unit circle map, it's helpful to see what the colors correspond to::
sage: f(t) = e^(I*t) sage: fprime(t) = I*e^(I*t) sage: m = Riemann_Map([f], [fprime], 0, 1000) sage: m.plot_colored() Graphics object consisting of 1 graphics primitive """ plot_points)
cdef comp_pt(clist, loop=True): """ Utility function to convert the list of complex numbers ``xderivs = get_derivatives(z_values, xstep, ystep)[0]`` to the plottable `(x,y)` form. If ``loop=True``, then the first point will be added as the last, i.e. to plot a closed circle.
INPUT:
- ``clist`` -- a list of complex numbers.
- ``loop`` -- boolean (default: ``True``) controls whether or not the first point will be added as the last to plot a closed circle.
EXAMPLES:
This tests it implicitly::
sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: m.plot_spiderweb() Graphics object consisting of 21 graphics primitives """
cpdef get_derivatives(np.ndarray[COMPLEX_T, ndim=2] z_values, FLOAT_T xstep, FLOAT_T ystep): """ Computes the r*e^(I*theta) form of derivatives from the grid of points. The derivatives are computed using quick-and-dirty taylor expansion and assuming analyticity. As such ``get_derivatives`` is primarily intended to be used for comparisons in ``plot_spiderweb`` and not for applications that require great precision.
INPUT:
- ``z_values`` -- The values for a complex function evaluated on a grid in the complex plane, usually from ``compute_on_grid``.
- ``xstep`` -- float, the spacing of the grid points in the real direction
OUTPUT:
- A tuple of arrays, [``dr``, ``dtheta``], with each array 2 less in both dimensions than ``z_values``
- ``dr`` - the abs of the derivative of the function in the +r direction - ``dtheta`` - the rate of accumulation of angle in the +theta direction
EXAMPLES:
Standard usage with compute_on_grid::
sage: from sage.calculus.riemann import get_derivatives sage: f(t) = e^(I*t) - 0.5*e^(-I*t) sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t) sage: m = Riemann_Map([f], [fprime], 0) sage: data = m.compute_on_grid([],19) sage: xstep = (data[2]-data[1])/19 sage: ystep = (data[4]-data[3])/19 sage: dr, dtheta = get_derivatives(data[0],xstep,ystep) sage: dr[8,8] 0.241... sage: dtheta[5,5] 5.907... """ cdef np.ndarray[COMPLEX_T, ndim=2] xderiv cdef np.ndarray[FLOAT_T, ndim = 2] dr, dtheta, zabs #(f(x+delta)-f(x-delta))/2delta #b/c the function is analytic, we know the magnitude of its #derivative is equal in all directions # the abs(derivative) scaled by distance from origin
cpdef complex_to_spiderweb(np.ndarray[COMPLEX_T, ndim = 2] z_values, np.ndarray[FLOAT_T, ndim = 2] dr, np.ndarray[FLOAT_T, ndim = 2] dtheta, spokes, circles, rgbcolor, thickness, withcolor, min_mag): """ Converts a grid of complex numbers into a matrix containing rgb data for the Riemann spiderweb plot.
INPUT:
- ``z_values`` -- A grid of complex numbers, as a list of lists.
- ``dr`` -- grid of floats, the r derivative of ``z_values``. Used to determine precision.
- ``dtheta`` -- grid of floats, the theta derivative of ``z_values``. Used to determine precision.
- ``spokes`` -- integer - the number of equally spaced radial lines to plot.
- ``circles`` -- integer - the number of equally spaced circles about the center to plot.
- ``rgbcolor`` -- float array - the red-green-blue color of the lines of the spiderweb.
- ``thickness`` -- positive float - the thickness of the lines or points in the spiderweb.
- ``withcolor`` -- boolean - If ``True`` the spiderweb will be overlaid on the basic color plot.
- ``min_mag`` -- float - The magnitude cutoff below which spiderweb points are not drawn. This only applies to multiply connected domains and is designed to prevent "fuzz" at the edge of the domain. Some complicated multiply connected domains (particularly those with corners) may require a larger value to look clean outside.
OUTPUT:
An `N x M x 3` floating point Numpy array ``X``, where ``X[i,j]`` is an (r,g,b) tuple.
EXAMPLES::
sage: from sage.calculus.riemann import complex_to_spiderweb sage: import numpy sage: zval = numpy.array([[0, 1, 1000],[.2+.3j,1,-.3j],[0,0,0]],dtype = numpy.complex128) sage: deriv = numpy.array([[.1]],dtype = numpy.float64) sage: complex_to_spiderweb(zval, deriv,deriv, 4,4,[0,0,0],1,False,0.001) array([[[ 1., 1., 1.], [ 1., 1., 1.], [ 1., 1., 1.]], <BLANKLINE> [[ 1., 1., 1.], [ 0., 0., 0.], [ 1., 1., 1.]], <BLANKLINE> [[ 1., 1., 1.], [ 1., 1., 1.], [ 1., 1., 1.]]])
sage: complex_to_spiderweb(zval, deriv,deriv, 4,4,[0,0,0],1,True,0.001) array([[[ 1. , 1. , 1. ], [ 1. , 0.05558355, 0.05558355], [ 0.17301243, 0. , 0. ]], <BLANKLINE> [[ 1. , 0.96804683, 0.48044583], [ 0. , 0. , 0. ], [ 0.77351965, 0.5470393 , 1. ]], <BLANKLINE> [[ 1. , 1. , 1. ], [ 1. , 1. , 1. ], [ 1. , 1. , 1. ]]]) """ cdef Py_ssize_t i, j, imax, jmax cdef FLOAT_T x, y, mag, arg, width, target, precision, dmag, darg cdef COMPLEX_T z cdef np.ndarray[FLOAT_T, ndim=3, mode="c"] rgb else: else: circ_radii = [] # both -pi and pi are included else: spoke_angles = [] #points that change too rapidly are presumed to be borders #points that are too small are presumed to be outside rgb[i+1,j+1] = rgbcolor break
cpdef complex_to_rgb(np.ndarray[COMPLEX_T, ndim = 2] z_values): r""" Convert from a (Numpy) array of complex numbers to its corresponding matrix of RGB values. For internal use of :meth:`~Riemann_Map.plot_colored` only.
INPUT:
- ``z_values`` -- A Numpy array of complex numbers.
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.calculus.riemann import complex_to_rgb sage: import numpy sage: complex_to_rgb(numpy.array([[0, 1, 1000]], dtype = numpy.complex128)) array([[[ 1. , 1. , 1. ], [ 1. , 0.05558355, 0.05558355], [ 0.17301243, 0. , 0. ]]])
sage: complex_to_rgb(numpy.array([[0, 1j, 1000j]], dtype = numpy.complex128)) array([[[ 1. , 1. , 1. ], [ 0.52779177, 1. , 0.05558355], [ 0.08650622, 0.17301243, 0. ]]])
TESTS::
sage: complex_to_rgb([[0, 1, 10]]) Traceback (most recent call last): ... TypeError: Argument 'z_values' has incorrect type (expected numpy.ndarray, got list) """ cdef Py_ssize_t i, j, imax, jmax cdef FLOAT_T x, y, mag, arg cdef FLOAT_T lightness, hue, top, bot cdef FLOAT_T r, g, b cdef int ihue cdef COMPLEX_T z
# tweak these levels to adjust how bright/dark the colors appear # output can range from -1 (black) to 1 (white) # in hsv, variable value, full saturation (s=1, v=1+lightness) # in hsv, variable saturation, full value (v=1, s=1-lightness) else: # Note that does same thing as colorsys module hsv_to_rgb for # this setup, but in Cython. # usual hsv hue is thus h=arg/(2*pi) for positive, # h=arg/(2*PI)+1 for negative elif ihue == 1: elif ihue == 2: elif ihue == 3: elif ihue == 4: else:
cpdef analytic_boundary(FLOAT_T t, int n, FLOAT_T epsilon): """ Provides an exact (for n = infinity) Riemann boundary correspondence for the ellipse with axes 1 + epsilon and 1 - epsilon. The boundary is therefore given by e^(I*t)+epsilon*e^(-I*t). It is primarily useful for testing the accuracy of the numerical :class:`Riemann_Map`.
INPUT:
- ``t`` -- The boundary parameter, from 0 to 2*pi
- ``n`` -- integer - the number of terms to include. 10 is fairly accurate, 20 is very accurate.
- ``epsilon`` -- float - the skew of the ellipse (0 is circular)
OUTPUT:
A theta value from 0 to 2*pi, corresponding to the point on the circle e^(I*theta)
TESTS:
Checking the accuracy of this function for different n values::
sage: from sage.calculus.riemann import analytic_boundary sage: t100 = analytic_boundary(pi/2, 100, .3) sage: abs(analytic_boundary(pi/2, 10, .3) - t100) < 10^-8 True sage: abs(analytic_boundary(pi/2, 20, .3) - t100) < 10^-15 True
Using this to check the accuracy of the Riemann_Map boundary::
sage: f(t) = e^(I*t)+.3*e^(-I*t) sage: fp(t) = I*e^(I*t)-I*.3*e^(-I*t) sage: m = Riemann_Map([f], [fp],0,200) sage: s = spline(m.get_theta_points()) sage: test_pt = uniform(0,2*pi) sage: s(test_pt) - analytic_boundary(test_pt,20, .3) < 10^-4 True """ cdef FLOAT_T i
cpdef cauchy_kernel(t, args): """ Intermediate function for the integration in :meth:`~Riemann_Map.analytic_interior`.
INPUT:
- ``t`` -- The boundary parameter, meant to be integrated over
- ``args`` -- a tuple containing:
- ``epsilon`` -- float - the skew of the ellipse (0 is circular)
- ``z`` -- complex - the point to be mapped.
- ``n`` -- integer - the number of terms to include. 10 is fairly accurate, 20 is very accurate.
- ``part`` -- will return the real ('r'), imaginary ('i') or complex ('c') value of the kernel
TESTS:
This is primarily tested implicitly by :meth:`~Riemann_Map.analytic_interior`. Here is a simple test::
sage: from sage.calculus.riemann import cauchy_kernel sage: cauchy_kernel(.5,(.3, .1+.2*I, 10,'c')) (-0.584136405997...+0.5948650858950...j) """ cdef COMPLEX_T result else: return None
cpdef analytic_interior(COMPLEX_T z, int n, FLOAT_T epsilon): """ Provides a nearly exact computation of the Riemann Map of an interior point of the ellipse with axes 1 + epsilon and 1 - epsilon. It is primarily useful for testing the accuracy of the numerical Riemann Map.
INPUT:
- ``z`` -- complex - the point to be mapped.
- ``n`` -- integer - the number of terms to include. 10 is fairly accurate, 20 is very accurate.
TESTS:
Testing the accuracy of :class:`Riemann_Map`::
sage: from sage.calculus.riemann import analytic_interior sage: f(t) = e^(I*t)+.3*e^(-I*t) sage: fp(t) = I*e^(I*t)-I*.3*e^(-I*t) sage: m = Riemann_Map([f],[fp],0,200) sage: abs(m.riemann_map(.5)-analytic_interior(.5, 20, .3)) < 10^-4 True sage: m = Riemann_Map([f],[fp],0,2000) sage: abs(m.riemann_map(.5)-analytic_interior(.5, 20, .3)) < 10^-6 True """ # evaluates the cauchy integral of the boundary, split into the real # and imaginary results because numerical_integral can't handle complex data. |