# Fractals/Iterations in the complex plane/MandelbrotSetExterior/ParameterExternalRay Editor's note This book is still under development. Please help us

Parameter external ray

• using Newton method
• description by Tomoki Kawahira 
• tracing inward ( from infinity toward Mandelbrot set) = ray-in
• arbitrary precision ( mpfr) with dynamic precision adjustment by Claude Heiland-Allen

Tuning external ray

• DOUADY’S MAGIC FORMULA: Claude : "Douady's magic formula maps rays from the cardioid to the real axis by prefixing binary expansion with 10 or 01 depending if the angle is below or above 1/2. The paper involves veins and Hubbard trees" 

# Q&A

## idea

• take a segment of straight ray ( near infinity )
• pull it back toward boundary of Mandelbrot set

## What means draw external ray ${\mathcal {R}}(\theta )$ ?

It means:

• calculate (approximate) DS points of ray. The result is the set of complex numbers $\{c_{m}:1\leq m\leq DS\}$  ( points on the parameter plane ), use numerical algotrithm
• join points by line segments, use graphical algorithm )

This will give an approximation of ray ${\mathcal {R}}$ .

## How to compute one point of the ray ?

By solving polynomial equation

  $P_{m}(c)=0$ with numerical methods. The root of above equation is point $c_{m}$ .

 $c_{m}=c:P_{m}(c)=0$ It is a point of the external parameter ray ${\mathcal {R}}(\theta )$

 $\arg(\Phi _{M}(c_{m}))=\theta$ or

 $c_{m}\in {\mathcal {R}}(\theta )$ Using Newton method ( iteration ) one can compute approximation of point $c_{m}$

What one needs to start :

• arbitrarily chosen external angle $\theta$  of the ray $R(\theta )$  one wants to draw. Angle is usually given in turns
• value of function P ( which approximates Boettcher mapping ) and its derivative P'
• starting point $c_{m,0}$  ( initial approximation )
• stopping rule ( criteria to stop iteration ): Ray tracing has a natural stopping condition: when the ray enters the atom domain with period p, Newton's method is very likely to converge to the nucleus at its center.

## When ray lands ?

"The rays get closer and closer to the boundary, but don't reach it in finite time - for a more exact boundary point you need to switch to different methods when the ray is close enough. For points on the boundary of hyperbolic components, split the internal angled address (computable from the angle) into island and child path components, when tracing the ray to the parent island use atom domain test (to see if Newton's method is likely to converge to the right place) and switch to Newton's method to find the nucleus of the parent island and then trace internal rays through the chain of connected components to the desired boundary point.

For Misiurewicz points, there is probably a similar test to the atom domain test after which point Newton's method will converge to the desired location (though rays to Misiurewicz points tend to converge much more quickly than rays to roots of hyperbolic components anyway). The atom domain test checks that the iteration count of the last minimum of |z| is the same as the period of the ray." Claude Heiland-Allen

So ray does not "land" in the finite time. Landing point can be denoted as $c_{\infty }$

# tracing rays

• outwards: "External Rays of the form 2pi*n/32, on top of the modulus of the potential gradient. For each point c, a path is created that follows the direction of the gradient of the potential. Each step size is proportional to the distance estimation to M. When the point is far enough of M, it's phase aproximates the phase of phi(c)." Inigo Quilez
• inwards : "The drawing method : ... the path is followed in reverse order: from the infinity towards M, following the minus gradient."

 "you need to trace a ray outwards, which means using different C values, and the bits come in reverse order, first the deepest bit from the iteration count of the start pixel, then move C outwards along the ray (perhaps using the newton's method of mandel-exray.pdf in reverse), repeat until no more bits left.  you move C a fractional iteration count each time, and collect bits when crossing integer dwell boundaries" Claude Heiland-Allen


# Newton method

## variables

• r = radial parameter = radius ( see complex potential )
• m = radial index = index of point along ray, integer
• j = sharpness = number of points on the dwell band, integer
• k = integer depth = number of dwell bands, integer
• d = m/S = real depth, floating point number ( name d is not used by Kawahira)
• l = index of Newton iteration ( name l is not used by Kawahira)
• n = index of iteration for computing Newton map

Names are from T Kawahira description

## constant values

• S = $j_{max}$  =
• D = $k_{max}$  =
• R = ER = Escape Radius
• DS = $m_{max}$  = number of points
• $L=L_{m}=l_{m,max}$

## ranges

• sharpness : $1\leq j\leq S$
• radial index : $1\leq m\leq DS$
• depth :
• $1\leq k\leq D$
• $1\leq d\leq D$
• radius : $R^{1/2^{D}}\leq r
• iterations :
• quadratic : $1\leq n\leq k$
• Newton : $1\leq l_{m}\leq L_{m}$

## sequences

m-sequences ( along the ray toward the Mandelbrot set):

• $r_{1},r_{2},\cdots ,r_{SD}$
• $t_{1},t_{2},\cdots ,t_{SD}$
• $c_{1},c_{2},\cdots ,c_{SD}$

Newton sequences = l-sequences, here m is constant:

• from initial value $c_{m,0}$  toward an approximation of $c_{m,L}$
• $c_{m,0},c_{m,1},\cdots ,c_{m,L}$
  $c_{m,0}{\xrightarrow {N}}c_{m,1}{\xrightarrow {N}}c_{m,1}{\xrightarrow {N}}\cdots {\xrightarrow {N}}c_{m,L}$ ## Maps

### r map

 $r_{m}=R^{1/2^{d}}$ compare it with inverse iteration on c=0 dynamic plane

#### Depth

Using fixed integer D (maximal depth ) :

$k_{max}=D>1$

and fixed maximal value of radial parameter ( escape radius = ER ) :

$r_{max}=R>1$

one can compute D points of ray using fomula :

$r=R^{1/2^{k}}$

which is :

$1

When $k\to k_{max}=D\$  then $r\to 1$  and radius reaches enough close to the boundary of Mandelbrot set


/*
Maxima CAS code

Number of ray points = depth
r = radial parametr : 1 < R^{1/{2^D}} <= r > ER
*/

block (
[ r, R: 65536],
r:R,
print("r = ER = ", float(R)),

for k:1   thru depth  do
(
r:er^(1/(2^k)),
print("k = ", k, " r = ", float(r))
)
)$compile(all)$

/* --------------------------- */

GiveRadius(10)$ Output :  r = ER = 65536.0 "k = "1" r = "256.0 "k = "2" r = "16.0 "k = "3" r = "4.0 "k = "4" r = "2.0 "k = "5" r = "1.414213562373095 "k = "6" r = "1.189207115002721 "k = "7" r = "1.090507732665258 "k = "8" r = "1.044273782427414 "k = "9" r = "1.021897148654117 "k = "10" r = "1.0108892860517  #### Depth and sharpness How to make ray more smooth ? Add more points between level sets. Using: • fixed integer S =maximal sharpness • fixed integer D = maximal depth  $S>0$ one can compute S*D points of ray using fomula :  $m=m_{S}(j,k)=(k-1)S+j$ $d=d_{S}(j,k)=d_{S}(m)={\frac {m}{S}}=(k-1)+{\frac {j}{S}}$ Note that k is equal to integer part of d :  $k=int(d)$ and last point is the same as in depth method  $r_{d_{max}}=r_{k_{max}}$ but there are more points here because :  $S*D>D$ /* Maxima CAS code */ kill(all); remvalue(all); GiveRadius( depth, sharpness):= block ( [ r, R: 65536, d ], r:R, print("r = ER = ", float(R)), for k:1 thru depth do ( for j:1 thru sharpness do ( d: (k-1) + j/sharpness, r:R^(1/(2^d)), print("k = ", k, " ; j = ", j , "; d = ", float(d), "; r = ", float(r)) ) ) )$

compile(all)$/* --------------------------- */ GiveRadius( 10, 4)$
compile(all)$ Output : r = ER = 65536.0 k = 1 ; j = 1 ; d = 0.25 ; r = 11224.33726645605 k = 1 ; j = 2 ; d = 0.5 ; r = 2545.456152628088 k = 1 ; j = 3 ; d = 0.75 ; r = 730.9641900482128 k = 1 ; j = 4 ; d = 1.0 ; r = 256.0 k = 2 ; j = 1 ; d = 1.25 ; r = 105.9449728229521 k = 2 ; j = 2 ; d = 1.5 ; r = 50.45251383854013 k = 2 ; j = 3 ; d = 1.75 ; r = 27.0363494216252 k = 2 ; j = 4 ; d = 2.0 ; r = 16.0 k = 3 ; j = 1 ; d = 2.25 ; r = 10.29295743812011 k = 3 ; j = 2 ; d = 2.5 ; r = 7.10299330131601 k = 3 ; j = 3 ; d = 2.75 ; r = 5.199648971000369 k = 3 ; j = 4 ; d = 3.0 ; r = 4.0 k = 4 ; j = 1 ; d = 3.25 ; r = 3.208263928999625 k = 4 ; j = 2 ; d = 3.5 ; r = 2.665144142690224 k = 4 ; j = 3 ; d = 3.75 ; r = 2.280273880699502 k = 4 ; j = 4 ; d = 4.0 ; r = 2.0 k = 5 ; j = 1 ; d = 4.25 ; r = 1.791162731021284 k = 5 ; j = 2 ; d = 4.5 ; r = 1.632526919438152 k = 5 ; j = 3 ; d = 4.75 ; r = 1.51005757529291 k = 5 ; j = 4 ; d = 5.0 ; r = 1.414213562373095 k = 6 ; j = 1 ; d = 5.25 ; r = 1.338343278468302 k = 6 ; j = 2 ; d = 5.5 ; r = 1.277703768264832 k = 6 ; j = 3 ; d = 5.75 ; r = 1.228843999575581 k = 6 ; j = 4 ; d = 6.0 ; r = 1.189207115002721 k = 7 ; j = 1 ; d = 6.25 ; r = 1.156867874248526 k = 7 ; j = 2 ; d = 6.5 ; r = 1.13035559372475 k = 7 ; j = 3 ; d = 6.75 ; r = 1.108532362890494 k = 7 ; j = 4 ; d = 7.0 ; r = 1.090507732665258 k = 8 ; j = 1 ; d = 7.25 ; r = 1.075577925697867 k = 8 ; j = 2 ; d = 7.5 ; r = 1.063181825335982 k = 8 ; j = 3 ; d = 7.75 ; r = 1.052868635153737 k = 8 ; j = 4 ; d = 8.0 ; r = 1.044273782427414 k = 9 ; j = 1 ; d = 8.25 ; r = 1.037100730738276 k = 9 ; j = 2 ; d = 8.5 ; r = 1.031107087230023 k = 9 ; j = 3 ; d = 8.75 ; r = 1.026093872486205 k = 9 ; j = 4 ; d = 9.0 ; r = 1.021897148654117 k = 10 ; j = 1 ; d = 9.25 ; r = 1.018381426940945 k = 10 ; j = 2 ; d = 9.5 ; r = 1.015434432757735 k = 10 ; j = 3 ; d = 9.75 ; r = 1.012962917626408 k = 10 ; j = 4 ; d = 10.0 ; r = 1.0108892860517  #### m One can use only one loop : m-loop and ccompute j,k and d from m /* Maxima CAS code */ kill(all); remvalue(all)$

block (
[ r, R: 65536, j, k, d, mMax ],

r:float(R),
mMax:depth*sharpness,

print("r = ER = ", r),
print( "m k j  r"),

for m:1   thru mMax  do
(
d: m/sharpness,
r:float(R^(1/(2^d))),

k: floor(d),
j: m - k*sharpness ,

print( m, k, j, r)

)
)$compile(all)$

/* --------------------------- */



output :

r = ER =  65536.0
m k j r
1 0 1 11224.33726645605
2 0 2 2545.456152628088
3 0 3 730.9641900482128
4 1 0 256.0
5 1 1 105.9449728229521
6 1 2 50.45251383854013
7 1 3 27.0363494216252
8 2 0 16.0
9 2 1 10.29295743812011
10 2 2 7.10299330131601
11 2 3 5.199648971000369
12 3 0 4.0
13 3 1 3.208263928999625
14 3 2 2.665144142690224
15 3 3 2.280273880699502
16 4 0 2.0
17 4 1 1.791162731021284
18 4 2 1.632526919438152
19 4 3 1.51005757529291
20 5 0 1.414213562373095
21 5 1 1.338343278468302
22 5 2 1.277703768264832
23 5 3 1.228843999575581
24 6 0 1.189207115002721
25 6 1 1.156867874248526
26 6 2 1.13035559372475
27 6 3 1.108532362890494
28 7 0 1.090507732665258
29 7 1 1.075577925697867
30 7 2 1.063181825335982
31 7 3 1.052868635153737
32 8 0 1.044273782427414
33 8 1 1.037100730738276
34 8 2 1.031107087230023
35 8 3 1.026093872486205
36 9 0 1.021897148654117
37 9 1 1.018381426940945
38 9 2 1.015434432757735
39 9 3 1.012962917626408
40 10 0 1.0108892860517


### Polynomial map q

 $q_{c}(z)=z^{2}+c$ iteration :

 $q_{c}^{n}(z)={q_{c}^{n-1}(z)}^{2}+c$ ### Map t

   $t_{m}=t_{m}(r,\theta ):=r_{m}^{2^{k}}e^{2\pi i2^{k}\theta }$ compare it with forward iteration on c=0 plane :

 $z=r_{m}e^{2\pi i\theta }$ $q_{0}^{k}(z_{m})$ ### Polynomial map P

P is a polynomial of degree $2^{k}$  in variable c.

 $P_{m}(c):=q_{c}^{k}(c)-t_{m}$ Derivative with respect to c :

 $P'_{m}(c):={\frac {dP_{m}(c)}{dc}}=2P'_{m-1}(c)P_{k-m}(c)+1$ ### Newton map N

 $N_{m}(c_{l})=c_{l}-{\frac {P_{m}(c_{l})}{P'_{m}(c_{l})}}$ note that here :

  $c_{l}=c_{m,l}$ #### How to compute new value $c_{m,l+1}$ ?

Arbitrary names :

  $C_{n}:=q^{n}(c_{m,l})$ $D_{n}:=P'_{n}(c_{m,l})=(q^{n}(c_{m,l})-t)'=(q^{n}(c_{m,l}))'$ Note that the derivative of a constant is zero.

The recursive formulae and initial values:

 $C_{n}=C_{n-1}^{2}+c_{m,l};\qquad \qquad C_{1}=c_{m,l}$ $D_{n}=2C_{n-1}D_{n-1}+1;\qquad D_{1}=1$ After k quadratic iterations compute new value $c_{m,l+1}$  using one Newton iteration

 $c_{m,l+1}=N_{m}(c_{m,l})=c_{m,l}-{\frac {C_{k}-t_{m}}{D_{k}}}$ It is implemented in :

#### Newton iteration

Formula :

 $c_{m,l+1}:=N(c_{m,l})$ Newton iterations gives Newton sequence ( = l-sequence, here m is constant):

• from initial value $c_{m,0}$  toward an approximation of $c_{m,L}$
• $c_{m,0},c_{m,1},\cdots ,c_{m,L}$

Sequence :

  $c_{m,0}{\xrightarrow {N}}c_{m,1}{\xrightarrow {N}}c_{m,1}{\xrightarrow {N}}\cdots {\xrightarrow {N}}c_{m,L}$ #### Initial points

• $c_{1,0}=Re^{2\pi i\theta }$
• The value $c_{m-1,L}$  is presumably a “neighbor” of $c_{m,L}$  on ray so use it as the initial value for $c_{m}$  which is $c_{m,0}$
  $c_{m,0}=c_{m-1,L}$ # Code

## Python

Ray in procedure

r"""
Mandelbrot and Julia sets (Cython helper)

This is the helper file providing functionality for mandel_julia.py.
https://git.sagemath.org/sage.git/tree/src/sage/dynamics/complex_dynamics/mandel_julia_helper.pyx?id=bf9df0d7ff4f272b19293fd0d04ef3a649d05863

AUTHORS:

- Ben Barros

"""

#*****************************************************************************
#       Copyright (C) 2017 BEN BARROS <bbarros@slu.edu>
#
# This program is free software: you can redistribute it and/or modify
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#*****************************************************************************

from __future__ import absolute_import, division
from sage.plot.colors import Color
from sage.repl.image import Image
from copy import copy
from cysignals.signals cimport sig_check
from sage.rings.complex_field import ComplexField
from sage.functions.log import exp, log
from sage.symbolic.constants import pi

def external_ray(theta, **kwds):
r"""
Draws the external ray(s) of a given angle (or list of angles)
by connecting a finite number of points that were approximated using
Newton's method. The algorithm used is described in a paper by
Tomoki Kawahira.
https://git.sagemath.org/sage.git/tree/src/sage/dynamics/complex_dynamics/mandel_julia.py?id=bf9df0d7ff4f272b19293fd0d04ef3a649d05863
REFERENCE:

[Kaw2009]_

INPUT:

- theta -- double or list of doubles, angles between 0 and 1 inclusive.

kwds:

- image -- 24-bit RGB image (optional - default: None) user specified image of Mandelbrot set.

- D -- long (optional - default: 25) depth of the approximation. As D increases, the external ray gets closer to the boundary of the Mandelbrot set. If the ray doesn't reach the boundary of the Mandelbrot set, increase D.

- S -- long (optional - default: 10) sharpness of the approximation. Adjusts the number of points used to approximate the external ray (number of points is equal to S*D). If ray looks jagged, increase S.

- R -- long (optional - default: 100) radial parameter. If R is large, the external ray reaches sufficiently close to infinity. If R is too small, Newton's method may not converge to the correct ray.

- prec -- long (optional - default: 300) specifies the bits of precision used by the Complex Field when using Newton's method to compute points on the external ray.

- ray_color -- RGB color (optional - default: [255, 255, 255]) color of the external ray(s).

OUTPUT:

24-bit RGB image of external ray(s) on the Mandelbrot set.

EXAMPLES::

sage: external_ray(1/3)
500x500px 24-bit RGB image

::

sage: external_ray(0.6, ray_color=[255, 0, 0])
500x500px 24-bit RGB image

::

sage: external_ray([0, 0.2, 0.4, 0.7]) # long time
500x500px 24-bit RGB image

::

sage: external_ray([i/5 for i in range(1,5)]) # long time
500x500px 24-bit RGB image

WARNING:

If you are passing in an image, make sure you specify
which parameters to use when drawing the external ray.
For example, the following is incorrect::

sage: M = mandelbrot_plot(x_center=0) # not tested
sage: external_ray(5/7, image=M) # not tested
500x500px 24-bit RGB image

To get the correct external ray, we adjust our parameters::

sage: M = mandelbrot_plot(x_center=0) # not tested
sage: external_ray(5/7, x_center=0, image=M) # not tested
500x500px 24-bit RGB image

TODO:

The copy() function for bitmap images needs to be implemented in Sage.
"""

x_0 = kwds.get("x_center", -1)
y_0 = kwds.get("y_center", 0)
plot_width = kwds.get("image_width", 4)
pixel_width = kwds.get("pixel_count", 500)
depth = kwds.get("D", 25)
sharpness = kwds.get("S", 10)
precision = kwds.get("prec", 300)
precision = max(precision, -log(pixel_width * 0.001, 2).round() + 10)
ray_color = kwds.get("ray_color", *3)
image = kwds.get("image", None)
if image is None:
image = mandelbrot_plot(**kwds)

# Make a copy of the bitmap image.
# M = copy(image)
old_pixel = image.pixels()
M = Image('RGB', (pixel_width, pixel_width))
pixel = M.pixels()
for i in range(pixel_width):
for j in range(pixel_width):
pixel[i,j] = old_pixel[i,j]

# Make sure that theta is a list so loop below works
if type(theta) != list:
theta = [theta]

# Check if theta is in the invterval [0,1]
for angle in theta:
if angle < 0 or angle > 1:
raise \
ValueError("values for theta must be in the closed interval [0,1].")

# Loop through each value for theta in list and plot the external ray.
for angle in theta:
E = fast_external_ray(angle, D=depth, S=sharpness, R=radial_parameter,
prec=precision, image_width=plot_width, pixel_count=pixel_width)

# Convert points to pixel coordinates.
pixel_list = convert_to_pixels(E, x_0, y_0, plot_width, pixel_width)

# Find the pixels between points in pixel_list.
extra_points = []
for i in range(len(pixel_list) - 1):
if min(pixel_list[i+1]) >= 0 and max(pixel_list[i+1]) < pixel_width:
for j in get_line(pixel_list[i], pixel_list[i+1]):
extra_points.append(j)

# Add these points to pixel_list to fill in gaps in the ray.
pixel_list += extra_points

# Remove duplicates from list.
pixel_list = list(set(pixel_list))

# Check if point is in window and if it is, plot it on the image to
# create an external ray.
for k in pixel_list:
if max(k) < pixel_width and min(k) >= 0:
pixel[int(k), int(k)] = tuple(ray_color)
return M

cpdef fast_external_ray(double theta, long D=30, long S=10, long R=100,
long pixel_count=500, double image_width=4, long prec=300):
r"""
Returns a list of points that approximate the external ray for a given angle.

INPUT:

- theta -- double, angle between 0 and 1 inclusive.

- D -- long (optional - default: 25) depth of the approximation. As D increases, the external ray gets closer to the boundary of the Mandelbrot set.

- S -- long (optional - default: 10) sharpness of the approximation. Adjusts the number of points used to approximate the external ray (number of points is equal to S*D).

- R -- long (optional - default: 100) radial parameter. If R is sufficiently large, the external ray reaches enough close to infinity.

- pixel_count -- long (optional - default: 500) side length of image in number of pixels.

- image_width -- double (optional - default: 4) width of the image in the complex plane.

- prec -- long (optional - default: 300) specifies the bits of precision used by the Complex Field when using Newton's method to compute points on the external ray.

OUTPUT:

List of tuples of Real Interval Field Elements

EXAMPLES::

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray
sage: fast_external_ray(0,S=1,D=1)
[(100.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000),
(9.51254777713729174697578576623132297117784691109499464854806785133621315075854778426714908,
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)]

::

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray
sage: fast_external_ray(1/3,S=1,D=1)
[(-49.9999999999999786837179271969944238662719726562500000000000000000000000000000000000000000,
86.6025403784438765342201804742217063903808593750000000000000000000000000000000000000000000),
(-5.50628047023173006234970878097113901879832542655926629309001652388544528575532346900138516,
8.64947510053972513843999918917106032664030380426885745306040284140385975750462108180377187)]

::

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray
sage: fast_external_ray(0.75234,S=1,D=1)
[(1.47021239172637052661229972727596759796142578125000000000000000000000000000000000000000000,
-99.9891917935294287644865107722580432891845703125000000000000000000000000000000000000000000),
(-0.352790406744857508500937144524776555433184352559852962308757189778284058275081335121601384,
-9.98646630765023514178761177926164047797465369576787921409326037870837930920646860774032363)]
"""

cdef:
CF = ComplexField(prec)
PI = CF.pi()
I = CF.gen()
c_0, r_m, t_m, temp_c, C_k, D_k, old_c, x, y, dist
int k, j, t
double difference, m
double error = pixel_count * 0.0001

double pixel_width = image_width / pixel_count

# initialize list with c_0
c_list = [CF(R*exp(2*PI*I*theta))]

# Loop through each subinterval and approximate point on external ray.
for k in range(1,D+1):
for j in range(1,S+1):
m = (k-1)*S + j
r_m = CF(R**(2**(-m/S)))
t_m = CF(r_m**(2**k) * exp(2*PI*I*theta * 2**k))
temp_c = c_list[-1]
difference = error

# Repeat Newton's method until points are close together.
while error <= difference:
sig_check()
old_c = temp_c
# Recursive formula for iterates of q(z) = z^2 + c
C_k, D_k = CF(old_c), CF(1)
for t in range(k):
C_k, D_k = C_k**2 + old_c, CF(2)*D_k*C_k + CF(1)
temp_c = old_c - (C_k - t_m) / D_k   # Newton map
difference = abs(old_c) - abs(temp_c)

dist = (2*C_k.abs()*(C_k.abs()).log()) / D_k.abs()
if dist < pixel_width:
break
c_list.append(CF(temp_c))
if dist < pixel_width:
break

# Convert Complex Field elements into tuples.
for k in range(len(c_list)):
x,y = c_list[k].real(), c_list[k].imag()
c_list[k] = (x, y)

return c_list


# Test

angle $\theta$  in turns landing point $c_{\infty }$  precision
0 1/4
1/2 -2
1/3 -3/4
1/4 -0.228155493653962 +1.115142508039937i
1/5 -0.154724526314600 +1.031047184160779i
1/6 i
1/10 0.384063957 +0.666805123i

See