Develop Reference

Python: maldelbrot set multithreaded

I'm using the following algorithm to generate a Mandelbrot set fractal using threads in Python. Something is not working properly because I got minimum difference in time processing using 2, 4 , 8 or even 10 threads. Can someone kinldy help me and take a look to the code so it work correctly, please?!
import threading
from PIL import Image
import matplotlib.pyplot as plt
from timeit import default_timer as timer
# Fractals are infinitely repeating patterns on different scales
xSize = ySize = 2096 # defines a finite number of pixels on axes x and y.
image = Image.new("RGB", (xSize, ySize))
imageSize = xSize * ySize
maxIt = 256 # defines the maximun number of interactions allowed
# drawing region (xa < xb & ya < yb): defined the domain plan
xa = -2.0 # defines the begin of the real axis X: from xa
xb = 0.5 # defines the end of the real axis X: to xb
ya = -1.25 # defines the begin of the imaginary axis Y: from ya
yb = 1.25 # defines the end of the imaginary axis Y: to yb
xd = xb - xa # defines x dimension
yd = yb - ya # defines y dimension
numThr = 4 # number of threads to run
class MandelbrotThread(threading.Thread):
print('Generating fractal image of size {0} x {1} with {2} interactions using {3} threads'.format(xSize, ySize, maxIt, numThr))
def __init__ (self, k):
self.k = k
threading.Thread.__init__(self)
print("Executing thread ", k)
def run(self):
# each thread only calculates its own share of pixels on the domain plan
for i in range(t, imageSize, numThr):
kx = i % xSize #calculates module to define pixels width
ky = int(i / xSize) #calculates module to define pixels height
xvalue = xa + xd * kx / (xSize - 1.0) # for every row: associates values to the axis x
yvalue = ya + yd * ky / (ySize - 1.0) # for every col: associates values to the axis Y
nReal = xvalue
nImag = yvalue
for n in range(maxIt):
nReal2 = nReal * nReal
nImag2 = nImag * nImag
nImag = 2.0 * nReal * nImag + yvalue
nReal = nReal2 - nImag2 + xvalue
if nReal2 + nImag2 > 4:
# various color palettes can be created here
red = (n % 8) * 32
green = (20 - n % 20) * 20
blue = (n % 15) * 15
image.putpixel((kx, ky), (red, green, blue))
break
if __name__ == "__main__":
tArr = []
for t in range(numThr): # create all threads
tArr.append(MandelbrotThread(t))
start = timer()
for t in range(numThr): # start all threads
tArr[t].start()
for t in range(numThr): # wait until all threads finished
tArr[t].join()
duration = timer() - start
print(f'Total processing time: {duration:.2f} seconds')
image.show()

How to increase FPS in ursina python

I want to create survival games with infinite block terrain(like Minecraft). So i using ursina python game engine, you can see it here
So i using perlin noise to create the terrain with build-in ursina block model. I test for first 25 block and it work pretty good with above 100 FPS, so i start increase to 250 block and more because I want a infinite terrain. But i ran to some problem, when i increase to 100 block or more, my FPS start to decrease below 30 FPS (With i create just one layer).
Here is my code:
#-------------------------------Noise.py(I got on the github)-------------------------
# Copyright (c) 2008, Casey Duncan (casey dot duncan at gmail dot com)
# see LICENSE.txt for details
"""Perlin noise -- pure python implementation"""
__version__ = '$Id: perlin.py 521 2008-12-15 03:03:52Z casey.duncan $'
from math import floor, fmod, sqrt
from random import randint
# 3D Gradient vectors
_GRAD3 = ((1,1,0),(-1,1,0),(1,-1,0),(-1,-1,0),
(1,0,1),(-1,0,1),(1,0,-1),(-1,0,-1),
(0,1,1),(0,-1,1),(0,1,-1),(0,-1,-1),
(1,1,0),(0,-1,1),(-1,1,0),(0,-1,-1),
)
# 4D Gradient vectors
_GRAD4 = ((0,1,1,1), (0,1,1,-1), (0,1,-1,1), (0,1,-1,-1),
(0,-1,1,1), (0,-1,1,-1), (0,-1,-1,1), (0,-1,-1,-1),
(1,0,1,1), (1,0,1,-1), (1,0,-1,1), (1,0,-1,-1),
(-1,0,1,1), (-1,0,1,-1), (-1,0,-1,1), (-1,0,-1,-1),
(1,1,0,1), (1,1,0,-1), (1,-1,0,1), (1,-1,0,-1),
(-1,1,0,1), (-1,1,0,-1), (-1,-1,0,1), (-1,-1,0,-1),
(1,1,1,0), (1,1,-1,0), (1,-1,1,0), (1,-1,-1,0),
(-1,1,1,0), (-1,1,-1,0), (-1,-1,1,0), (-1,-1,-1,0))
# A lookup table to traverse the simplex around a given point in 4D.
# Details can be found where this table is used, in the 4D noise method.
_SIMPLEX = (
(0,1,2,3),(0,1,3,2),(0,0,0,0),(0,2,3,1),(0,0,0,0),(0,0,0,0),(0,0,0,0),(1,2,3,0),
(0,2,1,3),(0,0,0,0),(0,3,1,2),(0,3,2,1),(0,0,0,0),(0,0,0,0),(0,0,0,0),(1,3,2,0),
(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),
(1,2,0,3),(0,0,0,0),(1,3,0,2),(0,0,0,0),(0,0,0,0),(0,0,0,0),(2,3,0,1),(2,3,1,0),
(1,0,2,3),(1,0,3,2),(0,0,0,0),(0,0,0,0),(0,0,0,0),(2,0,3,1),(0,0,0,0),(2,1,3,0),
(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0),
(2,0,1,3),(0,0,0,0),(0,0,0,0),(0,0,0,0),(3,0,1,2),(3,0,2,1),(0,0,0,0),(3,1,2,0),
(2,1,0,3),(0,0,0,0),(0,0,0,0),(0,0,0,0),(3,1,0,2),(0,0,0,0),(3,2,0,1),(3,2,1,0))
# Simplex skew constants
_F2 = 0.5 * (sqrt(3.0) - 1.0)
_G2 = (3.0 - sqrt(3.0)) / 6.0
_F3 = 1.0 / 3.0
_G3 = 1.0 / 6.0
class BaseNoise:
"""Noise abstract base class"""
permutation = (151,160,137,91,90,15,
131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23,
190,6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33,
88,237,149,56,87,174,20,125,136,171,168,68,175,74,165,71,134,139,48,27,166,
77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244,
102,143,54,65,25,63,161,1,216,80,73,209,76,132,187,208,89,18,169,200,196,
135,130,116,188,159,86,164,100,109,198,173,186,3,64,52,217,226,250,124,123,
5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42,
223,183,170,213,119,248,152,2,44,154,163,70,221,153,101,155,167,43,172,9,
129,22,39,253,9,98,108,110,79,113,224,232,178,185,112,104,218,246,97,228,
251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107,
49,192,214,31,181,199,106,157,184,84,204,176,115,121,50,45,127,4,150,254,
138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180)
period = len(permutation)
# Double permutation array so we don't need to wrap
permutation = permutation * 2
randint_function = randint
def __init__(self, period=None, permutation_table=None, randint_function=None):
"""Initialize the noise generator. With no arguments, the default
period and permutation table are used (256). The default permutation
table generates the exact same noise pattern each time.
An integer period can be specified, to generate a random permutation
table with period elements. The period determines the (integer)
interval that the noise repeats, which is useful for creating tiled
textures. period should be a power-of-two, though this is not
enforced. Note that the speed of the noise algorithm is indpendent of
the period size, though larger periods mean a larger table, which
consume more memory.
A permutation table consisting of an iterable sequence of whole
numbers can be specified directly. This should have a power-of-two
length. Typical permutation tables are a sequnce of unique integers in
the range [0,period) in random order, though other arrangements could
prove useful, they will not be "pure" simplex noise. The largest
element in the sequence must be no larger than period-1.
period and permutation_table may not be specified together.
A substitute for the method random.randint(a, b) can be chosen. The
method must take two integer parameters a and b and return an integer N
such that a <= N <= b.
"""
if randint_function is not None: # do this before calling randomize()
if not hasattr(randint_function, '__call__'):
raise TypeError(
'randint_function has to be a function')
self.randint_function = randint_function
if period is None:
period = self.period # enforce actually calling randomize()
if period is not None and permutation_table is not None:
raise ValueError(
'Can specify either period or permutation_table, not both')
if period is not None:
self.randomize(period)
elif permutation_table is not None:
self.permutation = tuple(permutation_table) * 2
self.period = len(permutation_table)
def randomize(self, period=None):
"""Randomize the permutation table used by the noise functions. This
makes them generate a different noise pattern for the same inputs.
"""
if period is not None:
self.period = period
perm = list(range(self.period))
perm_right = self.period - 1
for i in list(perm):
j = self.randint_function(0, perm_right)
perm[i], perm[j] = perm[j], perm[i]
self.permutation = tuple(perm) * 2
class SimplexNoise(BaseNoise):
"""Perlin simplex noise generator
Adapted from Stefan Gustavson's Java implementation described here:
http://staffwww.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf
To summarize:
"In 2001, Ken Perlin presented 'simplex noise', a replacement for his classic
noise algorithm. Classic 'Perlin noise' won him an academy award and has
become an ubiquitous procedural primitive for computer graphics over the
years, but in hindsight it has quite a few limitations. Ken Perlin himself
designed simplex noise specifically to overcome those limitations, and he
spent a lot of good thinking on it. Therefore, it is a better idea than his
original algorithm. A few of the more prominent advantages are:
* Simplex noise has a lower computational complexity and requires fewer
multiplications.
* Simplex noise scales to higher dimensions (4D, 5D and up) with much less
computational cost, the complexity is O(N) for N dimensions instead of
the O(2^N) of classic Noise.
* Simplex noise has no noticeable directional artifacts. Simplex noise has
a well-defined and continuous gradient everywhere that can be computed
quite cheaply.
* Simplex noise is easy to implement in hardware."
"""
def noise2(self, x, y):
"""2D Perlin simplex noise.
Return a floating point value from -1 to 1 for the given x, y coordinate.
The same value is always returned for a given x, y pair unless the
permutation table changes (see randomize above).
"""
# Skew input space to determine which simplex (triangle) we are in
s = (x + y) * _F2
i = floor(x + s)
j = floor(y + s)
t = (i + j) * _G2
x0 = x - (i - t) # "Unskewed" distances from cell origin
y0 = y - (j - t)
if x0 > y0:
i1 = 1; j1 = 0 # Lower triangle, XY order: (0,0)->(1,0)->(1,1)
else:
i1 = 0; j1 = 1 # Upper triangle, YX order: (0,0)->(0,1)->(1,1)
x1 = x0 - i1 + _G2 # Offsets for middle corner in (x,y) unskewed coords
y1 = y0 - j1 + _G2
x2 = x0 + _G2 * 2.0 - 1.0 # Offsets for last corner in (x,y) unskewed coords
y2 = y0 + _G2 * 2.0 - 1.0
# Determine hashed gradient indices of the three simplex corners
perm = self.permutation
ii = int(i) % self.period
jj = int(j) % self.period
gi0 = perm[ii + perm[jj]] % 12
gi1 = perm[ii + i1 + perm[jj + j1]] % 12
gi2 = perm[ii + 1 + perm[jj + 1]] % 12
# Calculate the contribution from the three corners
tt = 0.5 - x0**2 - y0**2
if tt > 0:
g = _GRAD3[gi0]
noise = tt**4 * (g[0] * x0 + g[1] * y0)
else:
noise = 0.0
tt = 0.5 - x1**2 - y1**2
if tt > 0:
g = _GRAD3[gi1]
noise += tt**4 * (g[0] * x1 + g[1] * y1)
tt = 0.5 - x2**2 - y2**2
if tt > 0:
g = _GRAD3[gi2]
noise += tt**4 * (g[0] * x2 + g[1] * y2)
return noise * 70.0 # scale noise to [-1, 1]
def noise3(self, x, y, z):
"""3D Perlin simplex noise.
Return a floating point value from -1 to 1 for the given x, y, z coordinate.
The same value is always returned for a given x, y, z pair unless the
permutation table changes (see randomize above).
"""
# Skew the input space to determine which simplex cell we're in
s = (x + y + z) * _F3
i = floor(x + s)
j = floor(y + s)
k = floor(z + s)
t = (i + j + k) * _G3
x0 = x - (i - t) # "Unskewed" distances from cell origin
y0 = y - (j - t)
z0 = z - (k - t)
# For the 3D case, the simplex shape is a slightly irregular tetrahedron.
# Determine which simplex we are in.
if x0 >= y0:
if y0 >= z0:
i1 = 1; j1 = 0; k1 = 0
i2 = 1; j2 = 1; k2 = 0
elif x0 >= z0:
i1 = 1; j1 = 0; k1 = 0
i2 = 1; j2 = 0; k2 = 1
else:
i1 = 0; j1 = 0; k1 = 1
i2 = 1; j2 = 0; k2 = 1
else: # x0 < y0
if y0 < z0:
i1 = 0; j1 = 0; k1 = 1
i2 = 0; j2 = 1; k2 = 1
elif x0 < z0:
i1 = 0; j1 = 1; k1 = 0
i2 = 0; j2 = 1; k2 = 1
else:
i1 = 0; j1 = 1; k1 = 0
i2 = 1; j2 = 1; k2 = 0
# Offsets for remaining corners
x1 = x0 - i1 + _G3
y1 = y0 - j1 + _G3
z1 = z0 - k1 + _G3
x2 = x0 - i2 + 2.0 * _G3
y2 = y0 - j2 + 2.0 * _G3
z2 = z0 - k2 + 2.0 * _G3
x3 = x0 - 1.0 + 3.0 * _G3
y3 = y0 - 1.0 + 3.0 * _G3
z3 = z0 - 1.0 + 3.0 * _G3
# Calculate the hashed gradient indices of the four simplex corners
perm = self.permutation
ii = int(i) % self.period
jj = int(j) % self.period
kk = int(k) % self.period
gi0 = perm[ii + perm[jj + perm[kk]]] % 12
gi1 = perm[ii + i1 + perm[jj + j1 + perm[kk + k1]]] % 12
gi2 = perm[ii + i2 + perm[jj + j2 + perm[kk + k2]]] % 12
gi3 = perm[ii + 1 + perm[jj + 1 + perm[kk + 1]]] % 12
# Calculate the contribution from the four corners
noise = 0.0
tt = 0.6 - x0**2 - y0**2 - z0**2
if tt > 0:
g = _GRAD3[gi0]
noise = tt**4 * (g[0] * x0 + g[1] * y0 + g[2] * z0)
else:
noise = 0.0
tt = 0.6 - x1**2 - y1**2 - z1**2
if tt > 0:
g = _GRAD3[gi1]
noise += tt**4 * (g[0] * x1 + g[1] * y1 + g[2] * z1)
tt = 0.6 - x2**2 - y2**2 - z2**2
if tt > 0:
g = _GRAD3[gi2]
noise += tt**4 * (g[0] * x2 + g[1] * y2 + g[2] * z2)
tt = 0.6 - x3**2 - y3**2 - z3**2
if tt > 0:
g = _GRAD3[gi3]
noise += tt**4 * (g[0] * x3 + g[1] * y3 + g[2] * z3)
return noise * 32.0
def lerp(t, a, b):
return a + t * (b - a)
def grad3(hash, x, y, z):
g = _GRAD3[hash % 16]
return x*g[0] + y*g[1] + z*g[2]
class TileableNoise(BaseNoise):
"""Tileable implemention of Perlin "improved" noise. This
is based on the reference implementation published here:
http://mrl.nyu.edu/~perlin/noise/
"""
def noise3(self, x, y, z, repeat, base=0.0):
"""Tileable 3D noise.
repeat specifies the integer interval in each dimension
when the noise pattern repeats.
base allows a different texture to be generated for
the same repeat interval.
"""
i = int(fmod(floor(x), repeat))
j = int(fmod(floor(y), repeat))
k = int(fmod(floor(z), repeat))
ii = (i + 1) % repeat
jj = (j + 1) % repeat
kk = (k + 1) % repeat
if base:
i += base; j += base; k += base
ii += base; jj += base; kk += base
x -= floor(x); y -= floor(y); z -= floor(z)
fx = x**3 * (x * (x * 6 - 15) + 10)
fy = y**3 * (y * (y * 6 - 15) + 10)
fz = z**3 * (z * (z * 6 - 15) + 10)
perm = self.permutation
A = perm[i]
AA = perm[A + j]
AB = perm[A + jj]
B = perm[ii]
BA = perm[B + j]
BB = perm[B + jj]
return lerp(fz, lerp(fy, lerp(fx, grad3(perm[AA + k], x, y, z),
grad3(perm[BA + k], x - 1, y, z)),
lerp(fx, grad3(perm[AB + k], x, y - 1, z),
grad3(perm[BB + k], x - 1, y - 1, z))),
lerp(fy, lerp(fx, grad3(perm[AA + kk], x, y, z - 1),
grad3(perm[BA + kk], x - 1, y, z - 1)),
lerp(fx, grad3(perm[AB + kk], x, y - 1, z - 1),
grad3(perm[BB + kk], x - 1, y - 1, z - 1))))
#--------------------------Math.py(For InverseLefp)--------------------------------
def Clamp(t: float, minimum: float, maximum: float):
"""Float result between a min and max values."""
value = t
if t < minimum:
value = minimum
elif t > maximum:
value = maximum
return value
def InverseLefp(a: float, b: float, value: float):
if a != b:
return Clamp((value - a) / (b - a), 0, 1)
return 0
#-----------------------------Game.py(Main code)----------------------
from ursina import *
from ursina.prefabs import *
from ursina.prefabs.first_person_controller import *
from Math import InverseLefp
import Noise
app = Ursina()
#The maximum height of the terrain
maxHeight = 10
#Control the width and height of the map
mapWidth = 10
mapHeight = 10
#A class that create a block
class Voxel(Button):
def __init__(self, position=(0,0,0)):
super().__init__(
parent = scene,
position = position,
model = 'cube',
origin_y = .5,
texture = 'white_cube',
color = color.color(0, 0, random.uniform(.9, 1.0)),
highlight_color = color.lime,
)
#Detect user key input
def input(self, key):
if self.hovered:
if key == 'right mouse down':
#Place block if user right click
voxel = Voxel(position=self.position + mouse.normal)
if key == 'left mouse down':
#Break block if user left click
destroy(self)
if key == 'escape':
#Exit the game if user press the esc key
app.userExit()
#Return perlin noise value between 0 and 1 with x, y position with scale = noiseScale
def GeneratedNoiseMap(y: int, x: int, noiseScale: float):
#Check if the noise scale was invalid or not
if noiseScale <= 0:
noiseScale = 0.001
sampleX = x / noiseScale
sampleY = y / noiseScale
#The Noise.SimplexNoise().noise2 will return the value between -1 and 1
perlinValue = Noise.SimplexNoise().noise2(sampleX, sampleY)
#The InverseLefp will make the value scale to between 0 and 1
perlinValue = InverseLefp(-1, 1, perlinValue)
return perlinValue
for z in range(mapHeight):
for x in range(mapWidth):
#Calculating the height of the block and round it to integer
height = round(GeneratedNoiseMap(z, x, 20) * maxHeight)
#Place the block and make it always below the player
block = Voxel(position=(x, height - maxHeight - 1, z))
#Set the collider of the block
block.collider = 'mesh'
#Character movement
player = FirstPersonController()
#Run the game
app.run()
All file in same folder.
It was working fine but the FPS is very low, so can anyone help?

I'm not able to test this code at the moment but this should serve as a starting point:
level_parent = Entity(model=Mesh(vertices=[], uvs=[]))
for z in range(mapHeight):
for x in range(mapWidth):
height = round(GeneratedNoiseMap(z, x, 20) * maxHeight)
block = Voxel(position=(x, height - maxHeight - 1, z))
level_parent.model.vertices.extend(block.model.vertices)
level_parent.collider = 'mesh' # call this only once after all vertices are set up
For texturing, you might have to add the block.uvs from each block to level_parent.model.uvs as well. Alternatively, call level_parent.model.project_uvs() after setting up the vertices.

On my version of ursina engine (5.0.0) only this code:
`
level_parent = Entity(model=Mesh(vertices=[], uvs=[]))
for z in range(mapHeight):
for x in range(mapWidth):
height = round(GeneratedNoiseMap(z, x, 20) * maxHeight)
block = Voxel(position=(x, height - maxHeight - 1, z))
#level_parent.model.vertices.extend(block.model.vertices)
level_parent.combine().vertices.extend(block.combine().vertices)
level_parent.collider = 'mesh'
`
is working.

I want to have the pendulum blob in my double pendulum

In this code I want to have animation something like this. But I dont want the other pendulums that come into picture later. Just the initial one. Currently this is my output. This is the image after the animation completes. In the animation, I want to have a ball(blob) which plots the red lines and another one which plots the green lines.
import numpy as np
from numpy import cos, sin, arange, pi
import matplotlib.pyplot as plt
import matplotlib.animation as animation
h = 0.0002 #the change in runge kutta
figsize = 6
dpi = 1000
N = 200000 # iterations
L1=1 #length 1
L2=1.5 #lenth 2
m1=50 #mass of bob 1
m2=1 #mass of bob2
g = 9.81#gravity
theta_01 = (np.pi/180)*90
theta_02 = (np.pi/180)*60
w_1 = 0
w_2 = 0
# dw/dt function oft theta 1
def funcdwdt1(theta1,theta2,w1,w2):
cos12 = cos(theta1 - theta2)#for wrirting the main equation in less complex manner
sin12 = sin(theta1 - theta2)
sin1 = sin(theta1)
sin2 = sin(theta2)
denom = cos12**2*m2 - m1 - m2
ans = ( L1*m2*cos12*sin12*w1**2 + L2*m2*sin12*w2**2
- m2*g*cos12*sin2 + (m1 + m2)*g*sin1)/(L1*denom)
return ans
# dw/dt function oft thetas 2
def funcdwdt2(theta2,theta1,w1,w2):
cos12 = cos(theta1 - theta2)
sin12 = sin(theta1 - theta2)
sin1 = sin(theta1)
sin2 = sin(theta2)
denom = cos12**2*m2 - m1 - m2
ans2 = -( L2*m2*cos12*sin12*w2**2 + L1*(m1 + m2)*sin12*w1**2
+ (m1 + m2)*g*sin1*cos12 - (m1 + m2)*g*sin2 )/(L2*denom)
return ans2
# d0/dt function for theta 1
def funcd0dt1(w0):
return w0
# d0/dt function for theta 2
def funcd0dt2(w0):
return w0
X1= []
X2= []
Y1= []
Y2= []
def func(w1,w2, theta1,theta2):
for i in range(N):
k1a = h * funcd0dt1(w1) # gives theta1
k1b = h * funcdwdt1(theta1,theta2,w1,w2) # gives omega1
k1c = h * funcd0dt2(w2) # gives theta2
k1d = h * funcdwdt2(theta2,theta1,w1,w2) # gives omega2
k2a = h * funcd0dt1(w1 + (0.5 * k1b))
k2b = h * funcdwdt1(theta1 + (0.5 * k1a),theta2,w1,w2)
k2c = h * funcd0dt2(w2 + (0.5 * k1d))
k2d = h * funcdwdt2(theta2 + (0.5 * k1c),theta1,w1,w2)
k3a = h * funcd0dt1(w1 + (0.5 * k2b))
k3b = h * funcdwdt1(theta1 + (0.5 * k2a),theta2,w1,w2)
k3c = h * funcd0dt2(w2 + (0.5 * k2d))
k3d = h * funcdwdt2(theta2 + (0.5 * k2c),theta1,w1,w2)
k4a = h * funcd0dt1(w1 + k3b)
k4b = h * funcdwdt1(theta1 + k3a,theta2,w1,w2)
k4c = h * funcd0dt2(w2 + k3d)
k4d = h * funcdwdt2(theta2 + k3c,theta1,w1,w2)
#addidng the vakue aftyer the iterartions
theta1 += 1 / 6 * (k1a + 2 * k2a + 2 * k3a + k4a)
w1 +=1 / 6 * (k1b + 2 * k2b + 2 * k3b + k4b)
theta2 += + 1 / 6 * (k1c + 2 * k2c + 2 * k3c + k4c)
w2 += 1 / 6 * (k1d + 2 * k2d + 2 * k3d + k4d)
x1 = L1 * sin(theta1)
y1 = -L1 * cos(theta1)
x2 = x1 + L2 * sin(theta2)
y2 = y1 - L2 * cos(theta2)
X1.append(x1)
X2.append(x2)
Y1.append(y1)
Y2.append(y2)
return x1,y1,x2,y2
print(func(w_1, w_2, theta_01, theta_02))
fig, ax = plt.subplots()
l1, = ax.plot([], [])
l2, = ax.plot([],[])
ax.set(xlim=(-3, 3), ylim=(-2,2))
def animate(i):
l1.set_data(X1[:i], Y2[:i])
l2.set_data(X2[:i], Y2[:i])
return l1,l2,
ani = animation.FuncAnimation(fig, animate, interval = 5, frames=len(X1))
# plt.show()
ani.save('save.mp4', writer='ffmpeg')

Just add another line
l3, = ax.plot([],[], '-ob', lw=2, ms=8)
and in the animate function set its values to
l3.set_data([0,X1[i],X2[i]], [0,Y1[i],Y2[i]])
Adapt line-width and marker-size as necessary. This should draw filled circles at the pendulum positions and the origin with lines connecting them.
You should use Y1 in the l1 data. With a total pendulum length of 2.5, the vertical limits are too small. It is sufficient to use
h = 0.005 #the change in runge kutta
N = 5000 # iterations
to get an animation with realistic speed. Or combine several RK4 steps for each frame. For minimum error you can use h=1e-3, smaller step sizes only lead to the accumulation of floating point errors dominating the method error.

Smooth coloring\Normalized Iteration with Julia set - Python

I've been working on making a julia set code with python. I didn't write the original code, but I modified it slightly and it achieves what I want it too... except smooth coloring! I've read on how to do it, but I'm still confused.
Code pre-color-smoothing attempt:
def julia():
while True:
try:
julwidth = int(input("What width would you like the image? ex. 256, 512, etc. \nLarger numbers take longer to compute.\n"))
julheight = int(input("What height would you like the image? ex. 256, 512, etc. \nLarger numbers take longer to compute.\n"))
iter = int(input("Max iterations? If you don't know, 255 is a good number \n"))
break
except ValueError:
print("Enter an integer, please")
print("Running Julia fractal... please wait...")
imgx = julwidth
imgy = julheight
image = Image.new("RGB", (imgx, imgy))
# drawing area
xa = -2.0
xb = 2.0
ya = -1.5
yb = 1.5
maxIt = iter # max iterations allowed
Time = time.clock()
#print(time)
# find a good Julia set point using the Mandelbrot set
while True:
cx = random.random() * (xb - xa) + xa
cy = random.random() * (yb - ya) + ya
c = cx + cy * 1j
z = c
for i in range(maxIt):
if abs(z) > 2.0:
break
z = z * z + c
if i > 10 and i < 100:
break
# draw the Julia set
for y in range(imgy):
zy = y * (yb - ya) / (imgy - 1) + ya
for x in range(imgx):
zx = x * (xb - xa) / (imgx - 1) + xa
z = zx + zy * 1j
for i in range(maxIt):
if abs(z) > 2.0:
break
z = z * z + c
red = i % 8 *32
green = i % 16 * 16
blue = i % 32 * 8
#print(red, green, blue)
image.putpixel((x, y), (red, green, blue))
print("Done computing. \n")
With smoothing attempt:
def julia():
while True:
try:
julwidth = int(input("What width would you like the image? ex. 256, 512, etc. \nLarger numbers take longer to compute.\n"))
julheight = int(input("What height would you like the image? ex. 256, 512, etc. \nLarger numbers take longer to compute.\n"))
iter = int(input("Max iterations? If you don't know, 255 is a good number \n"))
break
except ValueError:
print("Enter an integer, please")
print("Running Julia fractal... please wait...")
imgx = julwidth
imgy = julheight
image = Image.new("RGB", (imgx, imgy))
# drawing area
xa = -2.0
xb = 2.0
ya = -1.5
yb = 1.5
maxIt = iter # max iterations allowed
Time = time.clock()
#print(time)
# find a good Julia set point using the Mandelbrot set
while True:
cx = random.random() * (xb - xa) + xa
cy = random.random() * (yb - ya) + ya
c = cx + cy * 1j
z = c
for i in range(maxIt):
if abs(z) > 2.0:
break
z = z * z + c
smooth = (abs(z))
if i > 10 and i < 100:
break
# draw the Julia set
for y in range(imgy):
zy = y * (yb - ya) / (imgy - 1) + ya
for x in range(imgx):
zx = x * (xb - xa) / (imgx - 1) + xa
z = zx + zy * 1j
for i in range(maxIt):
if abs(z) > 2.0:
break
z = z * z + c
smooth += (abs(z))
#print(smooth)
red = ceil(smooth % 8 * 32)
green = ceil(red*red)
blue = ceil(green*red)
#print(red, green, blue)
image.putpixel((x, y), (red, green, blue))
print("Done computing. \n")
If you can't tell, I added a variable called smooth that I'm using to try and smooth the colors with by using the complex number/variable, z
Running my code with the smoothing attempt yields something..... not what I want. It's a weird blue image and it's not good. How can I add smoothing to this? I probably don't have a good understanding of smoothing, so an explanation might help.

Compute fractional escape counts and set the color according to that:
ec = N + 1 - log (log |Z(N)|) / log 2 where N is the escape count when the value of 'z' escaped, Z(N) is the value of zwhen it escaped and |z| is the modulus of z which is just sqrt (x*x + y*y).

Simplex Noise Function: Unexpected Results

I'm trying to adapt this noise module to a project I'm working on but I'm not getting the results I was expecting:
https://pypi.python.org/pypi/noise/
Instead of a varied height-map, each row tends to have the exact same value. If I create something 250x250 it will generate the same value 250 times and then generate a new value 250 times until it ends.
Here is the function I'm currently using. I understand this function fairly well but I'm just not sure how to get more "interesting" results. Thank you for your help.
class SimplexNoise(BaseNoise):
def noise2(self, x, y):
"""2D Perlin simplex noise.
Return a floating point value from -1 to 1 for the given x, y coordinate.
The same value is always returned for a given x, y pair unless the
permutation table changes (see randomize above).
"""
# Skew input space to determine which simplex (triangle) we are in
s = (x + y) * _F2
i = floor(x + s)
j = floor(y + s)
t = (i + j) * _G2
x0 = x - (i - t) # "Unskewed" distances from cell origin
y0 = y - (j - t)
if x0 > y0:
i1 = 1; j1 = 0 # Lower triangle, XY order: (0,0)->(1,0)->(1,1)
else:
i1 = 0; j1 = 1 # Upper triangle, YX order: (0,0)->(0,1)->(1,1)
x1 = x0 - i1 + _G2 # Offsets for middle corner in (x,y) unskewed coords
y1 = y0 - j1 + _G2
x2 = x0 + _G2 * 2.0 - 1.0 # Offsets for last corner in (x,y) unskewed coords
y2 = y0 + _G2 * 2.0 - 1.0
# Determine hashed gradient indices of the three simplex corners
perm = BaseNoise.permutation
ii = int(i) % BaseNoise.period
jj = int(j) % BaseNoise.period
gi0 = perm[ii + perm[jj]] % 12
gi1 = perm[ii + i1 + perm[jj + j1]] % 12
gi2 = perm[ii + 1 + perm[jj + 1]] % 12
# Calculate the contribution from the three corners
tt = 0.5 - x0**2 - y0**2
if tt > 0:
g = _GRAD3[gi0]
noise = tt**4 * (g[0] * x0 + g[1] * y0)
else:
noise = 0.0
tt = 0.5 - x1**2 - y1**2
if tt > 0:
g = _GRAD3[gi1]
noise += tt**4 * (g[0] * x1 + g[1] * y1)
tt = 0.5 - x2**2 - y2**2
if tt > 0:
g = _GRAD3[gi2]
noise += tt**4 * (g[0] * x2 + g[1] * y2)
return noise * 70.0 # scale noise to [-1, 1]
win = pygcurse.PygcurseWindow(85, 70, 'Generate')
octaves = 2
ysize = 150
xsize = 150
freq = 32.0 * octaves
for y in range(ysize):
for x in range(xsize):
tile = SimplexNoise.noise2(x / freq, y / freq, octaves)
win.write(str(tile) + "\n")