Develop Reference

Orbit spirals using 4th order Yoshida integration

I am attempting to use the 4th order Yoshida integration technique to model the orbit of satellites in circular orbits around the Earth.
However, the orbits I achieve spiral away quite quickly. The code for a Moon like satellite is below. Interestingly, the particles behaved when I use Euler method, however, I wanted to try a more accurate method. The issue could then be within how I have implemented the algorithm itself.
I have tried using the gravitational parameter rather then computing G*M, but this did not help. I also reduced the time-step, messed around with units, printed and checked values for various integration steps etc., but could not find anything.
Is this the correct use of this algorithm?
G = 6.674e-20 # km^3 kg^-1 s^-2
day = 60.0 * 60.0 * 24.0 # length of day
dt = day / 10.0
M = 5.972e24 # kg
N = 1
delta = np.random.random(1) * 2.0 * np.pi / N
angles = np.linspace(0.0, 2.0 * np.pi, N) + delta
rad = np.random.uniform(low = 384e3, high = 384e3, size = (N))
x, y = rad * np.cos(angles), ringrad * np.sin(angles)
vx, vy = np.sqrt(G*M / rad) * -np.sin(angles), np.sqrt(G*M / rad) * np.cos(angles)
def update(frame):
global x, y, vx, vy, dt, day
positions.set_data(x, y)
# coefficients
q = 2**(1/3)
w1 = 1 / (2 - q)
w0 = -q * w1
d1 = w1
d3 = w1
d2 = w0
c1 = w1 / 2
c2 = (w0 + w1) / 2
c3 = c2
c4 = c1
# Step 1
x1 = x + c1*vx*dt
y1 = y + c1*vy*dt
dist1 = np.hypot(x1, y1)
acc1 = -(G*M) / (dist1**2.0)
dx1 = x1 - x
dy1 = y1 - y
accx1 = (acc1*dx1)/(x1)
accy1 = (acc1*dy1)/(y1)
vx1 = vx + d1*accx1*dt
vy1 = vy + d1*accy1*dt
# Step 2
x2 = x1 + c2*vx1*dt
y2 = y1 + c2*vy1*dt
dist2 = np.hypot(x2, y2)
acc2 = -(G*M) / (dist2**2.0)
dx2 = x2 - x1
dy2 = y2 - y1
accx2 = (acc2*dx2)/(x2)
accy2 = (acc2*dy2)/(y2)
vx2 = vx1 + d2*accx2*dt
vy2 = vy1 + d2*accy2*dt
# Step 3
x3 = x2 + c3*vx2*dt
y3 = y2 + c3*vy2*dt
dist3 = np.hypot(x3, y3)
acc3 = -(G*M) / (dist3**2.0)
dx3 = x3 - x2
dy3 = y3 - y2
accx3 = (acc3*dx3)/(x3)
accy3 = (acc3*dy3)/(y3)
vx3 = vx2 + d3*accx3*dt
vy3 = vy2 + d3*accy3*dt
# Full step
x = x3 + c4*vx3*dt
y = y3 + c4*vy3*dt
vx = vx3
vy = vy3
return positions

Compute Fourier Series for a discrete set of points

I'm trying to compute the continuous function hidden behind the points, but it shows a graph that looks like it actually coutns the points in-between as zeros.
Here's the plot that shows up (100 vectors, red dots - data set, blue plot - my Fourier series):
Here's the python code:
import matplotlib.pyplot as plt
import numpy as np
import math
step = (np.pi * 2) / 5
start = -np.pi
xDiscrete = [start, start + step, start + 2 * step, start + 3 * step, start + 4 * step, np.pi]
yDiscrete = [2.88, 2.98, 3.24, 3.42, 3.57, 3.79]
ak = []
bk = []
a0 = 0
precisionSize = 0.001
n = 100
avgError = 0
def getAN(k):
sum = 0
for ind in range(1, len(yDiscrete)):
sum += yDiscrete[ind] * math.cos(k * xDiscrete[ind])
an = (2.0 / n) * sum
print('a' + str(k) + ' = ' + str(an))
return an
def getBN(k):
sum = 0
for ind in range(1, len(yDiscrete)):
sum += yDiscrete[ind] * math.sin(k * xDiscrete[ind])
bn = (2.0 / n) * sum
print('b' + str(k) + ' = ' + str(bn))
return bn
def getA0():
sum = 0
for ind in range(1, len(yDiscrete)):
sum += yDiscrete[ind]
a0 = (2.0 / n) * sum
print('a0 = ' + str(a0))
return a0
def getFourierOneSum(x, i):
return ak[i - 1] * math.cos(i * x) + bk[i - 1] * math.sin(i * x)
def getFourierAtPoint(x):
sum = a0 / 2
for i in range(1, n + 1):
sum += getFourierOneSum(x, i)
return sum
for i in range(1, n + 1):
ak.append(getAN(i))
bk.append(getBN(i))
a0 = getA0()
x2 = np.arange(-np.pi, np.pi, precisionSize)
y2 = []
for coor in x2:
y2.append(getFourierAtPoint(coor))
plt.plot(xDiscrete, yDiscrete, 'ro', alpha=0.6)
plt.plot(x2, y2)
plt.grid()
plt.title('Approximation')
plt.show()
I've checked where is the problem, and I'm pretty sure it's with the coefficients (functions getAN, getBN, getA0), but I'm not sure how to fix it.

Fading a Line Exponentially

I'd like to fade the values of a line on the Y axis using a gradient that I can control.
This is a simplified version of what I'm trying right now.
y_values = [1,1,1,1,1,1]
for i, y in enumerate(y_values): #i is the index, y is the individual y value
perc = i/(len(y_values)-1) #progress the fade by the percentage of the index divided by the total values
amt = y * perc #y is the target amount to decrease multiplied by index percentage
y -= amt
print(i,y)
This code produces this:
1.0
0.8
0.6
0.4
0.2
0.0
It's creating a linear fade, but how do I increase the fade with an exponential fade like this?
Thank you!

To make exponential fading, you have to provide two coefficients to provide initial factor 1.0 at the start value x1 and desired final factor k at the end of interval x2
y = y * f(x)
f(x) = A * exp(-B * x)
So
f(x1) = 1 = A * exp(B * x1)
f(x2) = k = A * exp(B * x2)
divide the second by the first
k = exp(B * (x2 - x1))
ln(k) = B * (x2 - x1)
so
B = ln(k) / (x2 - x1)
A = exp(B * x1)
Example for x1 = 0, x2 = 60, k = 0.01
B = -4.6/60= -0.076
A = 1
f(x) = exp(-0.076*x)
f(30) = exp(-0.076*20) = 0.1
Python example:
import math
def calcfading(x1, x2, ratio):
B = math.log(ratio) / (x2 - x1)
A = math.exp(B * x1)
return A, B
def coef(x, fade):
return fade[0]*math.exp(x*fade[1])
cosine = [[x, math.cos(x)] for x in range(0, 11)]
print(cosine)
print()
fade = calcfading(0, 10, 0.01)
expcosine = [[s[0], coef(s[0], fade)*s[1]] for s in cosine]
print(expcosine)

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.

multithreaded mandelbrot set

Is it possible to change the formula of the mandelbrot set (which is f(z) = z^2 + c by default) to a different one ( f(z) = z^2 + c * e^(-z) is what i need) when using the escape time algorithm and if possible how?
I'm currently using this code by FB36
# Multi-threaded Mandelbrot Fractal (Do not run using IDLE!)
# FB - 201104306
import threading
from PIL import Image
w = 512 # image width
h = 512 # image height
image = Image.new("RGB", (w, h))
wh = w * h
maxIt = 256 # max number of iterations allowed
# drawing region (xa < xb & ya < yb)
xa = -2.0
xb = 1.0
ya = -1.5
yb = 1.5
xd = xb - xa
yd = yb - ya
numThr = 5 # number of threads to run
# lock = threading.Lock()
class ManFrThread(threading.Thread):
def __init__ (self, k):
self.k = k
threading.Thread.__init__(self)
def run(self):
# each thread only calculates its own share of pixels
for i in range(k, wh, numThr):
kx = i % w
ky = int(i / w)
a = xa + xd * kx / (w - 1.0)
b = ya + yd * ky / (h - 1.0)
x = a
y = b
for kc in range(maxIt):
x0 = x * x - y * y + a
y = 2.0 * x * y + b
x = x0
if x * x + y * y > 4:
# various color palettes can be created here
red = (kc % 8) * 32
green = (16 - kc % 16) * 16
blue = (kc % 16) * 16
# lock.acquire()
global image
image.putpixel((kx, ky), (red, green, blue))
# lock.release()
break
if __name__ == "__main__":
tArr = []
for k in range(numThr): # create all threads
tArr.append(ManFrThread(k))
for k in range(numThr): # start all threads
tArr[k].start()
for k in range(numThr): # wait until all threads finished
tArr[k].join()
image.save("MandelbrotFractal.png", "PNG")

From the code I infer that z = x + y * i and c = a + b * i. That corresponds f(z) - z ^2 + c. You want f(z) = z ^2 + c * e^(-z).
Recall that e^(-z) = e^-(x + yi) = e^(-x) * e^i(-y) = e^(-x)(cos(y) - i*sin(y)) = e^(-x)cos(y) - i (e^(-x)sin(y)). Thus you should update your lines to be the following:
x0 = x * x - y * y + a * exp(-x) * cos(y) + b * exp(-x) * sin(y);
y = 2.0 * x * y + a * exp(-x) * sin(y) - b * exp(-x) * cos(y)
x = x0
You might need to adjust maxIt if you don't get the level of feature differentiation you're after (it might take more or fewer iterations to escape now, on average) but this should be the mathematical expression you're after.
As pointed out in the comments, you might need to adjust the criterion itself and not just the maximum iterations in order to get the desired level of differentiation: changing the max doesn't help for ones that never escape.
You can try deriving a good escape condition or just try out some things and see what you get.