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opensimplex.py
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opensimplex.py
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# -*- coding: utf-8 -*-
"""
Created on Mon Feb 27 13:26:45 2017
@author: ktritz
Modified by Christopher Kohl on Wed Jul 28 16:37:58 2021
Originally ported by A. Svensson in 2015
"""
# Based on: https://gist.github.com/KdotJPG/b1270127455a94ac5d19
# A. Svensson's python port: https://github.com/lmas/opensimplex
# ktritz's numba optimized fork: https://github.com/ktritz/opensimplex
import numpy as np
from numba import njit
@njit(cache=True)
def fastfloor(x):
xi = int(x)
return xi - 1 if x < xi else xi
STRETCH_CONSTANT_2D = -0.211324865405187 # (1/Math.sqrt(2+1)-1)/2
SQUISH_CONSTANT_2D = 0.366025403784439 # (Math.sqrt(2+1)-1)/2
STRETCH_CONSTANT_3D = -1.0 / 6 # (1/Math.sqrt(3+1)-1)/3
SQUISH_CONSTANT_3D = 1.0 / 3 # (Math.sqrt(3+1)-1)/3
STRETCH_CONSTANT_4D = -0.138196601125011 # (1/Math.sqrt(4+1)-1)/4
SQUISH_CONSTANT_4D = 0.309016994374947 # (Math.sqrt(4+1)-1)/4
NORM_CONSTANT_2D = 47
NORM_CONSTANT_3D = 103
NORM_CONSTANT_4D = 30
DEFAULT_SEED = 0
# Gradients for 2D. They approximate the directions to the
# vertices of an octagon from the center.
GRADIENTS_2D = (
5, 2, 2, 5,
-5, 2, -2, 5,
5, -2, 2, -5,
-5, -2, -2, -5,
)
# Gradients for 3D. They approximate the directions to the
# vertices of a rhombicuboctahedron from the center, skewed so
# that the triangular and square facets can be inscribed inside
# circles of the same radius.
GRADIENTS_3D = (
-11, 4, 4, -4, 11, 4, -4, 4, 11,
11, 4, 4, 4, 11, 4, 4, 4, 11,
-11, -4, 4, -4, -11, 4, -4, -4, 11,
11, -4, 4, 4, -11, 4, 4, -4, 11,
-11, 4, -4, -4, 11, -4, -4, 4, -11,
11, 4, -4, 4, 11, -4, 4, 4, -11,
-11, -4, -4, -4, -11, -4, -4, -4, -11,
11, -4, -4, 4, -11, -4, 4, -4, -11,
)
# Gradients for 4D. They approximate the directions to the
# vertices of a disprismatotesseractihexadecachoron from the center,
# skewed so that the tetrahedral and cubic facets can be inscribed inside
# spheres of the same radius.
GRADIENTS_4D = (
3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3,
-3, 1, 1, 1, -1, 3, 1, 1, -1, 1, 3, 1, -1, 1, 1, 3,
3, -1, 1, 1, 1, -3, 1, 1, 1, -1, 3, 1, 1, -1, 1, 3,
-3, -1, 1, 1, -1, -3, 1, 1, -1, -1, 3, 1, -1, -1, 1, 3,
3, 1, -1, 1, 1, 3, -1, 1, 1, 1, -3, 1, 1, 1, -1, 3,
-3, 1, -1, 1, -1, 3, -1, 1, -1, 1, -3, 1, -1, 1, -1, 3,
3, -1, -1, 1, 1, -3, -1, 1, 1, -1, -3, 1, 1, -1, -1, 3,
-3, -1, -1, 1, -1, -3, -1, 1, -1, -1, -3, 1, -1, -1, -1, 3,
3, 1, 1, -1, 1, 3, 1, -1, 1, 1, 3, -1, 1, 1, 1, -3,
-3, 1, 1, -1, -1, 3, 1, -1, -1, 1, 3, -1, -1, 1, 1, -3,
3, -1, 1, -1, 1, -3, 1, -1, 1, -1, 3, -1, 1, -1, 1, -3,
-3, -1, 1, -1, -1, -3, 1, -1, -1, -1, 3, -1, -1, -1, 1, -3,
3, 1, -1, -1, 1, 3, -1, -1, 1, 1, -3, -1, 1, 1, -1, -3,
-3, 1, -1, -1, -1, 3, -1, -1, -1, 1, -3, -1, -1, 1, -1, -3,
3, -1, -1, -1, 1, -3, -1, -1, 1, -1, -3, -1, 1, -1, -1, -3,
-3, -1, -1, -1, -1, -3, -1, -1, -1, -1, -3, -1, -1, -1, -1, -3,
)
_PERM = np.zeros(256, dtype=np.int32)
_PERM_GRAD_INDEX_3D = np.zeros(256, dtype=np.int32)
@njit(cache=True)
def init(seed=DEFAULT_SEED):
perm = np.zeros(256, dtype=np.int32)
perm_grad_index_3D = np.zeros(256, dtype=np.int32)
source = np.arange(256)
seed = over(seed * 6364136223846793005 + 1442695040888963407)
seed = over(seed * 6364136223846793005 + 1442695040888963407)
seed = over(seed * 6364136223846793005 + 1442695040888963407)
for i in range(255, -1, -1):
seed = over(seed * 6364136223846793005 + 1442695040888963407)
r = int((seed + 31) % (i + 1))
if r < 0:
r += i + 1
perm[i] = source[r]
perm_grad_index_3D[i] = int((perm[i] % (len(GRADIENTS_3D) / 3)) * 3)
source[r] = source[i]
return (perm, perm_grad_index_3D)
@njit(["int32(int32)"], cache=True)
def over(x):
return x
@njit(cache=True)
def extrapolate2d(xsb, ysb, dx, dy, perm):
index = perm[(perm[xsb & 0xFF] + ysb) & 0xFF] & 0x0E
g1 = GRADIENTS_2D[index]
g2 = GRADIENTS_2D[index + 1]
return g1 * dx + g2 * dy
@njit(cache=True)
def extrapolate3d(xsb, ysb, zsb, dx, dy, dz, perm, perm_grad_index_3d):
index = perm_grad_index_3d[(perm[(perm[xsb & 0xFF] + ysb) & 0xFF] + zsb) & 0xFF]
g1 = GRADIENTS_3D[index]
g2 = GRADIENTS_3D[index + 1]
g3 = GRADIENTS_3D[index + 2]
return g1 * dx + g2 * dy + g3 * dz
@njit(cache=True)
def extrapolate4d(xsb, ysb, zsb, wsb, dx, dy, dz, dw, perm):
index = perm[(perm[(perm[(perm[xsb & 0xFF] + ysb) & 0xFF] + zsb) & 0xFF]+ wsb) & 0xFF] & 0xFC
g1 = GRADIENTS_4D[index]
g2 = GRADIENTS_4D[index + 1]
g3 = GRADIENTS_4D[index + 2]
g4 = GRADIENTS_4D[index + 3]
return g1 * dx + g2 * dy + g3 * dz + g4 * dw
@njit(cache=True)
def noisearr2d(x, y, perm):
num = x.size
noise = np.zeros(num, dtype=np.double)
for i in range(num):
noise[i] = noise2d(x[i], y[i], perm)
return noise
@njit(cache=True)
def noise2d(x, y, perm):
"""
Generate 2D OpenSimplex noise from X,Y coordinates.
"""
# Place input coordinates onto grid.
stretch_offset = (x + y) * STRETCH_CONSTANT_2D
xs = x + stretch_offset
ys = y + stretch_offset
# Floor to get grid coordinates of rhombus (stretched square) super-cell origin.
xsb = fastfloor(xs)
ysb = fastfloor(ys)
# Skew out to get actual coordinates of rhombus origin. We'll need these later.
squish_offset = (xsb + ysb) * SQUISH_CONSTANT_2D
xb = xsb + squish_offset
yb = ysb + squish_offset
# Compute grid coordinates relative to rhombus origin.
xins = xs - xsb
yins = ys - ysb
# Sum those together to get a value that determines which region we're in.
in_sum = xins + yins
# Positions relative to origin point.
dx0 = x - xb
dy0 = y - yb
value = 0
# Contribution (1,0)
dx1 = dx0 - 1 - SQUISH_CONSTANT_2D
dy1 = dy0 - 0 - SQUISH_CONSTANT_2D
attn1 = 2 - dx1 * dx1 - dy1 * dy1
if attn1 > 0:
attn1 *= attn1
value += attn1 * attn1 * extrapolate2d(xsb + 1, ysb + 0, dx1, dy1, perm)
# Contribution (0,1)
dx2 = dx0 - 0 - SQUISH_CONSTANT_2D
dy2 = dy0 - 1 - SQUISH_CONSTANT_2D
attn2 = 2 - dx2 * dx2 - dy2 * dy2
if attn2 > 0:
attn2 *= attn2
value += attn2 * attn2 * extrapolate2d(xsb + 0, ysb + 1, dx2, dy2, perm)
if in_sum <= 1: # We're inside the triangle (2-Simplex) at (0,0)
zins = 1 - in_sum
if zins > xins or zins > yins: # (0,0) is one of the closest two triangular vertices
if xins > yins:
xsv_ext = xsb + 1
ysv_ext = ysb - 1
dx_ext = dx0 - 1
dy_ext = dy0 + 1
else:
xsv_ext = xsb - 1
ysv_ext = ysb + 1
dx_ext = dx0 + 1
dy_ext = dy0 - 1
else: # (1,0) and (0,1) are the closest two vertices.
xsv_ext = xsb + 1
ysv_ext = ysb + 1
dx_ext = dx0 - 1 - 2 * SQUISH_CONSTANT_2D
dy_ext = dy0 - 1 - 2 * SQUISH_CONSTANT_2D
else: # We're inside the triangle (2-Simplex) at (1,1)
zins = 2 - in_sum
if zins < xins or zins < yins: # (0,0) is one of the closest two triangular vertices
if xins > yins:
xsv_ext = xsb + 2
ysv_ext = ysb + 0
dx_ext = dx0 - 2 - 2 * SQUISH_CONSTANT_2D
dy_ext = dy0 + 0 - 2 * SQUISH_CONSTANT_2D
else:
xsv_ext = xsb + 0
ysv_ext = ysb + 2
dx_ext = dx0 + 0 - 2 * SQUISH_CONSTANT_2D
dy_ext = dy0 - 2 - 2 * SQUISH_CONSTANT_2D
else: # (1,0) and (0,1) are the closest two vertices.
dx_ext = dx0
dy_ext = dy0
xsv_ext = xsb
ysv_ext = ysb
xsb += 1
ysb += 1
dx0 = dx0 - 1 - 2 * SQUISH_CONSTANT_2D
dy0 = dy0 - 1 - 2 * SQUISH_CONSTANT_2D
# Contribution (0,0) or (1,1)
attn0 = 2 - dx0 * dx0 - dy0 * dy0
if attn0 > 0:
attn0 *= attn0
value += attn0 * attn0 * extrapolate2d(xsb, ysb, dx0, dy0, perm)
# Extra Vertex
attn_ext = 2 - dx_ext * dx_ext - dy_ext * dy_ext
if attn_ext > 0:
attn_ext *= attn_ext
value += attn_ext * attn_ext * extrapolate2d(xsv_ext, ysv_ext, dx_ext, dy_ext, perm)
return value / NORM_CONSTANT_2D
@njit(cache=True)
def noisearr3d(x, y, z, perm, perm_grad_index_3D):
num = x.size
noise = np.zeros(num, dtype=np.double)
for i in range(num):
noise[i] = noise3d(x[i], y[i], z[i], perm, perm_grad_index_3D)
return noise
@njit(cache=True)
def noise3d(x, y, z, perm, perm_grad_index_3D):
"""
Generate 3D OpenSimplex noise from X,Y,Z coordinates.
"""
# Place input coordinates on simplectic honeycomb.
stretch_offset = (x + y + z) * STRETCH_CONSTANT_3D
xs = x + stretch_offset
ys = y + stretch_offset
zs = z + stretch_offset
# Floor to get simplectic honeycomb coordinates of rhombohedron (stretched cube) super-cell origin.
xsb = fastfloor(xs)
ysb = fastfloor(ys)
zsb = fastfloor(zs)
# Skew out to get actual coordinates of rhombohedron origin. We'll need these later.
squish_offset = (xsb + ysb + zsb) * SQUISH_CONSTANT_3D
xb = xsb + squish_offset
yb = ysb + squish_offset
zb = zsb + squish_offset
# Compute simplectic honeycomb coordinates relative to rhombohedral origin.
xins = xs - xsb
yins = ys - ysb
zins = zs - zsb
# Sum those together to get a value that determines which region we're in.
in_sum = xins + yins + zins
# Positions relative to origin point.
dx0 = x - xb
dy0 = y - yb
dz0 = z - zb
value = 0
if in_sum <= 1: # We're inside the tetrahedron (3-Simplex) at (0,0,0)
# Determine which two of (0,0,1), (0,1,0), (1,0,0) are closest.
a_point = 0x01
a_score = xins
b_point = 0x02
b_score = yins
if a_score >= b_score and zins > b_score:
b_score = zins
b_point = 0x04
elif a_score < b_score and zins > a_score:
a_score = zins
a_point = 0x04
# Now we determine the two lattice points not part of the tetrahedron that may contribute.
# This depends on the closest two tetrahedral vertices, including (0,0,0)
wins = 1 - in_sum
if wins > a_score or wins > b_score: # (0,0,0) is one of the closest two tetrahedral vertices.
c = b_point if (b_score > a_score) else a_point # Our other closest vertex is the closest out of a and b.
if (c & 0x01) == 0:
xsv_ext0 = xsb - 1
xsv_ext1 = xsb
dx_ext0 = dx0 + 1
dx_ext1 = dx0
else:
xsv_ext0 = xsv_ext1 = xsb + 1
dx_ext0 = dx_ext1 = dx0 - 1
if (c & 0x02) == 0:
ysv_ext0 = ysv_ext1 = ysb
dy_ext0 = dy_ext1 = dy0
if (c & 0x01) == 0:
ysv_ext1 -= 1
dy_ext1 += 1
else:
ysv_ext0 -= 1
dy_ext0 += 1
else:
ysv_ext0 = ysv_ext1 = ysb + 1
dy_ext0 = dy_ext1 = dy0 - 1
if (c & 0x04) == 0:
zsv_ext0 = zsb
zsv_ext1 = zsb - 1
dz_ext0 = dz0
dz_ext1 = dz0 + 1
else:
zsv_ext0 = zsv_ext1 = zsb + 1
dz_ext0 = dz_ext1 = dz0 - 1
else: # (0,0,0) is not one of the closest two tetrahedral vertices.
c = (a_point | b_point) # Our two extra vertices are determined by the closest two.
if (c & 0x01) == 0:
xsv_ext0 = xsb
xsv_ext1 = xsb - 1
dx_ext0 = dx0 - 2 * SQUISH_CONSTANT_3D
dx_ext1 = dx0 + 1 - SQUISH_CONSTANT_3D
else:
xsv_ext0 = xsv_ext1 = xsb + 1
dx_ext0 = dx0 - 1 - 2 * SQUISH_CONSTANT_3D
dx_ext1 = dx0 - 1 - SQUISH_CONSTANT_3D
if (c & 0x02) == 0:
ysv_ext0 = ysb
ysv_ext1 = ysb - 1
dy_ext0 = dy0 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 + 1 - SQUISH_CONSTANT_3D
else:
ysv_ext0 = ysv_ext1 = ysb + 1
dy_ext0 = dy0 - 1 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 1 - SQUISH_CONSTANT_3D
if (c & 0x04) == 0:
zsv_ext0 = zsb
zsv_ext1 = zsb - 1
dz_ext0 = dz0 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 + 1 - SQUISH_CONSTANT_3D
else:
zsv_ext0 = zsv_ext1 = zsb + 1
dz_ext0 = dz0 - 1 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 1 - SQUISH_CONSTANT_3D
# Contribution (0,0,0)
attn0 = 2 - dx0 * dx0 - dy0 * dy0 - dz0 * dz0
if attn0 > 0:
attn0 *= attn0
value += attn0 * attn0 * extrapolate3d(xsb + 0, ysb + 0, zsb + 0, dx0, dy0, dz0, perm, perm_grad_index_3D)
# Contribution (1,0,0)
dx1 = dx0 - 1 - SQUISH_CONSTANT_3D
dy1 = dy0 - 0 - SQUISH_CONSTANT_3D
dz1 = dz0 - 0 - SQUISH_CONSTANT_3D
attn1 = 2 - dx1 * dx1 - dy1 * dy1 - dz1 * dz1
if attn1 > 0:
attn1 *= attn1
value += attn1 * attn1 * extrapolate3d(xsb + 1, ysb + 0, zsb + 0, dx1, dy1, dz1, perm, perm_grad_index_3D)
# Contribution (0,1,0)
dx2 = dx0 - 0 - SQUISH_CONSTANT_3D
dy2 = dy0 - 1 - SQUISH_CONSTANT_3D
dz2 = dz1
attn2 = 2 - dx2 * dx2 - dy2 * dy2 - dz2 * dz2
if attn2 > 0:
attn2 *= attn2
value += attn2 * attn2 * extrapolate3d(xsb + 0, ysb + 1, zsb + 0, dx2, dy2, dz2, perm, perm_grad_index_3D)
# Contribution (0,0,1)
dx3 = dx2
dy3 = dy1
dz3 = dz0 - 1 - SQUISH_CONSTANT_3D
attn3 = 2 - dx3 * dx3 - dy3 * dy3 - dz3 * dz3
if attn3 > 0:
attn3 *= attn3
value += attn3 * attn3 * extrapolate3d(xsb + 0, ysb + 0, zsb + 1, dx3, dy3, dz3, perm, perm_grad_index_3D)
elif in_sum >= 2: # We're inside the tetrahedron (3-Simplex) at (1,1,1)
# Determine which two tetrahedral vertices are the closest, out of (1,1,0), (1,0,1), (0,1,1) but not (1,1,1).
a_point = 0x06
a_score = xins
b_point = 0x05
b_score = yins
if a_score <= b_score and zins < b_score:
b_score = zins
b_point = 0x03
elif a_score > b_score and zins < a_score:
a_score = zins
a_point = 0x03
# Now we determine the two lattice points not part of the tetrahedron that may contribute.
# This depends on the closest two tetrahedral vertices, including (1,1,1)
wins = 3 - in_sum
if wins < a_score or wins < b_score: # (1,1,1) is one of the closest two tetrahedral vertices.
c = b_point if (b_score < a_score) else a_point # Our other closest vertex is the closest out of a and b.
if (c & 0x01) != 0:
xsv_ext0 = xsb + 2
xsv_ext1 = xsb + 1
dx_ext0 = dx0 - 2 - 3 * SQUISH_CONSTANT_3D
dx_ext1 = dx0 - 1 - 3 * SQUISH_CONSTANT_3D
else:
xsv_ext0 = xsv_ext1 = xsb
dx_ext0 = dx_ext1 = dx0 - 3 * SQUISH_CONSTANT_3D
if (c & 0x02) != 0:
ysv_ext0 = ysv_ext1 = ysb + 1
dy_ext0 = dy_ext1 = dy0 - 1 - 3 * SQUISH_CONSTANT_3D
if (c & 0x01) != 0:
ysv_ext1 += 1
dy_ext1 -= 1
else:
ysv_ext0 += 1
dy_ext0 -= 1
else:
ysv_ext0 = ysv_ext1 = ysb
dy_ext0 = dy_ext1 = dy0 - 3 * SQUISH_CONSTANT_3D
if (c & 0x04) != 0:
zsv_ext0 = zsb + 1
zsv_ext1 = zsb + 2
dz_ext0 = dz0 - 1 - 3 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 - 3 * SQUISH_CONSTANT_3D
else:
zsv_ext0 = zsv_ext1 = zsb
dz_ext0 = dz_ext1 = dz0 - 3 * SQUISH_CONSTANT_3D
else: # (1,1,1) is not one of the closest two tetrahedral vertices.
c = (a_point & b_point) # Our two extra vertices are determined by the closest two.
if (c & 0x01) != 0:
xsv_ext0 = xsb + 1
xsv_ext1 = xsb + 2
dx_ext0 = dx0 - 1 - SQUISH_CONSTANT_3D
dx_ext1 = dx0 - 2 - 2 * SQUISH_CONSTANT_3D
else:
xsv_ext0 = xsv_ext1 = xsb
dx_ext0 = dx0 - SQUISH_CONSTANT_3D
dx_ext1 = dx0 - 2 * SQUISH_CONSTANT_3D
if (c & 0x02) != 0:
ysv_ext0 = ysb + 1
ysv_ext1 = ysb + 2
dy_ext0 = dy0 - 1 - SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 - 2 * SQUISH_CONSTANT_3D
else:
ysv_ext0 = ysv_ext1 = ysb
dy_ext0 = dy0 - SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 * SQUISH_CONSTANT_3D
if (c & 0x04) != 0:
zsv_ext0 = zsb + 1
zsv_ext1 = zsb + 2
dz_ext0 = dz0 - 1 - SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 - 2 * SQUISH_CONSTANT_3D
else:
zsv_ext0 = zsv_ext1 = zsb
dz_ext0 = dz0 - SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 * SQUISH_CONSTANT_3D
# Contribution (1,1,0)
dx3 = dx0 - 1 - 2 * SQUISH_CONSTANT_3D
dy3 = dy0 - 1 - 2 * SQUISH_CONSTANT_3D
dz3 = dz0 - 0 - 2 * SQUISH_CONSTANT_3D
attn3 = 2 - dx3 * dx3 - dy3 * dy3 - dz3 * dz3
if attn3 > 0:
attn3 *= attn3
value += attn3 * attn3 * extrapolate3d(xsb + 1, ysb + 1, zsb + 0, dx3, dy3, dz3, perm, perm_grad_index_3D)
# Contribution (1,0,1)
dx2 = dx3
dy2 = dy0 - 0 - 2 * SQUISH_CONSTANT_3D
dz2 = dz0 - 1 - 2 * SQUISH_CONSTANT_3D
attn2 = 2 - dx2 * dx2 - dy2 * dy2 - dz2 * dz2
if attn2 > 0:
attn2 *= attn2
value += attn2 * attn2 * extrapolate3d(xsb + 1, ysb + 0, zsb + 1, dx2, dy2, dz2, perm, perm_grad_index_3D)
# Contribution (0,1,1)
dx1 = dx0 - 0 - 2 * SQUISH_CONSTANT_3D
dy1 = dy3
dz1 = dz2
attn1 = 2 - dx1 * dx1 - dy1 * dy1 - dz1 * dz1
if attn1 > 0:
attn1 *= attn1
value += attn1 * attn1 * extrapolate3d(xsb + 0, ysb + 1, zsb + 1, dx1, dy1, dz1, perm, perm_grad_index_3D)
# Contribution (1,1,1)
dx0 = dx0 - 1 - 3 * SQUISH_CONSTANT_3D
dy0 = dy0 - 1 - 3 * SQUISH_CONSTANT_3D
dz0 = dz0 - 1 - 3 * SQUISH_CONSTANT_3D
attn0 = 2 - dx0 * dx0 - dy0 * dy0 - dz0 * dz0
if attn0 > 0:
attn0 *= attn0
value += attn0 * attn0 * extrapolate3d(xsb + 1, ysb + 1, zsb + 1, dx0, dy0, dz0, perm, perm_grad_index_3D)
else: # We're inside the octahedron (Rectified 3-Simplex) in between.
# Decide between point (0,0,1) and (1,1,0) as closest
p1 = xins + yins
if p1 > 1:
a_score = p1 - 1
a_point = 0x03
a_is_further_side = True
else:
a_score = 1 - p1
a_point = 0x04
a_is_further_side = False
# Decide between point (0,1,0) and (1,0,1) as closest
p2 = xins + zins
if p2 > 1:
b_score = p2 - 1
b_point = 0x05
b_is_further_side = True
else:
b_score = 1 - p2
b_point = 0x02
b_is_further_side = False
# The closest out of the two (1,0,0) and (0,1,1) will replace the furthest out of the two decided above, if closer.
p3 = yins + zins
if p3 > 1:
score = p3 - 1
if a_score <= b_score and a_score < score:
a_point = 0x06
a_is_further_side = True
elif a_score > b_score and b_score < score:
b_point = 0x06
b_is_further_side = True
else:
score = 1 - p3
if a_score <= b_score and a_score < score:
a_point = 0x01
a_is_further_side = False
elif a_score > b_score and b_score < score:
b_point = 0x01
b_is_further_side = False
# Where each of the two closest points are determines how the extra two vertices are calculated.
if a_is_further_side == b_is_further_side:
if a_is_further_side: # Both closest points on (1,1,1) side
# One of the two extra points is (1,1,1)
dx_ext0 = dx0 - 1 - 3 * SQUISH_CONSTANT_3D
dy_ext0 = dy0 - 1 - 3 * SQUISH_CONSTANT_3D
dz_ext0 = dz0 - 1 - 3 * SQUISH_CONSTANT_3D
xsv_ext0 = xsb + 1
ysv_ext0 = ysb + 1
zsv_ext0 = zsb + 1
# Other extra point is based on the shared axis.
c = (a_point & b_point)
if (c & 0x01) != 0:
dx_ext1 = dx0 - 2 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 * SQUISH_CONSTANT_3D
xsv_ext1 = xsb + 2
ysv_ext1 = ysb
zsv_ext1 = zsb
elif (c & 0x02) != 0:
dx_ext1 = dx0 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 * SQUISH_CONSTANT_3D
xsv_ext1 = xsb
ysv_ext1 = ysb + 2
zsv_ext1 = zsb
else:
dx_ext1 = dx0 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 - 2 * SQUISH_CONSTANT_3D
xsv_ext1 = xsb
ysv_ext1 = ysb
zsv_ext1 = zsb + 2
else: # Both closest points on (0,0,0) side
# One of the two extra points is (0,0,0)
dx_ext0 = dx0
dy_ext0 = dy0
dz_ext0 = dz0
xsv_ext0 = xsb
ysv_ext0 = ysb
zsv_ext0 = zsb
# Other extra point is based on the omitted axis.
c = (a_point | b_point)
if (c & 0x01) == 0:
dx_ext1 = dx0 + 1 - SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 1 - SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 1 - SQUISH_CONSTANT_3D
xsv_ext1 = xsb - 1
ysv_ext1 = ysb + 1
zsv_ext1 = zsb + 1
elif (c & 0x02) == 0:
dx_ext1 = dx0 - 1 - SQUISH_CONSTANT_3D
dy_ext1 = dy0 + 1 - SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 1 - SQUISH_CONSTANT_3D
xsv_ext1 = xsb + 1
ysv_ext1 = ysb - 1
zsv_ext1 = zsb + 1
else:
dx_ext1 = dx0 - 1 - SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 1 - SQUISH_CONSTANT_3D
dz_ext1 = dz0 + 1 - SQUISH_CONSTANT_3D
xsv_ext1 = xsb + 1
ysv_ext1 = ysb + 1
zsv_ext1 = zsb - 1
else: # One point on (0,0,0) side, one point on (1,1,1) side
if a_is_further_side:
c1 = a_point
c2 = b_point
else:
c1 = b_point
c2 = a_point
# One contribution is a _permutation of (1,1,-1)
if (c1 & 0x01) == 0:
dx_ext0 = dx0 + 1 - SQUISH_CONSTANT_3D
dy_ext0 = dy0 - 1 - SQUISH_CONSTANT_3D
dz_ext0 = dz0 - 1 - SQUISH_CONSTANT_3D
xsv_ext0 = xsb - 1
ysv_ext0 = ysb + 1
zsv_ext0 = zsb + 1
elif (c1 & 0x02) == 0:
dx_ext0 = dx0 - 1 - SQUISH_CONSTANT_3D
dy_ext0 = dy0 + 1 - SQUISH_CONSTANT_3D
dz_ext0 = dz0 - 1 - SQUISH_CONSTANT_3D
xsv_ext0 = xsb + 1
ysv_ext0 = ysb - 1
zsv_ext0 = zsb + 1
else:
dx_ext0 = dx0 - 1 - SQUISH_CONSTANT_3D
dy_ext0 = dy0 - 1 - SQUISH_CONSTANT_3D
dz_ext0 = dz0 + 1 - SQUISH_CONSTANT_3D
xsv_ext0 = xsb + 1
ysv_ext0 = ysb + 1
zsv_ext0 = zsb - 1
# One contribution is a _permutation of (0,0,2)
dx_ext1 = dx0 - 2 * SQUISH_CONSTANT_3D
dy_ext1 = dy0 - 2 * SQUISH_CONSTANT_3D
dz_ext1 = dz0 - 2 * SQUISH_CONSTANT_3D
xsv_ext1 = xsb
ysv_ext1 = ysb
zsv_ext1 = zsb
if (c2 & 0x01) != 0:
dx_ext1 -= 2
xsv_ext1 += 2
elif (c2 & 0x02) != 0:
dy_ext1 -= 2
ysv_ext1 += 2
else:
dz_ext1 -= 2
zsv_ext1 += 2
# Contribution (1,0,0)
dx1 = dx0 - 1 - SQUISH_CONSTANT_3D
dy1 = dy0 - 0 - SQUISH_CONSTANT_3D
dz1 = dz0 - 0 - SQUISH_CONSTANT_3D
attn1 = 2 - dx1 * dx1 - dy1 * dy1 - dz1 * dz1
if attn1 > 0:
attn1 *= attn1
value += attn1 * attn1 * extrapolate3d(xsb + 1, ysb + 0, zsb + 0, dx1, dy1, dz1, perm, perm_grad_index_3D)
# Contribution (0,1,0)
dx2 = dx0 - 0 - SQUISH_CONSTANT_3D
dy2 = dy0 - 1 - SQUISH_CONSTANT_3D
dz2 = dz1
attn2 = 2 - dx2 * dx2 - dy2 * dy2 - dz2 * dz2
if attn2 > 0:
attn2 *= attn2
value += attn2 * attn2 * extrapolate3d(xsb + 0, ysb + 1, zsb + 0, dx2, dy2, dz2, perm, perm_grad_index_3D)
# Contribution (0,0,1)
dx3 = dx2
dy3 = dy1
dz3 = dz0 - 1 - SQUISH_CONSTANT_3D
attn3 = 2 - dx3 * dx3 - dy3 * dy3 - dz3 * dz3
if attn3 > 0:
attn3 *= attn3
value += attn3 * attn3 * extrapolate3d(xsb + 0, ysb + 0, zsb + 1, dx3, dy3, dz3, perm, perm_grad_index_3D)
# Contribution (1,1,0)
dx4 = dx0 - 1 - 2 * SQUISH_CONSTANT_3D
dy4 = dy0 - 1 - 2 * SQUISH_CONSTANT_3D
dz4 = dz0 - 0 - 2 * SQUISH_CONSTANT_3D
attn4 = 2 - dx4 * dx4 - dy4 * dy4 - dz4 * dz4
if attn4 > 0:
attn4 *= attn4
value += attn4 * attn4 * extrapolate3d(xsb + 1, ysb + 1, zsb + 0, dx4, dy4, dz4, perm, perm_grad_index_3D)
# Contribution (1,0,1)
dx5 = dx4
dy5 = dy0 - 0 - 2 * SQUISH_CONSTANT_3D
dz5 = dz0 - 1 - 2 * SQUISH_CONSTANT_3D
attn5 = 2 - dx5 * dx5 - dy5 * dy5 - dz5 * dz5
if attn5 > 0:
attn5 *= attn5
value += attn5 * attn5 * extrapolate3d(xsb + 1, ysb + 0, zsb + 1, dx5, dy5, dz5, perm, perm_grad_index_3D)
# Contribution (0,1,1)
dx6 = dx0 - 0 - 2 * SQUISH_CONSTANT_3D
dy6 = dy4
dz6 = dz5
attn6 = 2 - dx6 * dx6 - dy6 * dy6 - dz6 * dz6
if attn6 > 0:
attn6 *= attn6
value += attn6 * attn6 * extrapolate3d(xsb + 0, ysb + 1, zsb + 1, dx6, dy6, dz6, perm, perm_grad_index_3D)
# First extra vertex
attn_ext0 = 2 - dx_ext0 * dx_ext0 - dy_ext0 * dy_ext0 - dz_ext0 * dz_ext0
if attn_ext0 > 0:
attn_ext0 *= attn_ext0
value += attn_ext0 * attn_ext0 * extrapolate3d(xsv_ext0, ysv_ext0, zsv_ext0, dx_ext0, dy_ext0, dz_ext0, perm, perm_grad_index_3D)
# Second extra vertex
attn_ext1 = 2 - dx_ext1 * dx_ext1 - dy_ext1 * dy_ext1 - dz_ext1 * dz_ext1
if attn_ext1 > 0:
attn_ext1 *= attn_ext1
value += attn_ext1 * attn_ext1 * extrapolate3d(xsv_ext1, ysv_ext1, zsv_ext1, dx_ext1, dy_ext1, dz_ext1, perm, perm_grad_index_3D)
return value / NORM_CONSTANT_3D
@njit(cache=True)
def noisearr4d(x, y, z, w, perm):
num = x.size
noise = np.zeros(num, dtype=np.double)
for i in range(num):
noise[i] = noise4d(x[i], y[i], z[i], w[i], perm)
return noise
@njit(cache=True)
def noise4d(x, y, z, w, perm):
"""
Generate 4D OpenSimplex noise from X,Y,Z,W coordinates.
"""
# Place input coordinates on simplectic honeycomb.
stretch_offset = (x + y + z + w) * STRETCH_CONSTANT_4D
xs = x + stretch_offset
ys = y + stretch_offset
zs = z + stretch_offset
ws = w + stretch_offset
# Floor to get simplectic honeycomb coordinates of rhombo-hypercube super-cell origin.
xsb = fastfloor(xs)
ysb = fastfloor(ys)
zsb = fastfloor(zs)
wsb = fastfloor(ws)
# Skew out to get actual coordinates of stretched rhombo-hypercube origin. We'll need these later.
squish_offset = (xsb + ysb + zsb + wsb) * SQUISH_CONSTANT_4D
xb = xsb + squish_offset
yb = ysb + squish_offset
zb = zsb + squish_offset
wb = wsb + squish_offset
# Compute simplectic honeycomb coordinates relative to rhombo-hypercube origin.
xins = xs - xsb
yins = ys - ysb
zins = zs - zsb
wins = ws - wsb
# Sum those together to get a value that determines which region we're in.
in_sum = xins + yins + zins + wins
# Positions relative to origin po.
dx0 = x - xb
dy0 = y - yb
dz0 = z - zb
dw0 = w - wb
value = 0
if in_sum <= 1: # We're inside the pentachoron (4-Simplex) at (0,0,0,0)
# Determine which two of (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0) are closest.
a_po = 0x01
a_score = xins
b_po = 0x02
b_score = yins
if a_score >= b_score and zins > b_score:
b_score = zins
b_po = 0x04
elif a_score < b_score and zins > a_score:
a_score = zins
a_po = 0x04
if a_score >= b_score and wins > b_score:
b_score = wins
b_po = 0x08
elif a_score < b_score and wins > a_score:
a_score = wins
a_po = 0x08
# Now we determine the three lattice pos not part of the pentachoron that may contribute.
# This depends on the closest two pentachoron vertices, including (0,0,0,0)
uins = 1 - in_sum
if uins > a_score or uins > b_score: # (0,0,0,0) is one of the closest two pentachoron vertices.
c = b_po if (b_score > a_score) else a_po # Our other closest vertex is the closest out of a and b.
if (c & 0x01) == 0:
xsv_ext0 = xsb - 1
xsv_ext1 = xsv_ext2 = xsb
dx_ext0 = dx0 + 1
dx_ext1 = dx_ext2 = dx0
else:
xsv_ext0 = xsv_ext1 = xsv_ext2 = xsb + 1
dx_ext0 = dx_ext1 = dx_ext2 = dx0 - 1
if (c & 0x02) == 0:
ysv_ext0 = ysv_ext1 = ysv_ext2 = ysb
dy_ext0 = dy_ext1 = dy_ext2 = dy0
if (c & 0x01) == 0x01:
ysv_ext0 -= 1
dy_ext0 += 1
else:
ysv_ext1 -= 1
dy_ext1 += 1
else:
ysv_ext0 = ysv_ext1 = ysv_ext2 = ysb + 1
dy_ext0 = dy_ext1 = dy_ext2 = dy0 - 1
if (c & 0x04) == 0:
zsv_ext0 = zsv_ext1 = zsv_ext2 = zsb
dz_ext0 = dz_ext1 = dz_ext2 = dz0
if (c & 0x03) != 0:
if (c & 0x03) == 0x03:
zsv_ext0 -= 1
dz_ext0 += 1
else:
zsv_ext1 -= 1
dz_ext1 += 1
else:
zsv_ext2 -= 1
dz_ext2 += 1
else:
zsv_ext0 = zsv_ext1 = zsv_ext2 = zsb + 1
dz_ext0 = dz_ext1 = dz_ext2 = dz0 - 1
if (c & 0x08) == 0:
wsv_ext0 = wsv_ext1 = wsb
wsv_ext2 = wsb - 1
dw_ext0 = dw_ext1 = dw0
dw_ext2 = dw0 + 1
else:
wsv_ext0 = wsv_ext1 = wsv_ext2 = wsb + 1
dw_ext0 = dw_ext1 = dw_ext2 = dw0 - 1
else: # (0,0,0,0) is not one of the closest two pentachoron vertices.
c = (a_po | b_po) # Our three extra vertices are determined by the closest two.
if (c & 0x01) == 0:
xsv_ext0 = xsv_ext2 = xsb
xsv_ext1 = xsb - 1
dx_ext0 = dx0 - 2 * SQUISH_CONSTANT_4D
dx_ext1 = dx0 + 1 - SQUISH_CONSTANT_4D
dx_ext2 = dx0 - SQUISH_CONSTANT_4D
else:
xsv_ext0 = xsv_ext1 = xsv_ext2 = xsb + 1
dx_ext0 = dx0 - 1 - 2 * SQUISH_CONSTANT_4D
dx_ext1 = dx_ext2 = dx0 - 1 - SQUISH_CONSTANT_4D
if (c & 0x02) == 0:
ysv_ext0 = ysv_ext1 = ysv_ext2 = ysb
dy_ext0 = dy0 - 2 * SQUISH_CONSTANT_4D
dy_ext1 = dy_ext2 = dy0 - SQUISH_CONSTANT_4D
if (c & 0x01) == 0x01:
ysv_ext1 -= 1
dy_ext1 += 1
else:
ysv_ext2 -= 1
dy_ext2 += 1
else:
ysv_ext0 = ysv_ext1 = ysv_ext2 = ysb + 1
dy_ext0 = dy0 - 1 - 2 * SQUISH_CONSTANT_4D
dy_ext1 = dy_ext2 = dy0 - 1 - SQUISH_CONSTANT_4D
if (c & 0x04) == 0:
zsv_ext0 = zsv_ext1 = zsv_ext2 = zsb
dz_ext0 = dz0 - 2 * SQUISH_CONSTANT_4D
dz_ext1 = dz_ext2 = dz0 - SQUISH_CONSTANT_4D
if (c & 0x03) == 0x03:
zsv_ext1 -= 1
dz_ext1 += 1
else:
zsv_ext2 -= 1
dz_ext2 += 1
else:
zsv_ext0 = zsv_ext1 = zsv_ext2 = zsb + 1
dz_ext0 = dz0 - 1 - 2 * SQUISH_CONSTANT_4D
dz_ext1 = dz_ext2 = dz0 - 1 - SQUISH_CONSTANT_4D
if (c & 0x08) == 0:
wsv_ext0 = wsv_ext1 = wsb
wsv_ext2 = wsb - 1
dw_ext0 = dw0 - 2 * SQUISH_CONSTANT_4D
dw_ext1 = dw0 - SQUISH_CONSTANT_4D
dw_ext2 = dw0 + 1 - SQUISH_CONSTANT_4D
else:
wsv_ext0 = wsv_ext1 = wsv_ext2 = wsb + 1
dw_ext0 = dw0 - 1 - 2 * SQUISH_CONSTANT_4D
dw_ext1 = dw_ext2 = dw0 - 1 - SQUISH_CONSTANT_4D
# Contribution (0,0,0,0)
attn0 = 2 - dx0 * dx0 - dy0 * dy0 - dz0 * dz0 - dw0 * dw0
if attn0 > 0:
attn0 *= attn0
value += attn0 * attn0 * extrapolate4d(xsb + 0, ysb + 0, zsb + 0, wsb + 0, dx0, dy0, dz0, dw0, perm)
# Contribution (1,0,0,0)
dx1 = dx0 - 1 - SQUISH_CONSTANT_4D
dy1 = dy0 - 0 - SQUISH_CONSTANT_4D
dz1 = dz0 - 0 - SQUISH_CONSTANT_4D
dw1 = dw0 - 0 - SQUISH_CONSTANT_4D
attn1 = 2 - dx1 * dx1 - dy1 * dy1 - dz1 * dz1 - dw1 * dw1
if attn1 > 0:
attn1 *= attn1
value += attn1 * attn1 * extrapolate4d(xsb + 1, ysb + 0, zsb + 0, wsb + 0, dx1, dy1, dz1, dw1, perm)
# Contribution (0,1,0,0)
dx2 = dx0 - 0 - SQUISH_CONSTANT_4D
dy2 = dy0 - 1 - SQUISH_CONSTANT_4D
dz2 = dz1
dw2 = dw1
attn2 = 2 - dx2 * dx2 - dy2 * dy2 - dz2 * dz2 - dw2 * dw2
if attn2 > 0:
attn2 *= attn2
value += attn2 * attn2 * extrapolate4d(xsb + 0, ysb + 1, zsb + 0, wsb + 0, dx2, dy2, dz2, dw2, perm)
# Contribution (0,0,1,0)
dx3 = dx2
dy3 = dy1
dz3 = dz0 - 1 - SQUISH_CONSTANT_4D
dw3 = dw1
attn3 = 2 - dx3 * dx3 - dy3 * dy3 - dz3 * dz3 - dw3 * dw3
if attn3 > 0:
attn3 *= attn3
value += attn3 * attn3 * extrapolate4d(xsb + 0, ysb + 0, zsb + 1, wsb + 0, dx3, dy3, dz3, dw3, perm)
# Contribution (0,0,0,1)
dx4 = dx2
dy4 = dy1
dz4 = dz1
dw4 = dw0 - 1 - SQUISH_CONSTANT_4D
attn4 = 2 - dx4 * dx4 - dy4 * dy4 - dz4 * dz4 - dw4 * dw4
if attn4 > 0:
attn4 *= attn4
value += attn4 * attn4 * extrapolate4d(xsb + 0, ysb + 0, zsb + 0, wsb + 1, dx4, dy4, dz4, dw4, perm)
elif in_sum >= 3: # We're inside the pentachoron (4-Simplex) at (1,1,1,1)
# Determine which two of (1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1) are closest.
a_po = 0x0E
a_score = xins
b_po = 0x0D
b_score = yins
if a_score <= b_score and zins < b_score:
b_score = zins
b_po = 0x0B