/*
* __ .__ .__ ._____.
* _/ |_ _______ __|__| ____ | | |__\_ |__ ______
* \ __\/ _ \ \/ / |/ ___\| | | || __ \ / ___/
* | | ( <_> > <| \ \___| |_| || \_\ \\___ \
* |__| \____/__/\_ \__|\___ >____/__||___ /____ >
* \/ \/ \/ \/
*
* Copyright (c) 2006-2011 Karsten Schmidt
*
* This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* http://creativecommons.org/licenses/LGPL/2.1/
*
* This library is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library; if not, write to the Free Software
* Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
*/
package spimedb.util.math.noise;
/**
* Simplex Noise in 2D, 3D and 4D. Based on the example code of this paper:
* http://staffwww.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf
*
* @author Stefan Gustavson, Linkping University, Sweden (stegu at itn dot liu
* dot se)
* @author Karsten Schmidt (slight optimizations & restructuring)
*/
public class SimplexNoise {
private static final double SQRT3 = Math.sqrt(3.0);
private static final double SQRT5 = Math.sqrt(5.0);
/**
* Skewing and unskewing factors for 2D, 3D and 4D, some of them
* pre-multiplied.
*/
private static final double F2 = 0.5 * (SQRT3 - 1.0);
private static final double G2 = (3.0 - SQRT3) / 6.0;
private static final double G22 = G2 * 2.0 - 1;
private static final double F3 = 1.0 / 3.0;
private static final double G3 = 1.0 / 6.0;
private static final double F4 = (SQRT5 - 1.0) / 4.0;
private static final double G4 = (5.0 - SQRT5) / 20.0;
private static final double G42 = G4 * 2.0;
private static final double G43 = G4 * 3.0;
private static final double G44 = G4 * 4.0 - 1.0;
/**
* Gradient vectors for 3D (pointing to mid points of all edges of a unit
* cube)
*/
private static final int[][] 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 } };
/**
* Gradient vectors for 4D (pointing to mid points of all edges of a unit 4D
* hypercube)
*/
private static final int[][] 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 } };
/**
* Permutation table
*/
private static final int[] p = { 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, 19, 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 };
/**
* To remove the need for index wrapping, double the permutation table
* length
*/
private static final int[] perm = new int[0x200];
/**
* 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.
*/
private static final int[][] 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 } };
static {
for (int i = 0; i < 0x200; i++) {
perm[i] = p[i & 0xff];
}
}
/**
* Computes dot product in 2D.
*
* @param g
* 2-vector (grid offset)
* @param x
* @param y
* @return dot product
*/
private static double dot(int g[], double x, double y) {
return g[0] * x + g[1] * y;
}
/**
* Computes dot product in 3D.
*
* @param g
* 3-vector (grid offset)
* @param x
* @param y
* @param z
* @return dot product
*/
private static double dot(int g[], double x, double y, double z) {
return g[0] * x + g[1] * y + g[2] * z;
}
/**
* Computes dot product in 4D.
*
* @param g
* 4-vector (grid offset)
* @param x
* @param y
* @param z
* @param w
* @return dot product
*/
private static double dot(int g[], double x, double y, double z, double w) {
return g[0] * x + g[1] * y + g[2] * z + g[3] * w;
}
/**
* This method is a *lot* faster than using (int)Math.floor(x).
*
* @param x
* value to be floored
* @return
*/
private static int fastfloor(double x) {
return x >= 0 ? (int) x : (int) x - 1;
}
/**
* Computes 2D Simplex Noise.
*
* @param x
* coordinate
* @param y
* coordinate
* @return noise value in range -1 ... +1.
*/
public static double noise(double x, double y) {
double n0 = 0, n1 = 0, n2 = 0; // Noise contributions from the three
// corners
// Skew the input space to determine which simplex cell we're in
double s = (x + y) * F2; // Hairy factor for 2D
int i = fastfloor(x + s);
int j = fastfloor(y + s);
double t = (i + j) * G2;
double x0 = x - (i - t); // The x,y distances from the cell origin
double y0 = y - (j - t);
// For the 2D case, the simplex shape is an equilateral triangle.
// Determine which simplex we are in.
int i1, j1; // Offsets for second (middle) corner of simplex in (i,j)
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)
// A step of (1,0) in (i,j) means a step of (1-c,-c) in (x,y), and
// a step of (0,1) in (i,j) means a step of (-c,1-c) in (x,y), where
// c = (3-sqrt(3))/6
double x1 = x0 - i1 + G2; // Offsets for middle corner in (x,y) unskewed
double y1 = y0 - j1 + G2;
double x2 = x0 + G22; // Offsets for last corner in (x,y) unskewed
double y2 = y0 + G22;
// Work out the hashed gradient indices of the three simplex corners
int ii = i & 0xff;
int jj = j & 0xff;
// Calculate the contribution from the three corners
double t0 = 0.5 - x0 * x0 - y0 * y0;
if (t0 > 0) {
t0 *= t0;
int gi0 = perm[ii + perm[jj]] % 12;
n0 = t0 * t0 * dot(grad3[gi0], x0, y0); // (x,y) of grad3 used for
// 2D gradient
}
double t1 = 0.5 - x1 * x1 - y1 * y1;
if (t1 > 0) {
t1 *= t1;
int gi1 = perm[ii + i1 + perm[jj + j1]] % 12;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1);
}
double t2 = 0.5 - x2 * x2 - y2 * y2;
if (t2 > 0) {
t2 *= t2;
int gi2 = perm[ii + 1 + perm[jj + 1]] % 12;
n2 = t2 * t2 * dot(grad3[gi2], x2, y2);
}
// Add contributions from each corner to get the final noise value.
// The result is scaled to return values in the interval [-1,1].
return 70.0 * (n0 + n1 + n2);
}
/**
* Computes 3D Simplex Noise.
*
* @param x
* coordinate
* @param y
* coordinate
* @param z
* coordinate
* @return noise value in range -1 ... +1
*/
public static double noise(double x, double y, double z) {
double n0 = 0, n1 = 0, n2 = 0, n3 = 0;
// Noise contributions from the
// four corners
// Skew the input space to determine which simplex cell we're in
// final double F3 = 1.0 / 3.0;
double s = (x + y + z) * F3; // Very nice and simple skew factor
// for 3D
int i = fastfloor(x + s);
int j = fastfloor(y + s);
int k = fastfloor(z + s);
// final double G3 = 1.0 / 6.0; // Very nice and simple unskew factor,
// too
double t = (i + j + k) * G3;
double x0 = x - (i - t); // The x,y,z distances from the cell origin
double y0 = y - (j - t);
double z0 = z - (k - t);
// For the 3D case, the simplex shape is a slightly irregular
// tetrahedron.
// Determine which simplex we are in.
int i1, j1, k1; // Offsets for second corner of simplex in (i,j,k)
// coords
int i2, j2, k2; // Offsets for third corner of simplex in (i,j,k) coords
if (x0 >= y0) {
if (y0 >= z0) {
i1 = 1;
j1 = 0;
k1 = 0;
i2 = 1;
j2 = 1;
k2 = 0;
} // X Y Z order
else if (x0 >= z0) {
i1 = 1;
j1 = 0;
k1 = 0;
i2 = 1;
j2 = 0;
k2 = 1;
} // X Z Y order
else {
i1 = 0;
j1 = 0;
k1 = 1;
i2 = 1;
j2 = 0;
k2 = 1;
} // Z X Y order
} else { // x0<y0
if (y0 < z0) {
i1 = 0;
j1 = 0;
k1 = 1;
i2 = 0;
j2 = 1;
k2 = 1;
} // Z Y X order
else if (x0 < z0) {
i1 = 0;
j1 = 1;
k1 = 0;
i2 = 0;
j2 = 1;
k2 = 1;
} // Y Z X order
else {
i1 = 0;
j1 = 1;
k1 = 0;
i2 = 1;
j2 = 1;
k2 = 0;
} // Y X Z order
}
// A step of (1,0,0) in (i,j,k) means a step of (1-c,-c,-c) in (x,y,z),
// a step of (0,1,0) in (i,j,k) means a step of (-c,1-c,-c) in (x,y,z),
// and
// a step of (0,0,1) in (i,j,k) means a step of (-c,-c,1-c) in (x,y,z),
// where
// c = 1/6.
double x1 = x0 - i1 + G3; // Offsets for second corner in (x,y,z) coords
double y1 = y0 - j1 + G3;
double z1 = z0 - k1 + G3;
double x2 = x0 - i2 + F3; // Offsets for third corner in (x,y,z)
double y2 = y0 - j2 + F3;
double z2 = z0 - k2 + F3;
double x3 = x0 - 0.5; // Offsets for last corner in (x,y,z)
double y3 = y0 - 0.5;
double z3 = z0 - 0.5;
// Work out the hashed gradient indices of the four simplex corners
int ii = i & 0xff;
int jj = j & 0xff;
int kk = k & 0xff;
// Calculate the contribution from the four corners
double t0 = 0.6 - x0 * x0 - y0 * y0 - z0 * z0;
if (t0 > 0) {
t0 *= t0;
int gi0 = perm[ii + perm[jj + perm[kk]]] % 12;
n0 = t0 * t0 * dot(grad3[gi0], x0, y0, z0);
}
double t1 = 0.6 - x1 * x1 - y1 * y1 - z1 * z1;
if (t1 > 0) {
t1 *= t1;
int gi1 = perm[ii + i1 + perm[jj + j1 + perm[kk + k1]]] % 12;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1, z1);
}
double t2 = 0.6 - x2 * x2 - y2 * y2 - z2 * z2;
if (t2 > 0) {
t2 *= t2;
int gi2 = perm[ii + i2 + perm[jj + j2 + perm[kk + k2]]] % 12;
n2 = t2 * t2 * dot(grad3[gi2], x2, y2, z2);
}
double t3 = 0.6 - x3 * x3 - y3 * y3 - z3 * z3;
if (t3 > 0) {
t3 *= t3;
int gi3 = perm[ii + 1 + perm[jj + 1 + perm[kk + 1]]] % 12;
n3 = t3 * t3 * dot(grad3[gi3], x3, y3, z3);
}
// Add contributions from each corner to get the final noise value.
// The result is scaled to stay just inside [-1,1]
return 32.0 * (n0 + n1 + n2 + n3);
}
/**
* Computes 4D Simplex Noise.
*
* @param x
* coordinate
* @param y
* coordinate
* @param z
* coordinate
* @param w
* coordinate
* @return noise value in range -1 ... +1
*/
public static double noise(double x, double y, double z, double w) {
// The skewing and unskewing factors are hairy again for the 4D case
double n0 = 0, n1 = 0, n2 = 0, n3 = 0, n4 = 0; // Noise contributions
// from the five corners
// Skew the (x,y,z,w) space to determine which cell of 24 simplices
double s = (x + y + z + w) * F4; // Factor for 4D skewing
int i = fastfloor(x + s);
int j = fastfloor(y + s);
int k = fastfloor(z + s);
int l = fastfloor(w + s);
double t = (i + j + k + l) * G4; // Factor for 4D unskewing
double x0 = x - (i - t); // The x,y,z,w distances from the cell origin
double y0 = y - (j - t);
double z0 = z - (k - t);
double w0 = w - (l - t);
// For the 4D case, the simplex is a 4D shape I won't even try to
// describe.
// To find out which of the 24 possible simplices we're in, we need to
// determine the magnitude ordering of x0, y0, z0 and w0.
// The method below is a good way of finding the ordering of x,y,z,w and
// then find the correct traversal order for the simplex were in.
// First, six pair-wise comparisons are performed between each possible
// pair of the four coordinates, and the results are used to add up
// binary bits for an integer index.
int c = 0;
if (x0 > y0) {
c = 0x20;
}
if (x0 > z0) {
c |= 0x10;
}
if (y0 > z0) {
c |= 0x08;
}
if (x0 > w0) {
c |= 0x04;
}
if (y0 > w0) {
c |= 0x02;
}
if (z0 > w0) {
c |= 0x01;
}
int i1, j1, k1, l1; // The integer offsets for the second simplex corner
int i2, j2, k2, l2; // The integer offsets for the third simplex corner
int i3, j3, k3, l3; // The integer offsets for the fourth simplex corner
// simplex[c] is a 4-vector with the numbers 0, 1, 2 and 3 in some
// order. Many values of c will never occur, since e.g. x>y>z>w makes
// x<z, y<w and x<w impossible. Only the 24 indices which have non-zero
// entries make any sense. We use a thresholding to set the coordinates
// in turn from the largest magnitude. The number 3 in the "simplex"
// array is at the position of the largest coordinate.
int[] sc = simplex[c];
i1 = sc[0] >= 3 ? 1 : 0;
j1 = sc[1] >= 3 ? 1 : 0;
k1 = sc[2] >= 3 ? 1 : 0;
l1 = sc[3] >= 3 ? 1 : 0;
// The number 2 in the "simplex" array is at the second largest
// coordinate.
i2 = sc[0] >= 2 ? 1 : 0;
j2 = sc[1] >= 2 ? 1 : 0;
k2 = sc[2] >= 2 ? 1 : 0;
l2 = sc[3] >= 2 ? 1 : 0;
// The number 1 in the "simplex" array is at the second smallest
// coordinate.
i3 = sc[0] >= 1 ? 1 : 0;
j3 = sc[1] >= 1 ? 1 : 0;
k3 = sc[2] >= 1 ? 1 : 0;
l3 = sc[3] >= 1 ? 1 : 0;
// The fifth corner has all coordinate offsets = 1, so no need to look
// that up.
double x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w)
double y1 = y0 - j1 + G4;
double z1 = z0 - k1 + G4;
double w1 = w0 - l1 + G4;
double x2 = x0 - i2 + G42; // Offsets for third corner in (x,y,z,w)
double y2 = y0 - j2 + G42;
double z2 = z0 - k2 + G42;
double w2 = w0 - l2 + G42;
double x3 = x0 - i3 + G43; // Offsets for fourth corner in (x,y,z,w)
double y3 = y0 - j3 + G43;
double z3 = z0 - k3 + G43;
double w3 = w0 - l3 + G43;
double x4 = x0 + G44; // Offsets for last corner in (x,y,z,w)
double y4 = y0 + G44;
double z4 = z0 + G44;
double w4 = w0 + G44;
// Work out the hashed gradient indices of the five simplex corners
int ii = i & 0xff;
int jj = j & 0xff;
int kk = k & 0xff;
int ll = l & 0xff;
// Calculate the contribution from the five corners
double t0 = 0.6 - x0 * x0 - y0 * y0 - z0 * z0 - w0 * w0;
if (t0 > 0) {
t0 *= t0;
int gi0 = perm[ii + perm[jj + perm[kk + perm[ll]]]] % 32;
n0 = t0 * t0 * dot(grad4[gi0], x0, y0, z0, w0);
}
double t1 = 0.6 - x1 * x1 - y1 * y1 - z1 * z1 - w1 * w1;
if (t1 > 0) {
t1 *= t1;
int gi1 = perm[ii + i1
+ perm[jj + j1 + perm[kk + k1 + perm[ll + l1]]]] % 32;
n1 = t1 * t1 * dot(grad4[gi1], x1, y1, z1, w1);
}
double t2 = 0.6 - x2 * x2 - y2 * y2 - z2 * z2 - w2 * w2;
if (t2 > 0) {
t2 *= t2;
int gi2 = perm[ii + i2
+ perm[jj + j2 + perm[kk + k2 + perm[ll + l2]]]] % 32;
n2 = t2 * t2 * dot(grad4[gi2], x2, y2, z2, w2);
}
double t3 = 0.6 - x3 * x3 - y3 * y3 - z3 * z3 - w3 * w3;
if (t3 > 0) {
t3 *= t3;
int gi3 = perm[ii + i3
+ perm[jj + j3 + perm[kk + k3 + perm[ll + l3]]]] % 32;
n3 = t3 * t3 * dot(grad4[gi3], x3, y3, z3, w3);
}
double t4 = 0.6 - x4 * x4 - y4 * y4 - z4 * z4 - w4 * w4;
if (t4 > 0) {
t4 *= t4;
int gi4 = perm[ii + 1 + perm[jj + 1 + perm[kk + 1 + perm[ll + 1]]]] % 32;
n4 = t4 * t4 * dot(grad4[gi4], x4, y4, z4, w4);
}
// Sum up and scale the result to cover the range [-1,1]
return 27.0 * (n0 + n1 + n2 + n3 + n4);
}
}