Any Simplex Noise Tutorials or Resources?
I want to create a terrain-like 3D noise generator and after doing some research I came to the conclusion that Simplex Noise is by far the best type of noise to do this.
I find the name quite misleading though as I have a lot of trouble finding resources on the subject and the resources I find are often not well written.
What I am basically looking for is a good resource/tutorial explaining step by step how simplex noise works, and explains how to implement that into a program.
I am not looking for resources explaining how to use a library or something.
Answer
In lue of a tutorial recommendation I will attempt to explain how to use an existing java source that creates a single octave of simplex noise.
Simplex noise code
This part of the code was created by Stefan Gustavson and was placed in the public domain. It can be found here. It is quoted here for convinience
import java.awt.Color;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import java.util.Random;
import javax.imageio.ImageIO;
/*
* A speed-improved simplex noise algorithm for 2D, 3D and 4D in Java.
*
* Based on example code by Stefan Gustavson ([email protected]).
* Optimisations by Peter Eastman ([email protected]).
* Better rank ordering method by Stefan Gustavson in 2012.
*
* This could be speeded up even further, but it's useful as it is.
*
* Version 2012-03-09
*
* This code was placed in the public domain by its original author,
* Stefan Gustavson. You may use it as you see fit, but
* attribution is appreciated.
*
*/
public class SimplexNoise_octave { // Simplex noise in 2D, 3D and 4D
public static int RANDOMSEED=0;
private static int NUMBEROFSWAPS=400;
private static Grad grad3[] = {new Grad(1,1,0),new Grad(-1,1,0),new Grad(1,-1,0),new Grad(-1,-1,0),
new Grad(1,0,1),new Grad(-1,0,1),new Grad(1,0,-1),new Grad(-1,0,-1),
new Grad(0,1,1),new Grad(0,-1,1),new Grad(0,1,-1),new Grad(0,-1,-1)};
private static Grad grad4[]= {new Grad(0,1,1,1),new Grad(0,1,1,-1),new Grad(0,1,-1,1),new Grad(0,1,-1,-1),
new Grad(0,-1,1,1),new Grad(0,-1,1,-1),new Grad(0,-1,-1,1),new Grad(0,-1,-1,-1),
new Grad(1,0,1,1),new Grad(1,0,1,-1),new Grad(1,0,-1,1),new Grad(1,0,-1,-1),
new Grad(-1,0,1,1),new Grad(-1,0,1,-1),new Grad(-1,0,-1,1),new Grad(-1,0,-1,-1),
new Grad(1,1,0,1),new Grad(1,1,0,-1),new Grad(1,-1,0,1),new Grad(1,-1,0,-1),
new Grad(-1,1,0,1),new Grad(-1,1,0,-1),new Grad(-1,-1,0,1),new Grad(-1,-1,0,-1),
new Grad(1,1,1,0),new Grad(1,1,-1,0),new Grad(1,-1,1,0),new Grad(1,-1,-1,0),
new Grad(-1,1,1,0),new Grad(-1,1,-1,0),new Grad(-1,-1,1,0),new Grad(-1,-1,-1,0)};
private static short p_supply[] = {151,160,137,91,90,15, //this contains all the numbers between 0 and 255, these are put in a random order depending upon the seed
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};
private short p[]=new short[p_supply.length];
// To remove the need for index wrapping, double the permutation table length
private short perm[] = new short[512];
private short permMod12[] = new short[512];
public SimplexNoise_octave(int seed) {
p=p_supply.clone();
if (seed==RANDOMSEED){
Random rand=new Random();
seed=rand.nextInt();
}
//the random for the swaps
Random rand=new Random(seed);
//the seed determines the swaps that occur between the default order and the order we're actually going to use
for(int i=0;i<NUMBEROFSWAPS;i++){
int swapFrom=rand.nextInt(p.length);
int swapTo=rand.nextInt(p.length);
short temp=p[swapFrom];
p[swapFrom]=p[swapTo];
p[swapTo]=temp;
}
for(int i=0; i<512; i++)
{
perm[i]=p[i & 255];
permMod12[i] = (short)(perm[i] % 12);
}
}
// Skewing and unskewing factors for 2, 3, and 4 dimensions
private static final double F2 = 0.5*(Math.sqrt(3.0)-1.0);
private static final double G2 = (3.0-Math.sqrt(3.0))/6.0;
private static final double F3 = 1.0/3.0;
private static final double G3 = 1.0/6.0;
private static final double F4 = (Math.sqrt(5.0)-1.0)/4.0;
private static final double G4 = (5.0-Math.sqrt(5.0))/20.0;
// This method is a *lot* faster than using (int)Math.floor(x)
private static int fastfloor(double x) {
int xi = (int)x;
return x<xi ? xi-1 : xi;
}
private static double dot(Grad g, double x, double y) {
return g.x*x + g.y*y; }
private static double dot(Grad g, double x, double y, double z) {
return g.x*x + g.y*y + g.z*z; }
private static double dot(Grad g, double x, double y, double z, double w) {
return g.x*x + g.y*y + g.z*z + g.w*w; }
// 2D simplex noise
public double noise(double xin, double yin) {
double n0, n1, n2; // Noise contributions from the three corners
// Skew the input space to determine which simplex cell we're in
double s = (xin+yin)*F2; // Hairy factor for 2D
int i = fastfloor(xin+s);
int j = fastfloor(yin+s);
double t = (i+j)*G2;
double X0 = i-t; // Unskew the cell origin back to (x,y) space
double Y0 = j-t;
double x0 = xin-X0; // The x,y distances from the cell origin
double y0 = yin-Y0;
// 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) coords
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 coords
double y1 = y0 - j1 + G2;
double x2 = x0 - 1.0 + 2.0 * G2; // Offsets for last corner in (x,y) unskewed coords
double y2 = y0 - 1.0 + 2.0 * G2;
// Work out the hashed gradient indices of the three simplex corners
int ii = i & 255;
int jj = j & 255;
int gi0 = permMod12[ii+perm[jj]];
int gi1 = permMod12[ii+i1+perm[jj+j1]];
int gi2 = permMod12[ii+1+perm[jj+1]];
// Calculate the contribution from the three corners
double t0 = 0.5 - x0*x0-y0*y0;
if(t0<0) n0 = 0.0;
else {
t0 *= t0;
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) n1 = 0.0;
else {
t1 *= t1;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1);
}
double t2 = 0.5 - x2*x2-y2*y2;
if(t2<0) n2 = 0.0;
else {
t2 *= t2;
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);
}
// 3D simplex noise
public double noise(double xin, double yin, double zin) {
double n0, n1, n2, n3; // Noise contributions from the four corners
// Skew the input space to determine which simplex cell we're in
double s = (xin+yin+zin)*F3; // Very nice and simple skew factor for 3D
int i = fastfloor(xin+s);
int j = fastfloor(yin+s);
int k = fastfloor(zin+s);
double t = (i+j+k)*G3;
double X0 = i-t; // Unskew the cell origin back to (x,y,z) space
double Y0 = j-t;
double Z0 = k-t;
double x0 = xin-X0; // The x,y,z distances from the cell origin
double y0 = yin-Y0;
double z0 = zin-Z0;
// 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 + 2.0*G3; // Offsets for third corner in (x,y,z) coords
double y2 = y0 - j2 + 2.0*G3;
double z2 = z0 - k2 + 2.0*G3;
double x3 = x0 - 1.0 + 3.0*G3; // Offsets for last corner in (x,y,z) coords
double y3 = y0 - 1.0 + 3.0*G3;
double z3 = z0 - 1.0 + 3.0*G3;
// Work out the hashed gradient indices of the four simplex corners
int ii = i & 255;
int jj = j & 255;
int kk = k & 255;
int gi0 = permMod12[ii+perm[jj+perm[kk]]];
int gi1 = permMod12[ii+i1+perm[jj+j1+perm[kk+k1]]];
int gi2 = permMod12[ii+i2+perm[jj+j2+perm[kk+k2]]];
int gi3 = permMod12[ii+1+perm[jj+1+perm[kk+1]]];
// Calculate the contribution from the four corners
double t0 = 0.6 - x0*x0 - y0*y0 - z0*z0;
if(t0<0) n0 = 0.0;
else {
t0 *= t0;
n0 = t0 * t0 * dot(grad3[gi0], x0, y0, z0);
}
double t1 = 0.6 - x1*x1 - y1*y1 - z1*z1;
if(t1<0) n1 = 0.0;
else {
t1 *= t1;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1, z1);
}
double t2 = 0.6 - x2*x2 - y2*y2 - z2*z2;
if(t2<0) n2 = 0.0;
else {
t2 *= t2;
n2 = t2 * t2 * dot(grad3[gi2], x2, y2, z2);
}
double t3 = 0.6 - x3*x3 - y3*y3 - z3*z3;
if(t3<0) n3 = 0.0;
else {
t3 *= t3;
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);
}
// 4D simplex noise, better simplex rank ordering method 2012-03-09
public double noise(double x, double y, double z, double w) {
double n0, n1, n2, n3, n4; // Noise contributions from the five corners
// Skew the (x,y,z,w) space to determine which cell of 24 simplices we're in
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 = i - t; // Unskew the cell origin back to (x,y,z,w) space
double Y0 = j - t;
double Z0 = k - t;
double W0 = l - t;
double x0 = x - X0; // The x,y,z,w distances from the cell origin
double y0 = y - Y0;
double z0 = z - Z0;
double w0 = w - W0;
// 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.
// Six pair-wise comparisons are performed between each possible pair
// of the four coordinates, and the results are used to rank the numbers.
int rankx = 0;
int ranky = 0;
int rankz = 0;
int rankw = 0;
if(x0 > y0) rankx++; else ranky++;
if(x0 > z0) rankx++; else rankz++;
if(x0 > w0) rankx++; else rankw++;
if(y0 > z0) ranky++; else rankz++;
if(y0 > w0) ranky++; else rankw++;
if(z0 > w0) rankz++; else rankw++;
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.
// Rank 3 denotes the largest coordinate.
i1 = rankx >= 3 ? 1 : 0;
j1 = ranky >= 3 ? 1 : 0;
k1 = rankz >= 3 ? 1 : 0;
l1 = rankw >= 3 ? 1 : 0;
// Rank 2 denotes the second largest coordinate.
i2 = rankx >= 2 ? 1 : 0;
j2 = ranky >= 2 ? 1 : 0;
k2 = rankz >= 2 ? 1 : 0;
l2 = rankw >= 2 ? 1 : 0;
// Rank 1 denotes the second smallest coordinate.
i3 = rankx >= 1 ? 1 : 0;
j3 = ranky >= 1 ? 1 : 0;
k3 = rankz >= 1 ? 1 : 0;
l3 = rankw >= 1 ? 1 : 0;
// The fifth corner has all coordinate offsets = 1, so no need to compute that.
double x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w) coords
double y1 = y0 - j1 + G4;
double z1 = z0 - k1 + G4;
double w1 = w0 - l1 + G4;
double x2 = x0 - i2 + 2.0*G4; // Offsets for third corner in (x,y,z,w) coords
double y2 = y0 - j2 + 2.0*G4;
double z2 = z0 - k2 + 2.0*G4;
double w2 = w0 - l2 + 2.0*G4;
double x3 = x0 - i3 + 3.0*G4; // Offsets for fourth corner in (x,y,z,w) coords
double y3 = y0 - j3 + 3.0*G4;
double z3 = z0 - k3 + 3.0*G4;
double w3 = w0 - l3 + 3.0*G4;
double x4 = x0 - 1.0 + 4.0*G4; // Offsets for last corner in (x,y,z,w) coords
double y4 = y0 - 1.0 + 4.0*G4;
double z4 = z0 - 1.0 + 4.0*G4;
double w4 = w0 - 1.0 + 4.0*G4;
// Work out the hashed gradient indices of the five simplex corners
int ii = i & 255;
int jj = j & 255;
int kk = k & 255;
int ll = l & 255;
int gi0 = perm[ii+perm[jj+perm[kk+perm[ll]]]] % 32;
int gi1 = perm[ii+i1+perm[jj+j1+perm[kk+k1+perm[ll+l1]]]] % 32;
int gi2 = perm[ii+i2+perm[jj+j2+perm[kk+k2+perm[ll+l2]]]] % 32;
int gi3 = perm[ii+i3+perm[jj+j3+perm[kk+k3+perm[ll+l3]]]] % 32;
int gi4 = perm[ii+1+perm[jj+1+perm[kk+1+perm[ll+1]]]] % 32;
// Calculate the contribution from the five corners
double t0 = 0.6 - x0*x0 - y0*y0 - z0*z0 - w0*w0;
if(t0<0) n0 = 0.0;
else {
t0 *= t0;
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) n1 = 0.0;
else {
t1 *= t1;
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) n2 = 0.0;
else {
t2 *= t2;
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) n3 = 0.0;
else {
t3 *= t3;
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) n4 = 0.0;
else {
t4 *= t4;
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);
}
// Inner class to speed upp gradient computations
// (array access is a lot slower than member access)
private static class Grad
{
double x, y, z, w;
Grad(double x, double y, double z)
{
this.x = x;
this.y = y;
this.z = z;
}
Grad(double x, double y, double z, double w)
{
this.x = x;
this.y = y;
this.z = z;
this.w = w;
}
}
}
Frankly I consider this whole class to be a black box with a public constructor public SimplexNoise_octave(int seed), and 3 public methods public double noise(double xin, double yin), public double noise(double xin, double yin, double zin) and public double noise(double x, double y, double z, double w).
You can use these methods exactly as you would the perlin noise equivalents.
SimplexNoise_octave(int seed)
Create 1 SimplexNoise_octave for each octave you want, each should have its own seed
public double noise(double xin, double yin)
Call to get the particular noise value for that octave at those co-ordinates. Note; the co-ordinates should be pre-scaled (more later). The other noise functions are the same but for higher dimentions.
Creating Octaves
Just like in perlin noise you will in general combine several octaves of noise to create fractal noise (which gives you terrain like features). Note that 3D terrain heights are created by 2D noise.
Several octaves are combined using the following ratios
frequency = 2^i
amplitude = persistence^i
For each octave (i) you divide the input co-ordinates by frequency and multiply the result by amplitude; this gives a terrain like appearance. Persistence is used to affect the appearance of the terrain, high persistance (towards 1) gives rocky mountainous terrain. low persistance (towards 0) gives slowly varying flat terrain. See the tag page for more details.
An example of how this could be used is shown below:
import java.util.Random;
public class SimplexNoise {
SimplexNoise_octave[] octaves;
double[] frequencys;
double[] amplitudes;
int largestFeature;
double persistence;
int seed;
public SimplexNoise(int largestFeature,double persistence, int seed){
this.largestFeature=largestFeature;
this.persistence=persistence;
this.seed=seed;
//recieves a number (eg 128) and calculates what power of 2 it is (eg 2^7)
int numberOfOctaves=(int)Math.ceil(Math.log10(largestFeature)/Math.log10(2));
octaves=new SimplexNoise_octave[numberOfOctaves];
frequencys=new double[numberOfOctaves];
amplitudes=new double[numberOfOctaves];
Random rnd=new Random(seed);
for(int i=0;i<numberOfOctaves;i++){
octaves[i]=new SimplexNoise_octave(rnd.nextInt());
frequencys[i] = Math.pow(2,i);
amplitudes[i] = Math.pow(persistence,octaves.length-i);
}
}
public double getNoise(int x, int y){
double result=0;
for(int i=0;i<octaves.length;i++){
//double frequency = Math.pow(2,i);
//double amplitude = Math.pow(persistence,octaves.length-i);
result=result+octaves[i].noise(x/frequencys[i], y/frequencys[i])* amplitudes[i];
}
return result;
}
public double getNoise(int x,int y, int z){
double result=0;
for(int i=0;i<octaves.length;i++){
double frequency = Math.pow(2,i);
double amplitude = Math.pow(persistence,octaves.length-i);
result=result+octaves[i].noise(x/frequency, y/frequency,z/frequency)* amplitude;
}
return result;
}
}
This creates octaves that give features of size between 1 and largestFeature, I found this to be useful but theres nothing special about 1 being the smallest size and you could modify this. It outputs between -1 and 1, scale as needed.
Usage
An example main method that would use this class is as follows
public static void main(String args[]){
SimplexNoise simplexNoise=new SimplexNoise(100,0.1,5000);
double xStart=0;
double XEnd=500;
double yStart=0;
double yEnd=500;
int xResolution=200;
int yResolution=200;
double[][] result=new double[xResolution][yResolution];
for(int i=0;i<xResolution;i++){
for(int j=0;j<yResolution;j++){
int x=(int)(xStart+i*((XEnd-xStart)/xResolution));
int y=(int)(yStart+j*((yEnd-yStart)/yResolution));
result[i][j]=0.5*(1+simplexNoise.getNoise(x,y));
}
}
ImageWriter.greyWriteImage(result);
}
This method makes use of my own ImageWriter class just to render the output to a file
import java.awt.Color;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import javax.imageio.ImageIO;
public class ImageWriter {
//just convinence methods for debug
public static void greyWriteImage(double[][] data){
//this takes and array of doubles between 0 and 1 and generates a grey scale image from them
BufferedImage image = new BufferedImage(data.length,data[0].length, BufferedImage.TYPE_INT_RGB);
for (int y = 0; y < data[0].length; y++)
{
for (int x = 0; x < data.length; x++)
{
if (data[x][y]>1){
data[x][y]=1;
}
if (data[x][y]<0){
data[x][y]=0;
}
Color col=new Color((float)data[x][y],(float)data[x][y],(float)data[x][y]);
image.setRGB(x, y, col.getRGB());
}
}
try {
// retrieve image
File outputfile = new File("saved.png");
outputfile.createNewFile();
ImageIO.write(image, "png", outputfile);
} catch (IOException e) {
//o no! Blank catches are bad
throw new RuntimeException("I didn't handle this very well");
}
}
}