/* SimplexNoise1234, Simplex noise with true analytic * derivative in 1D to 4D. * * Author: Stefan Gustavson, 2003-2005 * Contact: stefan.gustavson@liu.se * * This code was GPL licensed until February 2011. * As the original author of this code, I hereby * release it into the public domain. * Please feel free to use it for whatever you want. * Credit is appreciated where appropriate, and I also * appreciate being told where this code finds any use, * but you may do as you like. */ /* * This implementation is "Simplex Noise" as presented by * Ken Perlin at a relatively obscure and not often cited course * session "Real-Time Shading" at Siggraph 2001 (before real * time shading actually took off), under the title "hardware noise". * The 3D function is numerically equivalent to his Java reference * code available in the PDF course notes, although I re-implemented * it from scratch to get more readable code. The 1D, 2D and 4D cases * were implemented from scratch by me from Ken Perlin's text. * * This file has no dependencies on any other file, not even its own * header file. The header file is made for use by external code only. */ // We don't really need to include this, but play nice and do it anyway. #include "simplexnoise1234.h" #define FASTFLOOR(x) ( ((int)(x)<=(x)) ? ((int)x) : (((int)x)-1) ) //--------------------------------------------------------------------- // Static data /* * Permutation table. This is just a random jumble of all numbers 0-255, * repeated twice to avoid wrapping the index at 255 for each lookup. * This needs to be exactly the same for all instances on all platforms, * so it's easiest to just keep it as static explicit data. * This also removes the need for any initialisation of this class. * * Note that making this an int[] instead of a char[] might make the * code run faster on platforms with a high penalty for unaligned single * byte addressing. Intel x86 is generally single-byte-friendly, but * some other CPUs are faster with 4-aligned reads. * However, a char[] is smaller, which avoids cache trashing, and that * is probably the most important aspect on most architectures. * This array is accessed a *lot* by the noise functions. * A vector-valued noise over 3D accesses it 96 times, and a * float-valued 4D noise 64 times. We want this to fit in the cache! */ unsigned char perm[512] = {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, 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 }; //--------------------------------------------------------------------- /* * Helper functions to compute gradients-dot-residualvectors (1D to 4D) * Note that these generate gradients of more than unit length. To make * a close match with the value range of classic Perlin noise, the final * noise values need to be rescaled to fit nicely within [-1,1]. * (The simplex noise functions as such also have different scaling.) * Note also that these noise functions are the most practical and useful * signed version of Perlin noise. To return values according to the * RenderMan specification from the SL noise() and pnoise() functions, * the noise values need to be scaled and offset to [0,1], like this: * float SLnoise = (noise(x,y,z) + 1.0) * 0.5; */ double grad1( int hash, double x ) { int h = hash & 15; double grad = 1.0 + (h & 7); // Gradient value 1.0, 2.0, ..., 8.0 if (h&8) grad = -grad; // Set a random sign for the gradient return ( grad * x ); // Multiply the gradient with the distance } double grad2( int hash, double x, double y ) { int h = hash & 7; // Convert low 3 bits of hash code double u = h<4 ? x : y; // into 8 simple gradient directions, double v = h<4 ? y : x; // and compute the dot product with (x,y). return ((h&1)? -u : u) + ((h&2)? -2.0*v : 2.0*v); } double grad3( int hash, double x, double y , double z ) { int h = hash & 15; // Convert low 4 bits of hash code into 12 simple double u = h<8 ? x : y; // gradient directions, and compute dot product. double v = h<4 ? y : h==12||h==14 ? x : z; // Fix repeats at h = 12 to 15 return ((h&1)? -u : u) + ((h&2)? -v : v); } double grad4( int hash, double x, double y, double z, double t ) { int h = hash & 31; // Convert low 5 bits of hash code into 32 simple double u = h<24 ? x : y; // gradient directions, and compute dot product. double v = h<16 ? y : z; double w = h<8 ? z : t; return ((h&1)? -u : u) + ((h&2)? -v : v) + ((h&4)? -w : w); } // 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. /* TODO: This should not be required, backport it from Bill's GLSL code! */ static unsigned char simplex[64][4] = { {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}}; // 1D simplex noise double snoise1(double x) { int i0 = FASTFLOOR(x); int i1 = i0 + 1; double x0 = x - i0; double x1 = x0 - 1.0; double n0, n1; double t0 = 1.0 - x0*x0; // if(t0 < 0.0) t0 = 0.0; // this never happens for the 1D case t0 *= t0; n0 = t0 * t0 * grad1(perm[i0 & 0xff], x0); double t1 = 1.0 - x1*x1; // if(t1 < 0.0) t1 = 0.0; // this never happens for the 1D case t1 *= t1; n1 = t1 * t1 * grad1(perm[i1 & 0xff], x1); // The maximum value of this noise is 8*(3/4)^4 = 2.53125 // A factor of 0.395 would scale to fit exactly within [-1,1], but // we want to match PRMan's 1D noise, so we scale it down some more. return 0.25f * (n0 + n1); } // 2D simplex noise double snoise2(double x, double y) { #define F2 0.366025403 // F2 = 0.5*(sqrt(3.0)-1.0) #define G2 0.211324865 // G2 = (3.0-Math.sqrt(3.0))/6.0 double n0, n1, n2; // 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 double xs = x + s; double ys = y + s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); double t = (double)(i+j)*G2; double X0 = i-t; // Unskew the cell origin back to (x,y) space double Y0 = j-t; double x0 = x-X0; // The x,y distances from the cell origin double y0 = y-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; // Wrap the integer indices at 256, to avoid indexing perm[] out of bounds int ii = i & 0xff; int jj = j & 0xff; // Calculate the contribution from the three corners double t0 = 0.5f - x0*x0-y0*y0; if(t0 < 0.0) n0 = 0.0; else { t0 *= t0; n0 = t0 * t0 * grad2(perm[ii+perm[jj]], x0, y0); } double t1 = 0.5f - x1*x1-y1*y1; if(t1 < 0.0) n1 = 0.0; else { t1 *= t1; n1 = t1 * t1 * grad2(perm[ii+i1+perm[jj+j1]], x1, y1); } double t2 = 0.5f - x2*x2-y2*y2; if(t2 < 0.0) n2 = 0.0; else { t2 *= t2; n2 = t2 * t2 * grad2(perm[ii+1+perm[jj+1]], 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 40.0 * (n0 + n1 + n2); // TODO: The scale factor is preliminary! } // 3D simplex noise double snoise3(double x, double y, double z) { // Simple skewing factors for the 3D case #define F3 0.333333333 #define G3 0.166666667 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 = (x+y+z)*F3; // Very nice and simple skew factor for 3D double xs = x+s; double ys = y+s; double zs = z+s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); int k = FASTFLOOR(zs); double t = (double)(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 = x-X0; // The x,y,z distances from the cell origin double y0 = y-Y0; double z0 = z-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 /* This code would benefit from a backport from the GLSL version! */ 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) ? 32 : 0; int c2 = (x0 > z0) ? 16 : 0; int c3 = (y0 > z0) ? 8 : 0; int c4 = (x0 > w0) ? 4 : 0; int c5 = (y0 > w0) ? 2 : 0; int c6 = (z0 > w0) ? 1 : 0; int c = c1 + c2 + c3 + c4 + c5 + c6; 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=3 ? 1 : 0; j1 = simplex[c][1]>=3 ? 1 : 0; k1 = simplex[c][2]>=3 ? 1 : 0; l1 = simplex[c][3]>=3 ? 1 : 0; // The number 2 in the "simplex" array is at the second largest coordinate. i2 = simplex[c][0]>=2 ? 1 : 0; j2 = simplex[c][1]>=2 ? 1 : 0; k2 = simplex[c][2]>=2 ? 1 : 0; l2 = simplex[c][3]>=2 ? 1 : 0; // The number 1 in the "simplex" array is at the second smallest coordinate. i3 = simplex[c][0]>=1 ? 1 : 0; j3 = simplex[c][1]>=1 ? 1 : 0; k3 = simplex[c][2]>=1 ? 1 : 0; l3 = simplex[c][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) 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; // Wrap the integer indices at 256, to avoid indexing perm[] out of bounds 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 = 0.5f - x0*x0 - y0*y0 - z0*z0 - w0*w0; if(t0 < 0.0) n0 = 0.0; else { t0 *= t0; n0 = t0 * t0 * grad4(perm[ii+perm[jj+perm[kk+perm[ll]]]], x0, y0, z0, w0); } double t1 = 0.5f - x1*x1 - y1*y1 - z1*z1 - w1*w1; if(t1 < 0.0) n1 = 0.0; else { t1 *= t1; n1 = t1 * t1 * grad4(perm[ii+i1+perm[jj+j1+perm[kk+k1+perm[ll+l1]]]], x1, y1, z1, w1); } double t2 = 0.5f - x2*x2 - y2*y2 - z2*z2 - w2*w2; if(t2 < 0.0) n2 = 0.0; else { t2 *= t2; n2 = t2 * t2 * grad4(perm[ii+i2+perm[jj+j2+perm[kk+k2+perm[ll+l2]]]], x2, y2, z2, w2); } double t3 = 0.5f - x3*x3 - y3*y3 - z3*z3 - w3*w3; if(t3 < 0.0) n3 = 0.0; else { t3 *= t3; n3 = t3 * t3 * grad4(perm[ii+i3+perm[jj+j3+perm[kk+k3+perm[ll+l3]]]], x3, y3, z3, w3); } double t4 = 0.5f - x4*x4 - y4*y4 - z4*z4 - w4*w4; if(t4 < 0.0) n4 = 0.0; else { t4 *= t4; n4 = t4 * t4 * grad4(perm[ii+1+perm[jj+1+perm[kk+1+perm[ll+1]]]], x4, y4, z4, w4); } // Sum up and scale the result to cover the range [-1,1] return 62.0 * (n0 + n1 + n2 + n3 + n4); } //---------------------------------------------------------------------