// 3D simplex noise float noise(float x, float y, float z) { // Simple skewing factors for the 3D case #define F3 0.333333333 #define G3 0.166666667 float n0, n1, n2, n3; // Noise contributions from the four corners // Skew the input space to determine which simplex cell we're in float s = (x+y+z)*F3; // Very nice and simple skew factor for 3D float xs = x+s; float ys = y+s; float zs = z+s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); int k = FASTFLOOR(zs); float t = (float)(i+j+k)*G3; float X0 = i-t; // Unskew the cell origin back to (x,y,z) space float Y0 = j-t; float Z0 = k-t; float x0 = x-X0; // The x,y,z distances from the cell origin float y0 = y-Y0; float 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 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. float x1 = x0 - i1 + G3; // Offsets for second corner in (x,y,z) coords float y1 = y0 - j1 + G3; float z1 = z0 - k1 + G3; float x2 = x0 - i2 + 2.0f*G3; // Offsets for third corner in (x,y,z) coords float y2 = y0 - j2 + 2.0f*G3; float z2 = z0 - k2 + 2.0f*G3; float x3 = x0 - 1.0f + 3.0f*G3; // Offsets for last corner in (x,y,z) coords float y3 = y0 - 1.0f + 3.0f*G3; float z3 = z0 - 1.0f + 3.0f*G3; // Wrap the integer indices at 256, to avoid indexing perm[] out of bounds int ii = i % 256; int jj = j % 256; int kk = k % 256; // Calculate the contribution from the four corners float t0 = 0.6f - x0*x0 - y0*y0 - z0*z0; if(t0 < 0.0f) n0 = 0.0f; else { t0 *= t0; n0 = t0 * t0 * grad(perm[ii+perm[jj+perm[kk]]], x0, y0, z0); } float t1 = 0.6f - x1*x1 - y1*y1 - z1*z1; if(t1 < 0.0f) n1 = 0.0f; else { t1 *= t1; n1 = t1 * t1 * grad(perm[ii+i1+perm[jj+j1+perm[kk+k1]]], x1, y1, z1); } float t2 = 0.6f - x2*x2 - y2*y2 - z2*z2; if(t2 < 0.0f) n2 = 0.0f; else { t2 *= t2; n2 = t2 * t2 * grad(perm[ii+i2+perm[jj+j2+perm[kk+k2]]], x2, y2, z2); } float t3 = 0.6f - x3*x3 - y3*y3 - z3*z3; if(t3<0.0f) n3 = 0.0f; else { t3 *= t3; n3 = t3 * t3 * grad(perm[ii+1+perm[jj+1+perm[kk+1]]], 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 32.0f * (n0 + n1 + n2 + n3); // TODO: The scale factor is preliminary! }
// 4D simplex noise float SimplexNoise1234::noise(float x, float y, float z, float w) { // The skewing and unskewing factors are hairy again for the 4D case #define F4 0.309016994f // F4 = (Math.sqrt(5.0)-1.0)/4.0 #define G4 0.138196601f // G4 = (5.0-Math.sqrt(5.0))/20.0 float 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 float s = (x + y + z + w) * F4; // Factor for 4D skewing float xs = x + s; float ys = y + s; float zs = z + s; float ws = w + s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); int k = FASTFLOOR(zs); int l = FASTFLOOR(ws); float t = (i + j + k + l) * G4; // Factor for 4D unskewing float X0 = i - t; // Unskew the cell origin back to (x,y,z,w) space float Y0 = j - t; float Z0 = k - t; float W0 = l - t; float x0 = x - X0; // The x,y,z,w distances from the cell origin float y0 = y - Y0; float z0 = z - Z0; float 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. // 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 we’re 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 c1 = (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<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. i1 = simplex[c][0]>=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. float x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w) coords float y1 = y0 - j1 + G4; float z1 = z0 - k1 + G4; float w1 = w0 - l1 + G4; float x2 = x0 - i2 + 2.0f*G4; // Offsets for third corner in (x,y,z,w) coords float y2 = y0 - j2 + 2.0f*G4; float z2 = z0 - k2 + 2.0f*G4; float w2 = w0 - l2 + 2.0f*G4; float x3 = x0 - i3 + 3.0f*G4; // Offsets for fourth corner in (x,y,z,w) coords float y3 = y0 - j3 + 3.0f*G4; float z3 = z0 - k3 + 3.0f*G4; float w3 = w0 - l3 + 3.0f*G4; float x4 = x0 - 1.0f + 4.0f*G4; // Offsets for last corner in (x,y,z,w) coords float y4 = y0 - 1.0f + 4.0f*G4; float z4 = z0 - 1.0f + 4.0f*G4; float w4 = w0 - 1.0f + 4.0f*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 float t0 = 0.6f - x0*x0 - y0*y0 - z0*z0 - w0*w0; if(t0 < 0.0f) n0 = 0.0f; else { t0 *= t0; n0 = t0 * t0 * grad(perm[ii+perm[jj+perm[kk+perm[ll]]]], x0, y0, z0, w0); } float t1 = 0.6f - x1*x1 - y1*y1 - z1*z1 - w1*w1; if(t1 < 0.0f) n1 = 0.0f; else { t1 *= t1; n1 = t1 * t1 * grad(perm[ii+i1+perm[jj+j1+perm[kk+k1+perm[ll+l1]]]], x1, y1, z1, w1); } float t2 = 0.6f - x2*x2 - y2*y2 - z2*z2 - w2*w2; if(t2 < 0.0f) n2 = 0.0f; else { t2 *= t2; n2 = t2 * t2 * grad(perm[ii+i2+perm[jj+j2+perm[kk+k2+perm[ll+l2]]]], x2, y2, z2, w2); } float t3 = 0.6f - x3*x3 - y3*y3 - z3*z3 - w3*w3; if(t3 < 0.0f) n3 = 0.0f; else { t3 *= t3; n3 = t3 * t3 * grad(perm[ii+i3+perm[jj+j3+perm[kk+k3+perm[ll+l3]]]], x3, y3, z3, w3); } float t4 = 0.6f - x4*x4 - y4*y4 - z4*z4 - w4*w4; if(t4 < 0.0f) n4 = 0.0f; else { t4 *= t4; n4 = t4 * t4 * grad(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 27.0f * (n0 + n1 + n2 + n3 + n4); // TODO: The scale factor is preliminary! }
// 2D simplex noise float noise(float x, float 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 float n0, n1, n2; // Noise contributions from the three corners // Skew the input space to determine which simplex cell we're in float s = (x+y)*F2; // Hairy factor for 2D float xs = x + s; float ys = y + s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); float t = (float)(i+j)*G2; float X0 = i-t; // Unskew the cell origin back to (x,y) space float Y0 = j-t; float x0 = x-X0; // The x,y distances from the cell origin float 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 float x1 = x0 - i1 + G2; // Offsets for middle corner in (x,y) unskewed coords float y1 = y0 - j1 + G2; float x2 = x0 - 1.0f + 2.0f * G2; // Offsets for last corner in (x,y) unskewed coords float y2 = y0 - 1.0f + 2.0f * G2; // Wrap the integer indices at 256, to avoid indexing perm[] out of bounds int ii = i % 256; int jj = j % 256; // Calculate the contribution from the three corners float t0 = 0.5f - x0*x0-y0*y0; if(t0 < 0.0f) n0 = 0.0f; else { t0 *= t0; n0 = t0 * t0 * grad(perm[ii+perm[jj]], x0, y0); } float t1 = 0.5f - x1*x1-y1*y1; if(t1 < 0.0f) n1 = 0.0f; else { t1 *= t1; n1 = t1 * t1 * grad(perm[ii+i1+perm[jj+j1]], x1, y1); } float t2 = 0.5f - x2*x2-y2*y2; if(t2 < 0.0f) n2 = 0.0f; else { t2 *= t2; n2 = t2 * t2 * grad(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 40.0f * (n0 + n1 + n2); // TODO: The scale factor is preliminary! }
static double _simplex_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 F3 = 1.0/3.0; 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 G3 = 1.0/6.0; // Very nice and simple unskew factor, too 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; // X Y Z order j1=0; k1=0; i2=1; j2=1; k2=0; } else if(x0>=z0) { i1=1; // X Z Y order j1=0; k1=0; i2=1; j2=0; k2=1; } else { i1=0; // Z X Y order j1=0; k1=1; i2=1; j2=0; k2=1; } } else // x0<y0 { if(y0<z0) { i1=0; // Z Y X order j1=0; k1=1; i2=0; j2=1; k2=1; } else if(x0<z0) { i1=0; // Y Z X order j1=1; k1=0; i2=0; j2=1; k2=1; } else { i1=0; // Y X Z order j1=1; k1=0; i2=1; j2=1; k2=0; } } // 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 = perm[ii+perm[jj+perm[kk]]] % 12; int gi1 = perm[ii+i1+perm[jj+j1+perm[kk+k1]]] % 12; int gi2 = perm[ii+i2+perm[jj+j2+perm[kk+k2]]] % 12; int gi3 = perm[ii+1+perm[jj+1+perm[kk+1]]] % 12; // 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); }
float Noise1234::pnoise( float x, float y, float z, float w, int px, int py, int pz, int pw ) { int ix0, iy0, iz0, iw0, ix1, iy1, iz1, iw1; float fx0, fy0, fz0, fw0, fx1, fy1, fz1, fw1; float s, t, r, q; float nxyz0, nxyz1, nxy0, nxy1, nx0, nx1, n0, n1; ix0 = FASTFLOOR( x ); // Integer part of x iy0 = FASTFLOOR( y ); // Integer part of y iz0 = FASTFLOOR( z ); // Integer part of y iw0 = FASTFLOOR( w ); // Integer part of w fx0 = x - ix0; // Fractional part of x fy0 = y - iy0; // Fractional part of y fz0 = z - iz0; // Fractional part of z fw0 = w - iw0; // Fractional part of w fx1 = fx0 - 1.0f; fy1 = fy0 - 1.0f; fz1 = fz0 - 1.0f; fw1 = fw0 - 1.0f; ix1 = (( ix0 + 1 ) % px ) & 0xff; // Wrap to 0..px-1 and wrap to 0..255 iy1 = (( iy0 + 1 ) % py ) & 0xff; // Wrap to 0..py-1 and wrap to 0..255 iz1 = (( iz0 + 1 ) % pz ) & 0xff; // Wrap to 0..pz-1 and wrap to 0..255 iw1 = (( iw0 + 1 ) % pw ) & 0xff; // Wrap to 0..pw-1 and wrap to 0..255 ix0 = ( ix0 % px ) & 0xff; iy0 = ( iy0 % py ) & 0xff; iz0 = ( iz0 % pz ) & 0xff; iw0 = ( iw0 % pw ) & 0xff; q = FADE( fw0 ); r = FADE( fz0 ); t = FADE( fy0 ); s = FADE( fx0 ); nxyz0 = grad(perm[ix0 + perm[iy0 + perm[iz0 + perm[iw0]]]], fx0, fy0, fz0, fw0); nxyz1 = grad(perm[ix0 + perm[iy0 + perm[iz0 + perm[iw1]]]], fx0, fy0, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad(perm[ix0 + perm[iy0 + perm[iz1 + perm[iw0]]]], fx0, fy0, fz1, fw0); nxyz1 = grad(perm[ix0 + perm[iy0 + perm[iz1 + perm[iw1]]]], fx0, fy0, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx0 = LERP ( r, nxy0, nxy1 ); nxyz0 = grad(perm[ix0 + perm[iy1 + perm[iz0 + perm[iw0]]]], fx0, fy1, fz0, fw0); nxyz1 = grad(perm[ix0 + perm[iy1 + perm[iz0 + perm[iw1]]]], fx0, fy1, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad(perm[ix0 + perm[iy1 + perm[iz1 + perm[iw0]]]], fx0, fy1, fz1, fw0); nxyz1 = grad(perm[ix0 + perm[iy1 + perm[iz1 + perm[iw1]]]], fx0, fy1, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx1 = LERP ( r, nxy0, nxy1 ); n0 = LERP( t, nx0, nx1 ); nxyz0 = grad(perm[ix1 + perm[iy0 + perm[iz0 + perm[iw0]]]], fx1, fy0, fz0, fw0); nxyz1 = grad(perm[ix1 + perm[iy0 + perm[iz0 + perm[iw1]]]], fx1, fy0, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad(perm[ix1 + perm[iy0 + perm[iz1 + perm[iw0]]]], fx1, fy0, fz1, fw0); nxyz1 = grad(perm[ix1 + perm[iy0 + perm[iz1 + perm[iw1]]]], fx1, fy0, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx0 = LERP ( r, nxy0, nxy1 ); nxyz0 = grad(perm[ix1 + perm[iy1 + perm[iz0 + perm[iw0]]]], fx1, fy1, fz0, fw0); nxyz1 = grad(perm[ix1 + perm[iy1 + perm[iz0 + perm[iw1]]]], fx1, fy1, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad(perm[ix1 + perm[iy1 + perm[iz1 + perm[iw0]]]], fx1, fy1, fz1, fw0); nxyz1 = grad(perm[ix1 + perm[iy1 + perm[iz1 + perm[iw1]]]], fx1, fy1, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx1 = LERP ( r, nxy0, nxy1 ); n1 = LERP( t, nx0, nx1 ); return 0.87f * ( LERP( s, n0, n1 ) ); }
/** 2D simplex noise */ GLfloat _mesa_noise2(GLfloat x, GLfloat y) { #define F2 0.366025403f /* F2 = 0.5*(sqrt(3.0)-1.0) */ #define G2 0.211324865f /* G2 = (3.0-Math.sqrt(3.0))/6.0 */ float n0, n1, n2; /* Noise contributions from the three corners */ /* Skew the input space to determine which simplex cell we're in */ float s = (x + y) * F2; /* Hairy factor for 2D */ float xs = x + s; float ys = y + s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); float t = (float) (i + j) * G2; float X0 = i - t; /* Unskew the cell origin back to (x,y) space */ float Y0 = j - t; float x0 = x - X0; /* The x,y distances from the cell origin */ float y0 = y - Y0; float x1, y1, x2, y2; int ii, jj; float t0, t1, t2; /* 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 */ x1 = x0 - i1 + G2; /* Offsets for middle corner in (x,y) unskewed coords */ y1 = y0 - j1 + G2; x2 = x0 - 1.0f + 2.0f * G2; /* Offsets for last corner in (x,y) unskewed coords */ y2 = y0 - 1.0f + 2.0f * G2; /* Wrap the integer indices at 256, to avoid indexing perm[] out of bounds */ ii = i % 256; jj = j % 256; /* Calculate the contribution from the three corners */ t0 = 0.5f - x0 * x0 - y0 * y0; if (t0 < 0.0f) n0 = 0.0f; else { t0 *= t0; n0 = t0 * t0 * grad2(perm[ii + perm[jj]], x0, y0); } t1 = 0.5f - x1 * x1 - y1 * y1; if (t1 < 0.0f) n1 = 0.0f; else { t1 *= t1; n1 = t1 * t1 * grad2(perm[ii + i1 + perm[jj + j1]], x1, y1); } t2 = 0.5f - x2 * x2 - y2 * y2; if (t2 < 0.0f) n2 = 0.0f; 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 40.0f * (n0 + n1 + n2); /* TODO: The scale factor is preliminary! */ }
float noise4( float x, float y, float z, float w ) { int ix0, iy0, iz0, iw0, ix1, iy1, iz1, iw1; float fx0, fy0, fz0, fw0, fx1, fy1, fz1, fw1; float s, t, r, q; float nxyz0, nxyz1, nxy0, nxy1, nx0, nx1, n0, n1; ix0 = FASTFLOOR( x ); // Integer part of x iy0 = FASTFLOOR( y ); // Integer part of y iz0 = FASTFLOOR( z ); // Integer part of y iw0 = FASTFLOOR( w ); // Integer part of w fx0 = x - ix0; // Fractional part of x fy0 = y - iy0; // Fractional part of y fz0 = z - iz0; // Fractional part of z fw0 = w - iw0; // Fractional part of w fx1 = fx0 - 1.0f; fy1 = fy0 - 1.0f; fz1 = fz0 - 1.0f; fw1 = fw0 - 1.0f; ix1 = ( ix0 + 1 ) & 0xff; // Wrap to 0..255 iy1 = ( iy0 + 1 ) & 0xff; iz1 = ( iz0 + 1 ) & 0xff; iw1 = ( iw0 + 1 ) & 0xff; ix0 = ix0 & 0xff; iy0 = iy0 & 0xff; iz0 = iz0 & 0xff; iw0 = iw0 & 0xff; q = FADE( fw0 ); r = FADE( fz0 ); t = FADE( fy0 ); s = FADE( fx0 ); nxyz0 = grad4(perm[ix0 + perm[iy0 + perm[iz0 + perm[iw0]]]], fx0, fy0, fz0, fw0); nxyz1 = grad4(perm[ix0 + perm[iy0 + perm[iz0 + perm[iw1]]]], fx0, fy0, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad4(perm[ix0 + perm[iy0 + perm[iz1 + perm[iw0]]]], fx0, fy0, fz1, fw0); nxyz1 = grad4(perm[ix0 + perm[iy0 + perm[iz1 + perm[iw1]]]], fx0, fy0, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx0 = LERP ( r, nxy0, nxy1 ); nxyz0 = grad4(perm[ix0 + perm[iy1 + perm[iz0 + perm[iw0]]]], fx0, fy1, fz0, fw0); nxyz1 = grad4(perm[ix0 + perm[iy1 + perm[iz0 + perm[iw1]]]], fx0, fy1, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad4(perm[ix0 + perm[iy1 + perm[iz1 + perm[iw0]]]], fx0, fy1, fz1, fw0); nxyz1 = grad4(perm[ix0 + perm[iy1 + perm[iz1 + perm[iw1]]]], fx0, fy1, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx1 = LERP ( r, nxy0, nxy1 ); n0 = LERP( t, nx0, nx1 ); nxyz0 = grad4(perm[ix1 + perm[iy0 + perm[iz0 + perm[iw0]]]], fx1, fy0, fz0, fw0); nxyz1 = grad4(perm[ix1 + perm[iy0 + perm[iz0 + perm[iw1]]]], fx1, fy0, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad4(perm[ix1 + perm[iy0 + perm[iz1 + perm[iw0]]]], fx1, fy0, fz1, fw0); nxyz1 = grad4(perm[ix1 + perm[iy0 + perm[iz1 + perm[iw1]]]], fx1, fy0, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx0 = LERP ( r, nxy0, nxy1 ); nxyz0 = grad4(perm[ix1 + perm[iy1 + perm[iz0 + perm[iw0]]]], fx1, fy1, fz0, fw0); nxyz1 = grad4(perm[ix1 + perm[iy1 + perm[iz0 + perm[iw1]]]], fx1, fy1, fz0, fw1); nxy0 = LERP( q, nxyz0, nxyz1 ); nxyz0 = grad4(perm[ix1 + perm[iy1 + perm[iz1 + perm[iw0]]]], fx1, fy1, fz1, fw0); nxyz1 = grad4(perm[ix1 + perm[iy1 + perm[iz1 + perm[iw1]]]], fx1, fy1, fz1, fw1); nxy1 = LERP( q, nxyz0, nxyz1 ); nx1 = LERP ( r, nxy0, nxy1 ); n1 = LERP( t, nx0, nx1 ); return 0.87f * ( LERP( s, n0, n1 ) ); }
/** 4D simplex noise with derivatives. * If the last four arguments are not null, the analytic derivative * (the 4D gradient of the scalar noise field) is also calculated. */ float sdnoise4( float x, float y, float z, float w, float *dnoise_dx, float *dnoise_dy, float *dnoise_dz, float *dnoise_dw) { float n0, n1, n2, n3, n4; // Noise contributions from the five corners float noise; // Return value float gx0, gy0, gz0, gw0, gx1, gy1, gz1, gw1; /* Gradients at simplex corners */ float gx2, gy2, gz2, gw2, gx3, gy3, gz3, gw3, gx4, gy4, gz4, gw4; float t20, t21, t22, t23, t24; float t40, t41, t42, t43, t44; // Skew the (x,y,z,w) space to determine which cell of 24 simplices we're in float s = (x + y + z + w) * F4; // Factor for 4D skewing float xs = x + s; float ys = y + s; float zs = z + s; float ws = w + s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); int k = FASTFLOOR(zs); int l = FASTFLOOR(ws); float t = (i + j + k + l) * G4; // Factor for 4D unskewing float X0 = i - t; // Unskew the cell origin back to (x,y,z,w) space float Y0 = j - t; float Z0 = k - t; float W0 = l - t; float x0 = x - X0; // The x,y,z,w distances from the cell origin float y0 = y - Y0; float z0 = z - Z0; float 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. // The method below is a reasonable way of finding the ordering of x,y,z,w // and then find the correct traversal order for the simplex we’re in. // First, six pair-wise comparisons are performed between each possible pair // of the four coordinates, and then the results are used to add up binary // bits for an integer index into a precomputed lookup table, simplex[]. int c1 = (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; // '&' is mostly faster than '+' 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. i1 = simplex[c][0]>=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. float x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w) coords float y1 = y0 - j1 + G4; float z1 = z0 - k1 + G4; float w1 = w0 - l1 + G4; float x2 = x0 - i2 + 2.0f * G4; // Offsets for third corner in (x,y,z,w) coords float y2 = y0 - j2 + 2.0f * G4; float z2 = z0 - k2 + 2.0f * G4; float w2 = w0 - l2 + 2.0f * G4; float x3 = x0 - i3 + 3.0f * G4; // Offsets for fourth corner in (x,y,z,w) coords float y3 = y0 - j3 + 3.0f * G4; float z3 = z0 - k3 + 3.0f * G4; float w3 = w0 - l3 + 3.0f * G4; float x4 = x0 - 1.0f + 4.0f * G4; // Offsets for last corner in (x,y,z,w) coords float y4 = y0 - 1.0f + 4.0f * G4; float z4 = z0 - 1.0f + 4.0f * G4; float w4 = w0 - 1.0f + 4.0f * 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 float t0 = 0.6f - x0*x0 - y0*y0 - z0*z0 - w0*w0; if(t0 < 0.0f) n0 = t0 = t20 = t40 = gx0 = gy0 = gz0 = gw0 = 0.0f; else { t20 = t0 * t0; t40 = t20 * t20; grad4(perm[ii+perm[jj+perm[kk+perm[ll]]]], &gx0, &gy0, &gz0, &gw0); n0 = t40 * ( gx0 * x0 + gy0 * y0 + gz0 * z0 + gw0 * w0 ); } float t1 = 0.6f - x1*x1 - y1*y1 - z1*z1 - w1*w1; if(t1 < 0.0f) n1 = t1 = t21 = t41 = gx1 = gy1 = gz1 = gw1 = 0.0f; else { t21 = t1 * t1; t41 = t21 * t21; grad4(perm[ii+i1+perm[jj+j1+perm[kk+k1+perm[ll+l1]]]], &gx1, &gy1, &gz1, &gw1); n1 = t41 * ( gx1 * x1 + gy1 * y1 + gz1 * z1 + gw1 * w1 ); } float t2 = 0.6f - x2*x2 - y2*y2 - z2*z2 - w2*w2; if(t2 < 0.0f) n2 = t2 = t22 = t42 = gx2 = gy2 = gz2 = gw2 = 0.0f; else { t22 = t2 * t2; t42 = t22 * t22; grad4(perm[ii+i2+perm[jj+j2+perm[kk+k2+perm[ll+l2]]]], &gx2, &gy2, &gz2, &gw2); n2 = t42 * ( gx2 * x2 + gy2 * y2 + gz2 * z2 + gw2 * w2 ); } float t3 = 0.6f - x3*x3 - y3*y3 - z3*z3 - w3*w3; if(t3 < 0.0f) n3 = t3 = t23 = t43 = gx3 = gy3 = gz3 = gw3 = 0.0f; else { t23 = t3 * t3; t43 = t23 * t23; grad4(perm[ii+i3+perm[jj+j3+perm[kk+k3+perm[ll+l3]]]], &gx3, &gy3, &gz3, &gw3); n3 = t43 * ( gx3 * x3 + gy3 * y3 + gz3 * z3 + gw3 * w3 ); } float t4 = 0.6f - x4*x4 - y4*y4 - z4*z4 - w4*w4; if(t4 < 0.0f) n4 = t4 = t24 = t44 = gx4 = gy4 = gz4 = gw4 = 0.0f; else { t24 = t4 * t4; t44 = t24 * t24; grad4(perm[ii+1+perm[jj+1+perm[kk+1+perm[ll+1]]]], &gx4, &gy4, &gz4, &gw4); n4 = t44 * ( gx4 * x4 + gy4 * y4 + gz4 * z4 + gw4 * w4 ); } // Sum up and scale the result to cover the range [-1,1] noise = 27.0f * (n0 + n1 + n2 + n3 + n4); // TODO: The scale factor is preliminary! /* Compute derivative, if requested by supplying non-null pointers * for the last four arguments */ if( ( dnoise_dx != 0 ) && ( dnoise_dy != 0 ) && ( dnoise_dz != 0 ) && ( dnoise_dw != 0 ) ) { /* A straight, unoptimised calculation would be like: * *dnoise_dx = -8.0f * t20 * t0 * x0 * dot(gx0, gy0, gz0, gw0, x0, y0, z0, w0) + t40 * gx0; * *dnoise_dy = -8.0f * t20 * t0 * y0 * dot(gx0, gy0, gz0, gw0, x0, y0, z0, w0) + t40 * gy0; * *dnoise_dz = -8.0f * t20 * t0 * z0 * dot(gx0, gy0, gz0, gw0, x0, y0, z0, w0) + t40 * gz0; * *dnoise_dw = -8.0f * t20 * t0 * w0 * dot(gx0, gy0, gz0, gw0, x0, y0, z0, w0) + t40 * gw0; * *dnoise_dx += -8.0f * t21 * t1 * x1 * dot(gx1, gy1, gz1, gw1, x1, y1, z1, w1) + t41 * gx1; * *dnoise_dy += -8.0f * t21 * t1 * y1 * dot(gx1, gy1, gz1, gw1, x1, y1, z1, w1) + t41 * gy1; * *dnoise_dz += -8.0f * t21 * t1 * z1 * dot(gx1, gy1, gz1, gw1, x1, y1, z1, w1) + t41 * gz1; * *dnoise_dw = -8.0f * t21 * t1 * w1 * dot(gx1, gy1, gz1, gw1, x1, y1, z1, w1) + t41 * gw1; * *dnoise_dx += -8.0f * t22 * t2 * x2 * dot(gx2, gy2, gz2, gw2, x2, y2, z2, w2) + t42 * gx2; * *dnoise_dy += -8.0f * t22 * t2 * y2 * dot(gx2, gy2, gz2, gw2, x2, y2, z2, w2) + t42 * gy2; * *dnoise_dz += -8.0f * t22 * t2 * z2 * dot(gx2, gy2, gz2, gw2, x2, y2, z2, w2) + t42 * gz2; * *dnoise_dw += -8.0f * t22 * t2 * w2 * dot(gx2, gy2, gz2, gw2, x2, y2, z2, w2) + t42 * gw2; * *dnoise_dx += -8.0f * t23 * t3 * x3 * dot(gx3, gy3, gz3, gw3, x3, y3, z3, w3) + t43 * gx3; * *dnoise_dy += -8.0f * t23 * t3 * y3 * dot(gx3, gy3, gz3, gw3, x3, y3, z3, w3) + t43 * gy3; * *dnoise_dz += -8.0f * t23 * t3 * z3 * dot(gx3, gy3, gz3, gw3, x3, y3, z3, w3) + t43 * gz3; * *dnoise_dw += -8.0f * t23 * t3 * w3 * dot(gx3, gy3, gz3, gw3, x3, y3, z3, w3) + t43 * gw3; * *dnoise_dx += -8.0f * t24 * t4 * x4 * dot(gx4, gy4, gz4, gw4, x4, y4, z4, w4) + t44 * gx4; * *dnoise_dy += -8.0f * t24 * t4 * y4 * dot(gx4, gy4, gz4, gw4, x4, y4, z4, w4) + t44 * gy4; * *dnoise_dz += -8.0f * t24 * t4 * z4 * dot(gx4, gy4, gz4, gw4, x4, y4, z4, w4) + t44 * gz4; * *dnoise_dw += -8.0f * t24 * t4 * w4 * dot(gx4, gy4, gz4, gw4, x4, y4, z4, w4) + t44 * gw4; */ float temp0 = t20 * t0 * ( gx0 * x0 + gy0 * y0 + gz0 * z0 + gw0 * w0 ); *dnoise_dx = temp0 * x0; *dnoise_dy = temp0 * y0; *dnoise_dz = temp0 * z0; *dnoise_dw = temp0 * w0; float temp1 = t21 * t1 * ( gx1 * x1 + gy1 * y1 + gz1 * z1 + gw1 * w1 ); *dnoise_dx += temp1 * x1; *dnoise_dy += temp1 * y1; *dnoise_dz += temp1 * z1; *dnoise_dw += temp1 * w1; float temp2 = t22 * t2 * ( gx2 * x2 + gy2 * y2 + gz2 * z2 + gw2 * w2 ); *dnoise_dx += temp2 * x2; *dnoise_dy += temp2 * y2; *dnoise_dz += temp2 * z2; *dnoise_dw += temp2 * w2; float temp3 = t23 * t3 * ( gx3 * x3 + gy3 * y3 + gz3 * z3 + gw3 * w3 ); *dnoise_dx += temp3 * x3; *dnoise_dy += temp3 * y3; *dnoise_dz += temp3 * z3; *dnoise_dw += temp3 * w3; float temp4 = t24 * t4 * ( gx4 * x4 + gy4 * y4 + gz4 * z4 + gw4 * w4 ); *dnoise_dx += temp4 * x4; *dnoise_dy += temp4 * y4; *dnoise_dz += temp4 * z4; *dnoise_dw += temp4 * w4; *dnoise_dx *= -8.0f; *dnoise_dy *= -8.0f; *dnoise_dz *= -8.0f; *dnoise_dw *= -8.0f; *dnoise_dx += t40 * gx0 + t41 * gx1 + t42 * gx2 + t43 * gx3 + t44 * gx4; *dnoise_dy += t40 * gy0 + t41 * gy1 + t42 * gy2 + t43 * gy3 + t44 * gy4; *dnoise_dz += t40 * gz0 + t41 * gz1 + t42 * gz2 + t43 * gz3 + t44 * gz4; *dnoise_dw += t40 * gw0 + t41 * gw1 + t42 * gw2 + t43 * gw3 + t44 * gw4; *dnoise_dx *= 27.0f; /* Scale derivative to match the noise scaling */ *dnoise_dy *= 27.0f; *dnoise_dz *= 27.0f; *dnoise_dw *= 27.0f; } return noise; }
/** 3D simplex noise with derivatives. * If the last tthree arguments are not null, the analytic derivative * (the 3D gradient of the scalar noise field) is also calculated. */ float sdnoise3( float x, float y, float z, float *dnoise_dx, float *dnoise_dy, float *dnoise_dz ) { float n0, n1, n2, n3; /* Noise contributions from the four simplex corners */ float noise; /* Return value */ float gx0, gy0, gz0, gx1, gy1, gz1; /* Gradients at simplex corners */ float gx2, gy2, gz2, gx3, gy3, gz3; /* Skew the input space to determine which simplex cell we're in */ float s = (x+y+z)*F3; /* Very nice and simple skew factor for 3D */ float xs = x+s; float ys = y+s; float zs = z+s; int i = FASTFLOOR(xs); int j = FASTFLOOR(ys); int k = FASTFLOOR(zs); float t = (float)(i+j+k)*G3; float X0 = i-t; /* Unskew the cell origin back to (x,y,z) space */ float Y0 = j-t; float Z0 = k-t; float x0 = x-X0; /* The x,y,z distances from the cell origin */ float y0 = y-Y0; float 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 */ /* TODO: 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 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. */ float x1 = x0 - i1 + G3; /* Offsets for second corner in (x,y,z) coords */ float y1 = y0 - j1 + G3; float z1 = z0 - k1 + G3; float x2 = x0 - i2 + 2.0f * G3; /* Offsets for third corner in (x,y,z) coords */ float y2 = y0 - j2 + 2.0f * G3; float z2 = z0 - k2 + 2.0f * G3; float x3 = x0 - 1.0f + 3.0f * G3; /* Offsets for last corner in (x,y,z) coords */ float y3 = y0 - 1.0f + 3.0f * G3; float z3 = z0 - 1.0f + 3.0f * G3; /* Wrap the integer indices at 256, to avoid indexing perm[] out of bounds */ int ii = i % 256; int jj = j % 256; int kk = k % 256; /* Calculate the contribution from the four corners */ float t0 = 0.6f - x0*x0 - y0*y0 - z0*z0; float t20, t40; if(t0 < 0.0f) n0 = t0 = t20 = t40 = gx0 = gy0 = gz0 = 0.0f; else { grad3( perm[ii + perm[jj + perm[kk]]], &gx0, &gy0, &gz0 ); t20 = t0 * t0; t40 = t20 * t20; n0 = t40 * ( gx0 * x0 + gy0 * y0 + gz0 * z0 ); } float t1 = 0.6f - x1*x1 - y1*y1 - z1*z1; float t21, t41; if(t1 < 0.0f) n1 = t1 = t21 = t41 = gx1 = gy1 = gz1 = 0.0f; else { grad3( perm[ii + i1 + perm[jj + j1 + perm[kk + k1]]], &gx1, &gy1, &gz1 ); t21 = t1 * t1; t41 = t21 * t21; n1 = t41 * ( gx1 * x1 + gy1 * y1 + gz1 * z1 ); } float t2 = 0.6f - x2*x2 - y2*y2 - z2*z2; float t22, t42; if(t2 < 0.0f) n2 = t2 = t22 = t42 = gx2 = gy2 = gz2 = 0.0f; else { grad3( perm[ii + i2 + perm[jj + j2 + perm[kk + k2]]], &gx2, &gy2, &gz2 ); t22 = t2 * t2; t42 = t22 * t22; n2 = t42 * ( gx2 * x2 + gy2 * y2 + gz2 * z2 ); } float t3 = 0.6f - x3*x3 - y3*y3 - z3*z3; float t23, t43; if(t3 < 0.0f) n3 = t3 = t23 = t43 = gx3 = gy3 = gz3 = 0.0f; else { grad3( perm[ii + 1 + perm[jj + 1 + perm[kk + 1]]], &gx3, &gy3, &gz3 ); t23 = t3 * t3; t43 = t23 * t23; n3 = t43 * ( gx3 * x3 + gy3 * y3 + gz3 * z3 ); } /* Add contributions from each corner to get the final noise value. * The result is scaled to return values in the range [-1,1] */ noise = 28.0f * (n0 + n1 + n2 + n3); /* Compute derivative, if requested by supplying non-null pointers * for the last three arguments */ if( ( dnoise_dx != 0 ) && ( dnoise_dy != 0 ) && ( dnoise_dz != 0 )) { /* A straight, unoptimised calculation would be like: * *dnoise_dx = -8.0f * t20 * t0 * x0 * dot(gx0, gy0, gz0, x0, y0, z0) + t40 * gx0; * *dnoise_dy = -8.0f * t20 * t0 * y0 * dot(gx0, gy0, gz0, x0, y0, z0) + t40 * gy0; * *dnoise_dz = -8.0f * t20 * t0 * z0 * dot(gx0, gy0, gz0, x0, y0, z0) + t40 * gz0; * *dnoise_dx += -8.0f * t21 * t1 * x1 * dot(gx1, gy1, gz1, x1, y1, z1) + t41 * gx1; * *dnoise_dy += -8.0f * t21 * t1 * y1 * dot(gx1, gy1, gz1, x1, y1, z1) + t41 * gy1; * *dnoise_dz += -8.0f * t21 * t1 * z1 * dot(gx1, gy1, gz1, x1, y1, z1) + t41 * gz1; * *dnoise_dx += -8.0f * t22 * t2 * x2 * dot(gx2, gy2, gz2, x2, y2, z2) + t42 * gx2; * *dnoise_dy += -8.0f * t22 * t2 * y2 * dot(gx2, gy2, gz2, x2, y2, z2) + t42 * gy2; * *dnoise_dz += -8.0f * t22 * t2 * z2 * dot(gx2, gy2, gz2, x2, y2, z2) + t42 * gz2; * *dnoise_dx += -8.0f * t23 * t3 * x3 * dot(gx3, gy3, gz3, x3, y3, z3) + t43 * gx3; * *dnoise_dy += -8.0f * t23 * t3 * y3 * dot(gx3, gy3, gz3, x3, y3, z3) + t43 * gy3; * *dnoise_dz += -8.0f * t23 * t3 * z3 * dot(gx3, gy3, gz3, x3, y3, z3) + t43 * gz3; */ float temp0 = t20 * t0 * ( gx0 * x0 + gy0 * y0 + gz0 * z0 ); *dnoise_dx = temp0 * x0; *dnoise_dy = temp0 * y0; *dnoise_dz = temp0 * z0; float temp1 = t21 * t1 * ( gx1 * x1 + gy1 * y1 + gz1 * z1 ); *dnoise_dx += temp1 * x1; *dnoise_dy += temp1 * y1; *dnoise_dz += temp1 * z1; float temp2 = t22 * t2 * ( gx2 * x2 + gy2 * y2 + gz2 * z2 ); *dnoise_dx += temp2 * x2; *dnoise_dy += temp2 * y2; *dnoise_dz += temp2 * z2; float temp3 = t23 * t3 * ( gx3 * x3 + gy3 * y3 + gz3 * z3 ); *dnoise_dx += temp3 * x3; *dnoise_dy += temp3 * y3; *dnoise_dz += temp3 * z3; *dnoise_dx *= -8.0f; *dnoise_dy *= -8.0f; *dnoise_dz *= -8.0f; *dnoise_dx += t40 * gx0 + t41 * gx1 + t42 * gx2 + t43 * gx3; *dnoise_dy += t40 * gy0 + t41 * gy1 + t42 * gy2 + t43 * gy3; *dnoise_dz += t40 * gz0 + t41 * gz1 + t42 * gz2 + t43 * gz3; *dnoise_dx *= 28.0f; /* Scale derivative to match the noise scaling */ *dnoise_dy *= 28.0f; *dnoise_dz *= 28.0f; } return noise; }
/** 2D simplex noise with derivatives. * If the last two arguments are not null, the analytic derivative * (the 2D gradient of the scalar noise field) is also calculated. */ float sdnoise2( float x, float y, float *dnoise_dx, float *dnoise_dy ) { float n0, n1, n2; /* Noise contributions from the three simplex corners */ float gx0, gy0, gx1, gy1, gx2, gy2; /* Gradients at simplex corners */ /* Skew the input space to determine which simplex cell we're in */ float s = ( x + y ) * F2; /* Hairy factor for 2D */ float xs = x + s; float ys = y + s; int i = FASTFLOOR( xs ); int j = FASTFLOOR( ys ); float t = ( float ) ( i + j ) * G2; float X0 = i - t; /* Unskew the cell origin back to (x,y) space */ float Y0 = j - t; float x0 = x - X0; /* The x,y distances from the cell origin */ float 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 */ float x1 = x0 - i1 + G2; /* Offsets for middle corner in (x,y) unskewed coords */ float y1 = y0 - j1 + G2; float x2 = x0 - 1.0f + 2.0f * G2; /* Offsets for last corner in (x,y) unskewed coords */ float y2 = y0 - 1.0f + 2.0f * G2; /* Wrap the integer indices at 256, to avoid indexing perm[] out of bounds */ int ii = i % 256; int jj = j % 256; /* Calculate the contribution from the three corners */ float t0 = 0.5f - x0 * x0 - y0 * y0; float t20, t40; if( t0 < 0.0f ) t40 = t20 = t0 = n0 = gx0 = gy0 = 0.0f; /* No influence */ else { grad2( perm[ii + perm[jj]], &gx0, &gy0 ); t20 = t0 * t0; t40 = t20 * t20; n0 = t40 * ( gx0 * x0 + gy0 * y0 ); } float t1 = 0.5f - x1 * x1 - y1 * y1; float t21, t41; if( t1 < 0.0f ) t21 = t41 = t1 = n1 = gx1 = gy1 = 0.0f; /* No influence */ else { grad2( perm[ii + i1 + perm[jj + j1]], &gx1, &gy1 ); t21 = t1 * t1; t41 = t21 * t21; n1 = t41 * ( gx1 * x1 + gy1 * y1 ); } float t2 = 0.5f - x2 * x2 - y2 * y2; float t22, t42; if( t2 < 0.0f ) t42 = t22 = t2 = n2 = gx2 = gy2 = 0.0f; /* No influence */ else { grad2( perm[ii + 1 + perm[jj + 1]], &gx2, &gy2 ); t22 = t2 * t2; t42 = t22 * t22; n2 = t42 * ( gx2 * x2 + gy2 * y2 ); } /* Add contributions from each corner to get the final noise value. * The result is scaled to return values in the interval [-1,1]. */ float noise = 40.0f * ( n0 + n1 + n2 ); /* Compute derivative, if requested by supplying non-null pointers * for the last two arguments */ if( ( dnoise_dx != 0 ) && ( dnoise_dy != 0 ) ) { /* A straight, unoptimised calculation would be like: * *dnoise_dx = -8.0f * t20 * t0 * x0 * ( gx0 * x0 + gy0 * y0 ) + t40 * gx0; * *dnoise_dy = -8.0f * t20 * t0 * y0 * ( gx0 * x0 + gy0 * y0 ) + t40 * gy0; * *dnoise_dx += -8.0f * t21 * t1 * x1 * ( gx1 * x1 + gy1 * y1 ) + t41 * gx1; * *dnoise_dy += -8.0f * t21 * t1 * y1 * ( gx1 * x1 + gy1 * y1 ) + t41 * gy1; * *dnoise_dx += -8.0f * t22 * t2 * x2 * ( gx2 * x2 + gy2 * y2 ) + t42 * gx2; * *dnoise_dy += -8.0f * t22 * t2 * y2 * ( gx2 * x2 + gy2 * y2 ) + t42 * gy2; */ float temp0 = t20 * t0 * ( gx0* x0 + gy0 * y0 ); *dnoise_dx = temp0 * x0; *dnoise_dy = temp0 * y0; float temp1 = t21 * t1 * ( gx1 * x1 + gy1 * y1 ); *dnoise_dx += temp1 * x1; *dnoise_dy += temp1 * y1; float temp2 = t22 * t2 * ( gx2* x2 + gy2 * y2 ); *dnoise_dx += temp2 * x2; *dnoise_dy += temp2 * y2; *dnoise_dx *= -8.0f; *dnoise_dy *= -8.0f; *dnoise_dx += t40 * gx0 + t41 * gx1 + t42 * gx2; *dnoise_dy += t40 * gy0 + t41 * gy1 + t42 * gy2; *dnoise_dx *= 40.0f; /* Scale derivative to match the noise scaling */ *dnoise_dy *= 40.0f; } return noise; }