// 4D simplex Noise
public double Noise4D(double x, double y, double z, double w)
{
// The skewing and unskewing factors are hairy again for the 4D case
double F4 = (Math.Sqrt(5.0) - 1.0) / 4.0;
double G4 = (5.0 - Math.Sqrt(5.0)) / 20.0;
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 = NoiseHelper.FastFloor(x + s);
int j = NoiseHelper.FastFloor(y + s);
int k = NoiseHelper.FastFloor(z + s);
int l = NoiseHelper.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.
// 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.
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;
}
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;
}
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;
}
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;
}
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;
}
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));
}
// 2D simplex Noise
public double Noise2D(double x, double y)
{
double n0, n1, n2; // Noise contributions from the three corners
// Skew the input space to determine which simplex cell we're in
double F2 = 0.5 * (Math.Sqrt(3.0) - 1.0);
double s = (x + y) * F2; // Hairy factor for 2D
int i = NoiseHelper.FastFloor(x + s);
int j = NoiseHelper.FastFloor(y + s);
double G2 = (3.0 - Math.Sqrt(3.0)) / 6.0;
double t = (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;
// Work out the hashed gradient indices of the three simplex corners
int ii = i & 255;
int jj = j & 255;
int gi0 = perm[ii + perm[jj]] % 12;
int gi1 = perm[ii + i1 + perm[jj + j1]] % 12;
int gi2 = perm[ii + 1 + perm[jj + 1]] % 12;
// Calculate the contribution from the three corners
double t0 = 0.5 - x0 * x0 - y0 * y0;
if (t0 < 0)
{
n0 = 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;
}
else
{
t1 *= t1;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1);
}
double t2 = 0.5 - x2 * x2 - y2 * y2;
if (t2 < 0)
{
n2 = 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 Noise3D(double x, double y, double z)
{
double n0, n1, n2, n3; // Noise contributions from the four corners
// Skew the input space to determine which simplex cell we're in
const double F3 = 1.0 / 3.0;
double s = (x + y + z) * F3; // Very nice and simple skew factor for 3D
int i = NoiseHelper.FastFloor(x + s);
int j = NoiseHelper.FastFloor(y + s);
int k = NoiseHelper.FastFloor(z + s);
const 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 = 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
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 = 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;
}
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;
}
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;
}
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;
}
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));
}
public double Noise3D(double x, double y, double z, NoiseQuality quality)
{
//returns a noise value between -1.5 and 1.5
int ix0, iy0, ix1, iy1, iz0, iz1;
double fx0, fy0, fz0, fx1, fy1, fz1;
double s, t, r;
double nxy0, nxy1, nx0, nx1, n0, n1;
ix0 = NoiseHelper.FastFloor(x); // Integer part of x
iy0 = NoiseHelper.FastFloor(y); // Integer part of y
iz0 = NoiseHelper.FastFloor(z); // Integer part of z
// Fractional part of x, y, z
if (quality == NoiseQuality.Low)
{
fx0 = x - ix0; fy0 = y - iy0; fz0 = z - iz0;
}
else if (quality == NoiseQuality.Standard)
{
fx0 = MathEx.SCurve3(x - ix0); fy0 = MathEx.SCurve3(y - iy0); fz0 = MathEx.SCurve3(z - iz0);
}
else
{
fx0 = MathEx.SCurve5(x - ix0); fy0 = MathEx.SCurve5(y - iy0); fz0 = MathEx.SCurve5(z - iz0);
}
fx1 = fx0 - 1.0f;
fy1 = fy0 - 1.0f;
fz1 = fz0 - 1.0f;
ix1 = (ix0 + 1) & 0xff; // Wrap to 0..255
iy1 = (iy0 + 1) & 0xff;
iz1 = (iz0 + 1) & 0xff;
ix0 = ix0 & 0xff;
iy0 = iy0 & 0xff;
iz0 = iz0 & 0xff;
r = FADE(fz0);
t = FADE(fy0);
s = FADE(fx0);
nxy0 = GRAD3(m_perm[ix0 + m_perm[iy0 + m_perm[iz0]]], fx0, fy0, fz0);
nxy1 = GRAD3(m_perm[ix0 + m_perm[iy0 + m_perm[iz1]]], fx0, fy0, fz1);
nx0 = LERP(r, nxy0, nxy1);
nxy0 = GRAD3(m_perm[ix0 + m_perm[iy1 + m_perm[iz0]]], fx0, fy1, fz0);
nxy1 = GRAD3(m_perm[ix0 + m_perm[iy1 + m_perm[iz1]]], fx0, fy1, fz1);
nx1 = LERP(r, nxy0, nxy1);
n0 = LERP(t, nx0, nx1);
nxy0 = GRAD3(m_perm[ix1 + m_perm[iy0 + m_perm[iz0]]], fx1, fy0, fz0);
nxy1 = GRAD3(m_perm[ix1 + m_perm[iy0 + m_perm[iz1]]], fx1, fy0, fz1);
nx0 = LERP(r, nxy0, nxy1);
nxy0 = GRAD3(m_perm[ix1 + m_perm[iy1 + m_perm[iz0]]], fx1, fy1, fz0);
nxy1 = GRAD3(m_perm[ix1 + m_perm[iy1 + m_perm[iz1]]], fx1, fy1, fz1);
nx1 = LERP(r, nxy0, nxy1);
n1 = LERP(t, nx0, nx1);
return(0.936f * LERP(s, n0, n1));
}