//recursively lerp the influences of the neighboring grid points to the pixel
//in the first iteration, the last index is split between 0 and 1
// (0, 0, ..., 0) (0, 1, ..., 0) vs. (0, 0, ..., 1) (0, 1, ..., 1)
// the weight by which these two groups will be lerped is the faded value
// weight = last coordinate of ((the map coordinate of v) - (the basis node))
//then for the second iteration, the second last index is split between 0 and 1
// weight = 2nd last of ...
//and so on
private float RecurLerpTree(
PerlinNode[] influencers, VecN[] deviations, VecN w,
int left, int right, int iteration
)
{
if (right - left == 1)
{
//only two elements to be processed
//by def. the iteration number will be d-1 at this point
//calculate left and right influences with dist dot grad
float lInf = w.Subtract(deviations[left]).Dot(influencers[left].gradient);
float rInf = w.Subtract(deviations[right]).Dot(influencers[right].gradient);
//lerp the two influences using the weight of the first digit
//this is because the 01 difference of the deviations are at the first digit
float lerped = Lerp(
lInf,
rInf,
Fade(w.coord[d - iteration - 1])
);
return(lerped);
}
//for more than one element, by def the length of influencers should always be even
//lerp the left and right influences using the last iteration'th value of w\
return(Lerp(
RecurLerpTree(influencers, deviations, w, left, left + (right - left) / 2, iteration + 1),
RecurLerpTree(influencers, deviations, w, left + (right - left) / 2 + 1, right, iteration + 1),
Fade(w.coord[d - iteration - 1])
));
}
//return a new VecN instance representing the result of a VecN instance multiplied
//by a scalar value
public static VecN Scale(VecN a, float b)
{
float[] c = new float[a.d];
for (int i = 0; i < a.d; i++)
{
c[i] = a.coord[i] * b;
}
return(new VecN(c));
}
//return the radian angle of a VecN instance
public static float Angle(VecN a)
{
if (a.d != 2)
{
throw new System.ArgumentException("angle only applies to 2D vectors, not "
+ a.d + "D");
}
return(Mathf.Atan2(a.coord[1], a.coord[0]));
}
//returns a new VecN instance representing the modulo b version of a VecN instance
//basically applies mod b to every coordinate of a
public static VecN Mod(VecN a, float b)
{
float[] coord = new float[a.d];
for (int i = 0; i < a.d; i++)
{
coord[i] = a.coord[i] % b;
}
return(new VecN(coord));
}
public PerlinNode(VecN coord)
{
this.coord = coord;
//generate random d-dimensional unit vector
gradient = Gaussian.UnitVecN(coord.d);
//gradient = new VecN(new float[] {
//1, 0
//});
}
//return a new VecN instance representing the normalized result of
//a VecN instance
public static VecN Normalize(VecN a)
{
float l = a.Norm();
if (l != 0.0f)
{
return(a.Scale(1 / l));
}
else
{
return(a.Scale(1.0f)); //not returning a for the sake of cloning
}
}
//return a new VecN instance (for N = 3) representing the cross product
//of two 3D vectors
public static VecN Cross(VecN a, VecN b)
{
if (a.d != 3 || b.d != 3)
{
throw new System.ArgumentException("cross product only takes 3D vectors, not "
+ a.d + " and " + b.d + "D vectors");
}
else
{
return(new VecN(new float[] {
a.coord[1] * b.coord[2] - a.coord[2] * b.coord[1],
a.coord[2] * b.coord[0] - a.coord[0] * b.coord[2],
a.coord[0] * b.coord[1] - a.coord[1] * b.coord[0],
}));
}
}
//find the base adjacent grid-coordinate for a given d-dimensional vector
//the first element returned is the basis node
//the second element is the deviation vector that goes from the pixel to the basis node
public ArrayList FindBasis(VecN a)
{
if (a.d != d)
{
throw new System.ArgumentException(
"dimension of pixel (" + a.d +
") does not match the dimension of this noise map (" + d + ")");
}
else
{
int[] coord = new int[a.d];
float[] deviation = new float[a.d];
for (int i = 0; i < a.d; i++)
{
//fit the i'th item of the coordinate into the range [0, w]
float temp = a.coord[i];
if (temp < 0) //for negative values of a coordinate
{
if (temp % w == 0.0f)
{
temp = 0;
}
else
{
temp += (int)(-1 * temp / w + 1) * w;
}
}
else if (temp > w) //for positive overflow (repeat)
{
temp = temp % w;
}
deviation[i] = temp % 1.0f;
//round the coordinate value down to get the basis value
coord[i] = (int)temp;
}
int l = Unwrap(coord);
if (l < 0 || l > nodes.Length)
{
throw new ArgumentException("length is out of bounds, " + l);
}
ArrayList result = new ArrayList();
result.Add(nodes[l]);
result.Add(new VecN(deviation));
return(result);
}
}
//convert a Vec-d instance to the corresponding
//length coordinate on the space filling curve
public int Unwrap(VecN a)
{
if (a.d != d)
{
throw new System.ArgumentException(
"the coordinate's dimension " + a.d +
" does not match with the noise map's dimension " + d);
}
int result = 0;
for (int i = 0; i < d; i++)
{
result += (int)a.coord[i] * (int)Mathf.Pow(w, i);
}
return(result);
}
//return a new VecN instance representing the difference of two
//VecN instances with the same dimensions (a - b), not (b - a)
public static VecN Subtract(VecN a, VecN b)
{
if (a.d != b.d)
{
throw new System.ArgumentException("dimensions do not agree when calculating difference: "
+ a.d + " and " + b.d);
}
else
{
float[] c = a.coord.Clone() as float[];
for (int i = 0; i < a.d; i++)
{
c[i] = a.coord[i] - b.coord[i];
}
return(new VecN(c));
}
}
//notice that none of the functions below modify the original VecN instance
//return the dot product of two vectors with the same dimensions
public static float Dot(VecN a, VecN b)
{
if (a.d != b.d)
{
throw new System.ArgumentException("dimensions do not agree when calculating dot product: "
+ a.d + " and " + b.d);
}
else
{
float result = 0.0f;
//Debug.Log("\t\tdot called with returning:");
for (int i = 0; i < a.d; i++)
{
result += a.coord[i] * b.coord[i];
//Debug.Log("\t\t\t" + i + "th value is" + (a.coord[i] * b.coord[i]));
}
return(result);
}
}
/*
* The noise map is noted by a grid of d-dimensional vectors going from:
* (0, 0, ..., 0), (1, 0, ..., 0), ..., (w-1, 0, ..., 0)
* (0, 1, ..., 0), (1, 1, ..., 0), ..., (w-1, 1, ..., 0)
* ...(w-1, w-1, ..., w-1)
* and each vector corresponds to a measurement of length for a space filling curve
* which works similar to a w-based number:
* digit 0 * 1 + digit 1 * w + ... + digit d-1 * w ^ d-1
*/
//beware that this method is not very efficient as it uses a rectangular grid instead of triangular
//initialize with flavor specification
public Perlin(int w, int d, int f)
{
this.w = w;
this.d = d;
this.f = f;
//initiate the relative positions of adjacent grids to the basis grid
//use a cursor going from 0 to 2^n-1 to iterate across all the
//possible deviations associated with adjacent coordinates
grid_deviations = new VecN[(int)Mathf.Pow(2, d)];
for (int i = 0; i < grid_deviations.Length; i++)
{
float[] c = new float[d];
for (int j = 0; j < d; j++)
{
c[j] = (i >> j) & 1; //gets the bit value of the j'th digit of i
}
grid_deviations[i] = new VecN(c);
}
//there are w * ... * w pivots in total, w^d
nodes = new PerlinNode[(int)Mathf.Pow(w, d)];
//iterate through the space filling curve by length
//populate the noise map and configure the adjacency relationships
for (int i = 0; i < nodes.Length; i++)
{
//populate the map and set the gradient vector (see constructor for pn)
PerlinNode pn = new PerlinNode(Wrap(i));
nodes[i] = pn;
}
for (int i = 0; i < nodes.Length; i++)
{
//setup adjacency relationships
nodes[i].neighbors = FindAdjacency(nodes[i]);
}
//Debug.Log(nodes[0].PrintNeighbors());
}
//find the noise in the noisespace corresponding to v
public float Noise(VecN v)
{
if (v.d != d)
{
throw new System.ArgumentException(
"dimension of pixel (" + v.d +
") does not match the dimension of this noise map (" + d + ")");
}
ArrayList found = FindBasis(v);
//the basis node
PerlinNode[] influencers = (found[0] as PerlinNode).neighbors;
//influences[0] is basically the basis grid node
//w is how much it deviates from that node, it's values will be
//for lerping purposes
VecN w = found[1] as VecN;
float n = RecurLerpTree(influencers, grid_deviations, w, 0, grid_deviations.Length - 1, 0);
return(Mathf.Min(Mathf.Max(0, n + 0.5f), 1.0f));
}
// update tex's content
private void FillTexture()
{
for (int i = 0; i < resolution; i++)
{
for (int j = 0; j < resolution; j++)
{
//calculate noise
VecN marker = new VecN(
new float[]
{
i / (float)pixelPerGrid,
j / (float)pixelPerGrid,
t
}
);
float n = map.Noise(marker);
//convert to color format
byte col = (byte)(n * 255);
tex.SetPixel(i, j, new Color32(col, col, col, 255));
}
}
tex.Apply();
}
//return a new VecN instance representing the sum of
//this VecN instance with another vector
public VecN Add(VecN a)
{
return(Add(this, a));
}
//return dot product of the current VecN instance with another vector
public float Dot(VecN a)
{
return(Dot(this, a));
}
//return a new VecN instance (for N = 3) representing the cross product
//of this VecN instance with another 3D vector
public VecN Cross(VecN a)
{
return(Cross(this, a));
}
//return a new VecN instance representing this VecN instance minus
//another VecN instance
public VecN Subtract(VecN a)
{
return(Subtract(this, a));
}
//return the length of a VecN instance
public static float Norm(VecN a)
{
return(Mathf.Sqrt(Dot(a, a)));
}
