//
// float3 psrdnoise(float2 pos, float2 per, float rot)
// float3 psrdnoise(float2 pos, float2 per)
// float psrnoise(float2 pos, float2 per, float rot)
// float psrnoise(float2 pos, float2 per)
// float3 srdnoise(float2 pos, float rot)
// float3 srdnoise(float2 pos)
// float srnoise(float2 pos, float rot)
// float srnoise(float2 pos)
//
// Periodic (tiling) 2-D simplex noise (hexagonal lattice gradient noise)
// with rotating gradients and analytic derivatives.
// Variants also without the derivative (no "d" in the name), without
// the tiling property (no "p" in the name) and without the rotating
// gradients (no "r" in the name).
//
// This is (yet) another variation on simplex noise. It's similar to the
// version presented by Ken Perlin, but the grid is axis-aligned and
// slightly stretched in the y direction to permit rectangular tiling.
//
// The noise can be made to tile seamlessly to any integer period in x and
// any even integer period in y. Odd periods may be specified for y, but
// then the actual tiling period will be twice that number.
//
// The rotating gradients give the appearance of a swirling motion, and can
// serve a similar purpose for animation as motion along z in 3-D noise.
// The rotating gradients in conjunction with the analytic derivatives
// can make "flow noise" effects as presented by Perlin and Neyret.
//
// float3 {p}s{r}dnoise(float2 pos {, float2 per} {, float rot})
// "pos" is the input (x,y) coordinate
// "per" is the x and y period, where per.x is a positive integer
// and per.y is a positive even integer
// "rot" is the angle to rotate the gradients (any float value,
// where 0.0 is no rotation and 1.0 is one full turn)
// The first component of the 3-element return vector is the noise value.
// The second and third components are the x and y partial derivatives.
//
// float {p}s{r}noise(float2 pos {, float2 per} {, float rot})
// "pos" is the input (x,y) coordinate
// "per" is the x and y period, where per.x is a positive integer
// and per.y is a positive even integer
// "rot" is the angle to rotate the gradients (any float value,
// where 0.0 is no rotation and 1.0 is one full turn)
// The return value is the noise value.
// Partial derivatives are not computed, making these functions faster.
//
// Author: Stefan Gustavson (stefan.gustavson@gmail.com)
// Version 2016-05-10.
//
// Many thanks to Ian McEwan of Ashima Arts for the
// idea of umath.sing a permutation polynomial.
//
// Copyright (c) 2016 Stefan Gustavson. All rights reserved.
// Distributed under the MIT license. See LICENSE file.
// https://github.com/stegu/webgl-noise
//
//
// TODO: One-pixel wide artefacts used to occur due to precision issues with
// the gradient indexing. This is specific to this variant of noise, because
// one axis of the simplex grid is perfectly aligned with the input x axis.
// The errors were rare, and they are now very unlikely to ever be visible
// after a quick fix was introduced: a small offset is added to the y coordinate.
// A proper fix would involve umath.sing round() instead of math.floor() in selected
// places, but the quick fix works fine.
// (If you run into problems with this, please let me know.)
//
using static Unity.Mathematics.math;
namespace Unity.Mathematics
{
public static partial class noise
{
/// <summary>
/// 2-D tiling simplex noise with rotating gradients and analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="per">The x and y period, where per.x is a positive integer and per.y is a positive even integer.</param>
/// <param name="rot">Angle to rotate the gradients.</param>
/// <returns>The first component of the 3-element return vector is the noise value, and the second and third components are the x and y partial derivatives.</returns>
public static float3 psrdnoise(float2 pos, float2 per, float rot)
{
// Hack: offset y slightly to hide some rare artifacts
pos.y += 0.01f;
// Skew to hexagonal grid
float2 uv = float2(pos.x + pos.y * 0.5f, pos.y);
float2 i0 = floor(uv);
float2 f0 = frac(uv);
// Traversal order
float2 i1 = (f0.x > f0.y) ? float2(1.0f, 0.0f) : float2(0.0f, 1.0f);
// Unskewed grid points in (x,y) space
float2 p0 = float2(i0.x - i0.y * 0.5f, i0.y);
float2 p1 = float2(p0.x + i1.x - i1.y * 0.5f, p0.y + i1.y);
float2 p2 = float2(p0.x + 0.5f, p0.y + 1.0f);
// Vectors in unskewed (x,y) coordinates from
// each of the simplex corners to the evaluation point
float2 d0 = pos - p0;
float2 d1 = pos - p1;
float2 d2 = pos - p2;
// Wrap p0, p1 and p2 to the desired period before gradient hashing:
// wrap points in (x,y), map to (u,v)
float3 xw = fmod(float3(p0.x, p1.x, p2.x), per.x);
float3 yw = fmod(float3(p0.y, p1.y, p2.y), per.y);
float3 iuw = xw + 0.5f * yw;
float3 ivw = yw;
// Create gradients from indices
float2 g0 = rgrad2(float2(iuw.x, ivw.x), rot);
float2 g1 = rgrad2(float2(iuw.y, ivw.y), rot);
float2 g2 = rgrad2(float2(iuw.z, ivw.z), rot);
// Gradients math.dot vectors to corresponding corners
// (The derivatives of this are simply the gradients)
float3 w = float3(dot(g0, d0), dot(g1, d1), dot(g2, d2));
// Radial weights from corners
// 0.8 is the square of 2/math.sqrt(5), the distance from
// a grid point to the nearest simplex boundary
float3 t = 0.8f - float3(dot(d0, d0), dot(d1, d1), dot(d2, d2));
// Partial derivatives for analytical gradient computation
float3 dtdx = -2.0f * float3(d0.x, d1.x, d2.x);
float3 dtdy = -2.0f * float3(d0.y, d1.y, d2.y);
// Set influence of each surflet to zero outside radius math.sqrt(0.8)
if (t.x < 0.0f)
{
dtdx.x = 0.0f;
dtdy.x = 0.0f;
t.x = 0.0f;
}
if (t.y < 0.0f)
{
dtdx.y = 0.0f;
dtdy.y = 0.0f;
t.y = 0.0f;
}
if (t.z < 0.0f)
{
dtdx.z = 0.0f;
dtdy.z = 0.0f;
t.z = 0.0f;
}
// Fourth power of t (and third power for derivative)
float3 t2 = t * t;
float3 t4 = t2 * t2;
float3 t3 = t2 * t;
// Final noise value is:
// sum of ((radial weights) times (gradient math.dot vector from corner))
float n = dot(t4, w);
// Final analytical derivative (gradient of a sum of scalar products)
float2 dt0 = float2(dtdx.x, dtdy.x) * 4.0f * t3.x;
float2 dn0 = t4.x * g0 + dt0 * w.x;
float2 dt1 = float2(dtdx.y, dtdy.y) * 4.0f * t3.y;
float2 dn1 = t4.y * g1 + dt1 * w.y;
float2 dt2 = float2(dtdx.z, dtdy.z) * 4.0f * t3.z;
float2 dn2 = t4.z * g2 + dt2 * w.z;
return 11.0f * float3(n, dn0 + dn1 + dn2);
}
/// <summary>
/// 2-D tiling simplex noise with fixed gradients and analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="per">The x and y period, where per.x is a positive integer and per.y is a positive even integer.</param>
/// <returns>The first component of the 3-element return vector is the noise value, and the second and third components are the x and y partial derivatives.</returns>
public static float3 psrdnoise(float2 pos, float2 per)
{
return psrdnoise(pos, per, 0.0f);
}
/// <summary>
/// 2-D tiling simplex noise with rotating gradients, but without the analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="per">The x and y period, where per.x is a positive integer and per.y is a positive even integer.</param>
/// <param name="rot">Angle to rotate the gradients.</param>
/// <returns>Noise value.</returns>
public static float psrnoise(float2 pos, float2 per, float rot)
{
// Offset y slightly to hide some rare artifacts
pos.y += 0.001f;
// Skew to hexagonal grid
float2 uv = float2(pos.x + pos.y * 0.5f, pos.y);
float2 i0 = floor(uv);
float2 f0 = frac(uv);
// Traversal order
float2 i1 = (f0.x > f0.y) ? float2(1.0f, 0.0f) : float2(0.0f, 1.0f);
// Unskewed grid points in (x,y) space
float2 p0 = float2(i0.x - i0.y * 0.5f, i0.y);
float2 p1 = float2(p0.x + i1.x - i1.y * 0.5f, p0.y + i1.y);
float2 p2 = float2(p0.x + 0.5f, p0.y + 1.0f);
// Vectors in unskewed (x,y) coordinates from
// each of the simplex corners to the evaluation point
float2 d0 = pos - p0;
float2 d1 = pos - p1;
float2 d2 = pos - p2;
// Wrap p0, p1 and p2 to the desired period before gradient hashing:
// wrap points in (x,y), map to (u,v)
float3 xw = fmod(float3(p0.x, p1.x, p2.x), per.x);
float3 yw = fmod(float3(p0.y, p1.y, p2.y), per.y);
float3 iuw = xw + 0.5f * yw;
float3 ivw = yw;
// Create gradients from indices
float2 g0 = rgrad2(float2(iuw.x, ivw.x), rot);
float2 g1 = rgrad2(float2(iuw.y, ivw.y), rot);
float2 g2 = rgrad2(float2(iuw.z, ivw.z), rot);
// Gradients math.dot vectors to corresponding corners
// (The derivatives of this are simply the gradients)
float3 w = float3(dot(g0, d0), dot(g1, d1), dot(g2, d2));
// Radial weights from corners
// 0.8 is the square of 2/math.sqrt(5), the distance from
// a grid point to the nearest simplex boundary
float3 t = 0.8f - float3(dot(d0, d0), dot(d1, d1), dot(d2, d2));
// Set influence of each surflet to zero outside radius math.sqrt(0.8)
t = max(t, 0.0f);
// Fourth power of t
float3 t2 = t * t;
float3 t4 = t2 * t2;
// Final noise value is:
// sum of ((radial weights) times (gradient math.dot vector from corner))
float n = dot(t4, w);
// Rescale to cover the range [-1,1] reasonably well
return 11.0f * n;
}
/// <summary>
/// 2-D tiling simplex noise with fixed gradients, without the analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="per">The x and y period, where per.x is a positive integer and per.y is a positive even integer.</param>
/// <returns>Noise value.</returns>
public static float psrnoise(float2 pos, float2 per)
{
return psrnoise(pos, per, 0.0f);
}
/// <summary>
/// 2-D non-tiling simplex noise with rotating gradients and analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="rot">Angle to rotate the gradients.</param>
/// <returns>The first component of the 3-element return vector is the noise value, and the second and third components are the x and y partial derivatives.</returns>
public static float3 srdnoise(float2 pos, float rot)
{
// Offset y slightly to hide some rare artifacts
pos.y += 0.001f;
// Skew to hexagonal grid
float2 uv = float2(pos.x + pos.y * 0.5f, pos.y);
float2 i0 = floor(uv);
float2 f0 = frac(uv);
// Traversal order
float2 i1 = (f0.x > f0.y) ? float2(1.0f, 0.0f) : float2(0.0f, 1.0f);
// Unskewed grid points in (x,y) space
float2 p0 = float2(i0.x - i0.y * 0.5f, i0.y);
float2 p1 = float2(p0.x + i1.x - i1.y * 0.5f, p0.y + i1.y);
float2 p2 = float2(p0.x + 0.5f, p0.y + 1.0f);
// Vectors in unskewed (x,y) coordinates from
// each of the simplex corners to the evaluation point
float2 d0 = pos - p0;
float2 d1 = pos - p1;
float2 d2 = pos - p2;
float3 x = float3(p0.x, p1.x, p2.x);
float3 y = float3(p0.y, p1.y, p2.y);
float3 iuw = x + 0.5f * y;
float3 ivw = y;
// Avoid precision issues in permutation
iuw = mod289(iuw);
ivw = mod289(ivw);
// Create gradients from indices
float2 g0 = rgrad2(float2(iuw.x, ivw.x), rot);
float2 g1 = rgrad2(float2(iuw.y, ivw.y), rot);
float2 g2 = rgrad2(float2(iuw.z, ivw.z), rot);
// Gradients math.dot vectors to corresponding corners
// (The derivatives of this are simply the gradients)
float3 w = float3(dot(g0, d0), dot(g1, d1), dot(g2, d2));
// Radial weights from corners
// 0.8 is the square of 2/math.sqrt(5), the distance from
// a grid point to the nearest simplex boundary
float3 t = 0.8f - float3(dot(d0, d0), dot(d1, d1), dot(d2, d2));
// Partial derivatives for analytical gradient computation
float3 dtdx = -2.0f * float3(d0.x, d1.x, d2.x);
float3 dtdy = -2.0f * float3(d0.y, d1.y, d2.y);
// Set influence of each surflet to zero outside radius math.sqrt(0.8)
if (t.x < 0.0f)
{
dtdx.x = 0.0f;
dtdy.x = 0.0f;
t.x = 0.0f;
}
if (t.y < 0.0f)
{
dtdx.y = 0.0f;
dtdy.y = 0.0f;
t.y = 0.0f;
}
if (t.z < 0.0f)
{
dtdx.z = 0.0f;
dtdy.z = 0.0f;
t.z = 0.0f;
}
// Fourth power of t (and third power for derivative)
float3 t2 = t * t;
float3 t4 = t2 * t2;
float3 t3 = t2 * t;
// Final noise value is:
// sum of ((radial weights) times (gradient math.dot vector from corner))
float n = dot(t4, w);
// Final analytical derivative (gradient of a sum of scalar products)
float2 dt0 = float2(dtdx.x, dtdy.x) * 4.0f * t3.x;
float2 dn0 = t4.x * g0 + dt0 * w.x;
float2 dt1 = float2(dtdx.y, dtdy.y) * 4.0f * t3.y;
float2 dn1 = t4.y * g1 + dt1 * w.y;
float2 dt2 = float2(dtdx.z, dtdy.z) * 4.0f * t3.z;
float2 dn2 = t4.z * g2 + dt2 * w.z;
return 11.0f * float3(n, dn0 + dn1 + dn2);
}
/// <summary>
/// 2-D non-tiling simplex noise with fixed gradients and analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <returns>The first component of the 3-element return vector is the noise value, and the second and third components are the x and y partial derivatives.</returns>
public static float3 srdnoise(float2 pos)
{
return srdnoise(pos, 0.0f);
}
/// <summary>
/// 2-D non-tiling simplex noise with rotating gradients, without the analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <param name="rot">Angle to rotate the gradients.</param>
/// <returns>Noise value.</returns>
public static float srnoise(float2 pos, float rot)
{
// Offset y slightly to hide some rare artifacts
pos.y += 0.001f;
// Skew to hexagonal grid
float2 uv = float2(pos.x + pos.y * 0.5f, pos.y);
float2 i0 = floor(uv);
float2 f0 = frac(uv);
// Traversal order
float2 i1 = (f0.x > f0.y) ? float2(1.0f, 0.0f) : float2(0.0f, 1.0f);
// Unskewed grid points in (x,y) space
float2 p0 = float2(i0.x - i0.y * 0.5f, i0.y);
float2 p1 = float2(p0.x + i1.x - i1.y * 0.5f, p0.y + i1.y);
float2 p2 = float2(p0.x + 0.5f, p0.y + 1.0f);
// Vectors in unskewed (x,y) coordinates from
// each of the simplex corners to the evaluation point
float2 d0 = pos - p0;
float2 d1 = pos - p1;
float2 d2 = pos - p2;
float3 x = float3(p0.x, p1.x, p2.x);
float3 y = float3(p0.y, p1.y, p2.y);
float3 iuw = x + 0.5f * y;
float3 ivw = y;
// Avoid precision issues in permutation
iuw = mod289(iuw);
ivw = mod289(ivw);
// Create gradients from indices
float2 g0 = rgrad2(float2(iuw.x, ivw.x), rot);
float2 g1 = rgrad2(float2(iuw.y, ivw.y), rot);
float2 g2 = rgrad2(float2(iuw.z, ivw.z), rot);
// Gradients math.dot vectors to corresponding corners
// (The derivatives of this are simply the gradients)
float3 w = float3(dot(g0, d0), dot(g1, d1), dot(g2, d2));
// Radial weights from corners
// 0.8 is the square of 2/math.sqrt(5), the distance from
// a grid point to the nearest simplex boundary
float3 t = 0.8f - float3(dot(d0, d0), dot(d1, d1), dot(d2, d2));
// Set influence of each surflet to zero outside radius math.sqrt(0.8)
t = max(t, 0.0f);
// Fourth power of t
float3 t2 = t * t;
float3 t4 = t2 * t2;
// Final noise value is:
// sum of ((radial weights) times (gradient math.dot vector from corner))
float n = dot(t4, w);
// Rescale to cover the range [-1,1] reasonably well
return 11.0f * n;
}
/// <summary>
/// 2-D non-tiling simplex noise with fixed gradients, without the analytical derivative.
/// </summary>
/// <param name="pos">Input (x,y) coordinate.</param>
/// <returns>Noise value.</returns>
public static float srnoise(float2 pos)
{
return srnoise(pos, 0.0f);
}
}
}