// // 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 { /// /// 2-D tiling simplex noise with rotating gradients and analytical derivative. /// /// Input (x,y) coordinate. /// The x and y period, where per.x is a positive integer and per.y is a positive even integer. /// Angle to rotate the gradients. /// 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. 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); } /// /// 2-D tiling simplex noise with fixed gradients and analytical derivative. /// /// Input (x,y) coordinate. /// The x and y period, where per.x is a positive integer and per.y is a positive even integer. /// 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. public static float3 psrdnoise(float2 pos, float2 per) { return psrdnoise(pos, per, 0.0f); } /// /// 2-D tiling simplex noise with rotating gradients, but without the analytical derivative. /// /// Input (x,y) coordinate. /// The x and y period, where per.x is a positive integer and per.y is a positive even integer. /// Angle to rotate the gradients. /// Noise value. 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; } /// /// 2-D tiling simplex noise with fixed gradients, without the analytical derivative. /// /// Input (x,y) coordinate. /// The x and y period, where per.x is a positive integer and per.y is a positive even integer. /// Noise value. public static float psrnoise(float2 pos, float2 per) { return psrnoise(pos, per, 0.0f); } /// /// 2-D non-tiling simplex noise with rotating gradients and analytical derivative. /// /// Input (x,y) coordinate. /// Angle to rotate the gradients. /// 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. 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); } /// /// 2-D non-tiling simplex noise with fixed gradients and analytical derivative. /// /// Input (x,y) coordinate. /// 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. public static float3 srdnoise(float2 pos) { return srdnoise(pos, 0.0f); } /// /// 2-D non-tiling simplex noise with rotating gradients, without the analytical derivative. /// /// Input (x,y) coordinate. /// Angle to rotate the gradients. /// Noise value. 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; } /// /// 2-D non-tiling simplex noise with fixed gradients, without the analytical derivative. /// /// Input (x,y) coordinate. /// Noise value. public static float srnoise(float2 pos) { return srnoise(pos, 0.0f); } } }