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# eevee/perlin.py

Last active May 18, 2021
Perlin noise in Python
 """Perlin noise implementation.""" # Licensed under ISC from itertools import product import math import random def smoothstep(t): """Smooth curve with a zero derivative at 0 and 1, making it useful for interpolating. """ return t * t * (3. - 2. * t) def lerp(t, a, b): """Linear interpolation between a and b, given a fraction t.""" return a + t * (b - a) class PerlinNoiseFactory(object): """Callable that produces Perlin noise for an arbitrary point in an arbitrary number of dimensions. The underlying grid is aligned with the integers. There is no limit to the coordinates used; new gradients are generated on the fly as necessary. """ def __init__(self, dimension, octaves=1, tile=(), unbias=False): """Create a new Perlin noise factory in the given number of dimensions, which should be an integer and at least 1. More octaves create a foggier and more-detailed noise pattern. More than 4 octaves is rather excessive. ``tile`` can be used to make a seamlessly tiling pattern. For example: pnf = PerlinNoiseFactory(2, tile=(0, 3)) This will produce noise that tiles every 3 units vertically, but never tiles horizontally. If ``unbias`` is true, the smoothstep function will be applied to the output before returning it, to counteract some of Perlin noise's significant bias towards the center of its output range. """ self.dimension = dimension self.octaves = octaves self.tile = tile + (0,) * dimension self.unbias = unbias # For n dimensions, the range of Perlin noise is ±sqrt(n)/2; multiply # by this to scale to ±1 self.scale_factor = 2 * dimension ** -0.5 self.gradient = {} def _generate_gradient(self): # Generate a random unit vector at each grid point -- this is the # "gradient" vector, in that the grid tile slopes towards it # 1 dimension is special, since the only unit vector is trivial; # instead, use a slope between -1 and 1 if self.dimension == 1: return (random.uniform(-1, 1),) # Generate a random point on the surface of the unit n-hypersphere; # this is the same as a random unit vector in n dimensions. Thanks # to: http://mathworld.wolfram.com/SpherePointPicking.html # Pick n normal random variables with stddev 1 random_point = [random.gauss(0, 1) for _ in range(self.dimension)] # Then scale the result to a unit vector scale = sum(n * n for n in random_point) ** -0.5 return tuple(coord * scale for coord in random_point) def get_plain_noise(self, *point): """Get plain noise for a single point, without taking into account either octaves or tiling. """ if len(point) != self.dimension: raise ValueError("Expected {} values, got {}".format( self.dimension, len(point))) # Build a list of the (min, max) bounds in each dimension grid_coords = [] for coord in point: min_coord = math.floor(coord) max_coord = min_coord + 1 grid_coords.append((min_coord, max_coord)) # Compute the dot product of each gradient vector and the point's # distance from the corresponding grid point. This gives you each # gradient's "influence" on the chosen point. dots = [] for grid_point in product(*grid_coords): if grid_point not in self.gradient: self.gradient[grid_point] = self._generate_gradient() gradient = self.gradient[grid_point] dot = 0 for i in range(self.dimension): dot += gradient[i] * (point[i] - grid_point[i]) dots.append(dot) # Interpolate all those dot products together. The interpolation is # done with smoothstep to smooth out the slope as you pass from one # grid cell into the next. # Due to the way product() works, dot products are ordered such that # the last dimension alternates: (..., min), (..., max), etc. So we # can interpolate adjacent pairs to "collapse" that last dimension. Then # the results will alternate in their second-to-last dimension, and so # forth, until we only have a single value left. dim = self.dimension while len(dots) > 1: dim -= 1 s = smoothstep(point[dim] - grid_coords[dim]) next_dots = [] while dots: next_dots.append(lerp(s, dots.pop(0), dots.pop(0))) dots = next_dots return dots * self.scale_factor def __call__(self, *point): """Get the value of this Perlin noise function at the given point. The number of values given should match the number of dimensions. """ ret = 0 for o in range(self.octaves): o2 = 1 << o new_point = [] for i, coord in enumerate(point): coord *= o2 if self.tile[i]: coord %= self.tile[i] * o2 new_point.append(coord) ret += self.get_plain_noise(*new_point) / o2 # Need to scale n back down since adding all those extra octaves has # probably expanded it beyond ±1 # 1 octave: ±1 # 2 octaves: ±1½ # 3 octaves: ±1¾ ret /= 2 - 2 ** (1 - self.octaves) if self.unbias: # The output of the plain Perlin noise algorithm has a fairly # strong bias towards the center due to the central limit theorem # -- in fact the top and bottom 1/8 virtually never happen. That's # a quarter of our entire output range! If only we had a function # in [0..1] that could introduce a bias towards the endpoints... r = (ret + 1) / 2 # Doing it this many times is a completely made-up heuristic. for _ in range(int(self.octaves / 2 + 0.5)): r = smoothstep(r) ret = r * 2 - 1 return ret

### liamHowatt commented May 24, 2018

 This is the best implementation for Python I've ever seen! I've been able to make random generated caves in a game.

### veluca93 commented Aug 23, 2018

 If anybody is interested, here is an implementation as a C++ python module (with a lot of speed hacks) that exposes (at least) the same interface (and also shares most of the algorithm). On my computer, it seems to be more or less 10-20 times faster.

### levibessa1 commented Nov 21, 2018

 If anybody is interested, here is an implementation as a C++ python module (with a lot of speed hacks) that exposes (at least) the same interface (and also shares most of the algorithm). On my computer, it seems to be more or less 10-20 times faster. Hello Veluca93 , what sequence should i execute the files in C++ ?

### thorhunter1 commented Nov 21, 2018

 Usage example? Gives me constantly values of 0.0.

### Aeium commented Jan 7, 2019

 @thorhunter1 Try calling the factory with values inside the interval [0-1]. For example ``````for i in range(frameSize): for j in range(frameSize): noise[i,j] = PNFactory(i/frameSize,j/frameSize) `````` instead of ``````for i in range(frameSize): for j in range(frameSize): noise[i,j] = PNFactory(i,j) ``````

### Wouterr0 commented Mar 25, 2020 • edited

 The improved version of Ken Perlin uses `6t**5 - 15t**4 + 10t**3` instead of `3t**2 - 2t**3` as smoothstep function.

### xavriley commented Apr 21, 2020

 Original blog post with instructions and explanation is here https://eev.ee/blog/2016/05/29/perlin-noise/

### KaufNation commented Jun 24, 2020

 Is this O(1) to calculate any random location on the continuous curve, or is this O(n)?

### Crowkk commented May 15, 2021

 Correct me if I'm wrong but there is an inconsistency in this snipet: `````` for grid_point in product(*grid_coords): if grid_point not in self.gradient: self.gradient[grid_point] = self._generate_gradient() gradient = self.gradient[grid_point] dot = 0 for i in range(self.dimension): dot += gradient[i] * (point[i] - grid_point[i]) dots.append(dot) `````` When you call self.gradient[grid_point] (say grid_point = [2,3]) you are calling the gradient at row 2 and column 3; But in "position" space it is actually the reverse: the second row is on y = 2 and third col at x = 3 so when you make (point[i] - grid_point[i]) you're calculating the wrong difference because it is reversed. It should be grid_point[::-1][i] at least for 2D;
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