## Simplex Noise
## シンプレックスノイズ
For Ken Perlin the success of his algorithm wasn't enough. He thought it could perform better. At Siggraph 2001 he presented the "simplex noise" in which he achieved the following improvements over the previous algorithm:
Ken Perlinにとって彼のアルゴリズムの成功は十分でありませんでした。彼はもっとパフォーマンスをあげられると考えたのです。2001年のSiggraphで発表したシンプレックスノイズでは以前のアルゴリズムに比べて下記に挙げるような改良を実現しました。
* An algorithm with lower computational complexity and fewer multiplications.
* よりシンプルで乗算の回数が少ない済むアルゴリズム。
* A noise that scales to higher dimensions with less computational cost.
* 次元の数が増えても計算量の増加が抑えられる。
* A noise without directional artifacts.
* 向きによって生じるアーティファクトがない。
* A noise with well-defined and continuous gradients that can be computed quite cheaply.
* An algorithm that is easy to implement in hardware.
* ハードウェアで実現しやすいアルゴリズム。
I know what you are thinking... "Who is this man?" Yes, his work is fantastic! But seriously, how did he improve the algorithm? Well, we saw how for two dimensions he was interpolating 4 points (corners of a square); so we can correctly guess that for [three (see an implementation here)](../edit.html#11/3d-noise.frag) and four dimensions we need to interpolate 8 and 16 points. Right? In other words for N dimensions you need to smoothly interpolate 2 to the N points (2^N). But Ken smartly noticed that although the obvious choice for a space-filling shape is a square, the simplest shape in 2D is the equilateral triangle. So he started by replacing the squared grid (we just learned how to use) for a simplex grid of equilateral triangles.
The simplex shape for N dimensions is a shape with N + 1 corners. In other words one fewer corner to compute in 2D, 4 fewer corners in 3D and 11 fewer corners in 4D! That's a huge improvement!
N次元に対して最も単純な形は N + 1 の頂点を持ちます。つまり二次元に対しては1頂点、三次元に対しては4頂点、四次元に対しては11頂点少ない計算で済むのです。これは大変な改善です。
In two dimensions the interpolation happens similarly to regular noise, by interpolating the values of the corners of a section. But in this case, by using a simplex grid, we only need to interpolate the sum of 3 corners.
How is the simplex grid made? In another brilliant and elegant move, the simplex grid can be obtained by subdividing the cells of a regular 4 cornered grid into two isosceles triangles and then skewing it until each triangle is equilateral.
Then, as [Stefan Gustavson describes in this paper](http://staffwww.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf): _"...by looking at the integer parts of the transformed coordinates (x,y) for the point we want to evaluate, we can quickly determine which cell of two simplices that contains the point. By also comparing the magnitudes of x and y, we can determine whether the point is in the upper or the lower simplex, and traverse the correct three corner points."_
In the following code you can uncomment line 44 to see how the grid is skewed, and then uncomment line 47 to see how a simplex grid can be constructed. Note how on line 22 we are subdividing the skewed square into two equilateral triangles just by detecting if ```x > y``` ("lower" triangle) or ```y > x``` ("upper" triangle).
Another improvement introduced by Perlin with **Simplex Noise**, is the replacement of the Cubic Hermite Curve ( _f(x) = 3x^2-2x^3_ , which is identical to the [```smoothstep()```](.../glossary/?search=smoothstep) function) for a Quintic Hermite Curve ( _f(x) = 6x^5-15x^4+10x^3_ ). This makes both ends of the curve more "flat" so each border gracefully stiches with the next one. In other words you get a more continuous transition between the cells. You can see this by uncommenting the second formula in the following graph example (or see the [two equations side by side here](https://www.desmos.com/calculator/2xvlk5xp8b)).
Ken Perlinがシンプレックスノイズで導入したもう1つの改善点は、3次エルミート曲線（f(x) = 3x^2-2x^3 これは [```smoothstep()```](.../glossary/?search=smoothstep)関数と同等です）を4次エルミート曲線（f(x) = 6x^5-15x^4+10x^3）で置き換えたことです。これにより曲線の両端がより平坦になり次の曲線により綺麗につながり、マス目間の変化がより滑らかになります。下記のグラフの2つ目の式のコメントを外すとこれを実際に見ることができます（もしくは[ここ](https://www.desmos.com/calculator/2xvlk5xp8b)で並べて比較することもできます）。
<divclass="simpleFunction"data="
// Cubic Hermine Curve. Same as SmoothStep()
y = x*x*(3.0-2.0*x);
@ -273,25 +309,42 @@ y = x*x*(3.0-2.0*x);
Note how the ends of the curve change. You can read more about this in [Ken's own words](http://mrl.nyu.edu/~perlin/paper445.pdf).
All these improvements result in an algorithmic masterpiece known as **Simplex Noise**. The following is a GLSL implementation of this algorithm made by Ian McEwan (and presented in [this paper](http://webstaff.itn.liu.se/~stegu/jgt2012/article.pdf)) which is overcomplicated for educational purposes, but you will be happy to click on it and see that it is less cryptic than you might expect.
[  ](../edit.html#11/2d-snoise-clear.frag)
Well... enough technicalities, it's time for you to use this resource in your own expressive way:
技術的な話は十分ですね。今度はこの財産をあなたの表現に使う番です。
* Contemplate how each noise implementation looks. Imagine them as a raw material, like a marble rock for a sculptor. What can you say about about the "feeling" that each one has? Squinch your eyes to trigger your imagination, like when you want to find shapes in a cloud. What do you see? What are you reminded of? What do you imagine each noise implementation could be made into? Following your guts and try to make it happen in code.
In the following chapters we will see some well known techniques to perfect your skills and get more out of your noise to design quality generative content with shaders. Until then enjoy some time outside contemplating nature and its intricate patterns. Your ability to observe needs equal (or probably more) dedication than your making skills. Go outside and enjoy the rest of the day!
<pstyle="text-align:center; font-style: italic;">"Talk to the tree, make friends with it." Bob Ross
kynd
9 years ago