@ -60,21 +60,21 @@ for (int i = 0; i < octaves; i++) {
* 当 octaves 大于 4 时，尝试改变 lacunarity 的值。
* 当 octaves 大于 4 时，改变 gain 的值，看看会发生什么。
Note how with each additional octave, the curve seems to get more detail. Also note the self-similarity while more octaves are added. If you zoom in on the curve, a smaller part looks about the same as the whole thing, and each section looks more or less the same as any other section. This is an important property of mathematical fractals, and we are simulating that property in our loop. We are not creating a *true* fractal, because we stop the summation after a few iterations, but theoretically speaking, we would get a true mathematical fractal if we allowed the loop to continue forever and add an infinite number of noise components. In computer graphics, we always have a limit to how small details we can resolve, for example when objects become smaller than a pixel, so there is no need to make infinite sums to create the appearance of a fractal. A lot of terms may be needed sometimes, but never an infinite number.
* Reduce the number of octaves by changing the value on line 37
* Modify the lacunarity of the fBm on line 47
* Explore by changing the gain on line 48
* 在 37 行减小八度（OCTAVES）的数量
* 在 47 行调试 fBm 的间隙度（lacumarity）
* 在 47 行调试 fBm 的增益（gain）
This technique is commonly used to construct procedural landscapes. The self-similarity of the fBm is perfect for mountains, because the erosion processes that create mountains work in a manner that yields this kind of self-similarity across a large range of scales. If you are interested in this, use you should definitly read [this great article by Inigo Quiles about advance noise](http://www.iquilezles.org/www/articles/morenoise/morenoise.htm).
Using more or less the same technique, it's also possible to obtain other effects like what is known as **turbulence**. It's essentially an fBm, but constructed from the absolute value of a signed noise to create sharp valleys in the function.
Another variant which can create useful variations is to multiply the noise components together instead of adding them. It's also interesting to scale subsequent noise functions with something that depends on the previous terms in the loop. When we do things like that, we are moving away from the strict definition of a fractal and into the relatively unknown field of "multifractals". Multifractals are not as strictly defined mathematically, but that doesn't make them less useful for graphics. In fact, multifractal simulations are very common in modern commercial software for terrain generation. For further reading, you could read chapter 16 of the book "Texturing and Modeling: a Procedural Approach" (3rd edition), by Kenton Musgrave. Sadly, that book is out of print since a few years back, but you can still find it in libraries and on the second hand market. (There's a PDF version of the 1st edition available for purchase online, but don't buy that - it's a waste of money. It's from 1994, and it doesn't contain any of the terrain modeling stuff from the 3rd edition.)
这个算法的另外一个变种，把噪声分量乘起来（而不是叠加）可以创造一些很有用的东西。另外一个方法是，根据前一次循环中的噪声来缩放后续的噪声。当我们这样做的时候，我们已经走出严格的分形定义了，走入了一个叫“多重分形”的未知领域。多重分形虽不是按数学方式严格定义，但这并不意味着它的用处会更少些。 实际上，多重分形模拟生成地形中在商业软件中非常常见。要了解更多，你可以去读 Kenton Musgrave 的“Texturing and Modeling: a Procedural Approach”（第三版）的 16 章。很可惜，这本书几年前已经绝版，不过你还可以从图书馆和二手市场找到。网上有卖这本书第一版的 PDF 版，但是别去买——只是浪费钱。这是 1994 年的版本，不包括第三版包含的地形建模的部分。
### 域翘曲（Domain Warping）
[Inigo Quiles wrote this other fascinating article](http://www.iquilezles.org/www/articles/warp/warp.htm) about how it's possible to use fBm to warp a space of a fBm. Mind blowing, Right? It's like the dream inside the dream of Inception.
A less extreme example of this technique is the following code where the wrap is used to produce this clouds-like texture. Note how the self-similarity property is still present in the result.
Warping the texture coordinates with noise in this manner can be very useful, a lot of fun, and fiendishly difficult to master. It's a powerful tool, but it takes quite a bit of experience to use it well. A useful tool for this is to displace the coordinates with the derivative (gradient) of the noise. [A famous article by Ken Perlin and Fabrice Neyret called "flow noise"](http://evasion.imag.fr/Publications/2001/PN01/) is based on this idea. Some modern implementations of Perlin noise include a variant that computes both the function and it's analytical gradient. If the "true" gradient is not available for a procedural function, you can always compute finite differences to approximate it, although this is less accurate and involves more work.
ray7551
7 years ago