opengl - 分形的连续着色

Question

我正在尝试使用 OpenGL 来可视化 Mandelbrot 集，并且在平滑着色方面发现了非常奇怪的行为。

让我们假设，对于当前的复数值 C , 算法在 n 后逃逸迭代时 Z已被证明大于2。

我编写了这样的着色部分:

if(n==maxIterations){
   color=0.0; //0.0 is black in OpenGL when put to each channel of RGB 
              //Points in M-brot set are colored black.
} else {
   color = (n + 1 - log(log(abs(Z)))/log(2.0) )/maxIterations; 
   //continuous coloring algorithm, color is between 0.0 and 1.0
   //Points outside M-brot set are colored depending of their absolute value,
   //from brightest near the edge of set to darkest far away from set.
}
glColor3f(color ,color ,color );
//OpenGL-command for making RGB-color from three channel values. 

问题是，这行不通。一些平滑是值得注意的，但并不完美。

但是当我添加两个额外的迭代时(只是在某处没有解释)

Z=Z*Z+C; 
n++; 

在计算颜色之前的“else”分支中，图像绝对流畅。

为什么会发生这种情况？为什么我们需要在检查要设置的点后在着色部分放置额外的迭代？

Answer 1

我实际上并不确定，但我猜这与一个数字( log(log(n)) )的对数的对数对于“小”数字 n 来说有点不稳定的事实有关。 ，在这种情况下，“小”意味着接近 2。如果您的 Z刚刚逃脱，接近2。如果继续迭代，你(很快)离2越来越远，log(log(abs(Z)))稳定，从而为您提供更可预测的值......然后，为您提供更平滑的值。

示例数据，任意选择:

 n            Z.real            Z.imag         |Z|  status color
-- ----------------- ----------------- ----------- ------- -----
 0             -0.74              -0.2    0.766551 bounded [nonsensical]
 1           -0.2324             0.096    0.251447 bounded [nonsensical]
 2       -0.69520624        -0.2446208    0.736988 bounded [nonsensical]
 3    -0.31652761966     0.14012381319    0.346157 bounded [nonsensical]
 4    -0.65944494902    -0.28870611409    0.719874 bounded [nonsensical]
 5    -0.38848357953     0.18077157738    0.428483 bounded [nonsensical]
 6    -0.62175887162    -0.34045357891    0.708867 bounded [nonsensical]
 7    -0.46932454495     0.22336006613    0.519765 bounded [nonsensical]
 8    -0.56962419064    -0.40965672279    0.701634 bounded [nonsensical]
 9    -0.58334691196     0.26670075833    0.641423 bounded [nonsensical]
10     -0.4708356748    -0.51115812757     0.69496 bounded [nonsensical]
11    -0.77959639873     0.28134296385    0.828809 bounded [nonsensical]
12     -0.2113833184    -0.63866792284     0.67274 bounded [nonsensical]
13     -1.1032138084    0.070007489775     1.10543 bounded 0.173185134517425
14     0.47217965836    -0.35446645882    0.590424 bounded [nonsensical]
15    -0.64269284066    -0.53474370285    0.836065 bounded [nonsensical]
16     -0.6128967403     0.48735189882    0.783042 bounded [nonsensical]
17    -0.60186945901    -0.79739278033    0.999041 bounded [nonsensical]
18     -1.0135884004     0.75985272263     1.26678 bounded 0.210802091344997
19    -0.29001471459     -1.7403558114     1.76435 bounded 0.208165835763602
20     -3.6847298156     0.80945758785     3.77259 ESCAPED 0.205910029166315
21      12.182012228     -6.1652650168     13.6533 ESCAPED 0.206137522227716
22      109.65092918     -150.41066764     186.136 ESCAPED 0.20614160700086
23     -10600.782669     -32985.538932     34647.1 ESCAPED 0.20614159039676
24     -975669186.18       699345058.7 1.20042e+09 ESCAPED 0.206141590396481
25  4.6284684972e+17 -1.3646588486e+18 1.44101e+18 ESCAPED 0.206141590396481
26 -1.6480665667e+36  -1.263256098e+36 2.07652e+36 ESCAPED 0.206141590396481

注意颜色值在 [20,22] 中的 n 处仍然波动，在 n=23 时稳定，从 n=24 开始保持一致。并注意 Z 值在半径 2 的圆之外，该圆限定了 Mandelbrot 集。

我实际上还没有做足够的数学来确定这实际上是一个可靠的解释，但这将是我的猜测。