如何执行简单放大到Mandelbrot集 [英] How to perform Simple Zoom into Mandelbrot Set

问题描述

我有一个普遍的问题，Mandelbrot设置缩放视图和与它有关的数学。我已经实现了256 X 256窗口大小的mandelbrot集合与值

  // ImageWidth = ImageHeight = 256; 
 
 double MinRe = -2.0; 
 double MaxRe = 1.0; 
 double MinIm = -1.2; 
 double MaxIm = 1.8; 
 
 ComputeMandelbrot（）; 
  

接下来，我选择一个正方形区域，这些是最左上角的坐标，55）和最右底部尖端（116,99）

所以，我选择 x2 = x1 + 44; y2 = y1 + 44;

如何将这些新坐标转换为复平面？以及新的真实值和虚数值如何改变以便为新的值集计算它？

这是我到目前为止所尝试的..

  double Re_factor =（MaxRe-MinRe）/（ImageWidth-1）; 
 double Im_factor =（MaxIm-MinIm）/（ImageHeight-1）; 
 
 double newMinRe = MinRe +（Re_factor * x1）; 
 double newMaxRe = MaxRe +（Re_factor * x2）; 
 double newMinIm = MinIm +（Im_factor * y1）; 
 double newMaxIm = MaxIm +（Im_factor * y2）; 
 
 //然后我计算c  -  real和c-imag值
 
 for（unsigned y = 0; y< ImageHeight; ++ y）
 { 
 double c_im = newMaxIm  -  y * Im_factor; 
 for（unsigned x = 0; x< ImageWidth; ++ x）
 {
 double c_re = newMinRe + x * Re_factor; 
 
 // ComputeMandelbrot（）; 
 
} 
 
} 
  

很难理解数学，并且关于生成'缩放'视图和任何帮助。

解决方案

这是一个线性缩放。让我们在1D中做。您有屏幕空间（屏幕坐标）和图像空间（在您的情况下，复杂的飞机）

• 屏幕空间= 0，255]


• image space => [-2，1]

将坐标X从屏幕空间转换为图像空间X'

X'=（X / 255）*（1 - （-2））+ / p>

使其更通用

• screen space => [SMin，SMax]


• image space => [IMin，IMax]

X'=（ SMin）/（SMax-SMin））*（IMax-IMin）+ IMin

---

在您的代码中， / p>

  double newMinRe = MinRe +（Re_factor * x1）; 
  

这相当于我所示。但是你这样做

  double newMaxRe = MaxRe +（Re_factor * x2）; 
  

这不正确，应为

  double newMaxRe = MinRe +（Re_factor * x2）; 
  

在您的循环中出现同样的问题，应该是

  for（unsigned y = 0; y< ImageHeight; ++ y）{
 double c_im = MinIm + y * Im_factor; 
 for（unsigned x = 0; x< ImageWidth; ++ x）{
 double c_re = MinRe + x * Re_factor; 
 // ComputeMandelbrot（）; 
} 
} 
  

---

额外的善良的额外细节：为了正确地采样图像空间，我建议这

  for（unsigned SX = SMin; x< ; SMax; ++ x）{
 double k =（double（SX + 0.5）-SMin）/（SMax-SMin）; 
 double IX =（k *（IMax-IMin））+ IMin; 
} 
  

+0.5项是在像素中间右取样。 。

I have a general question with the Mandelbrot set "zoom" view and the math pertaining to it. I have implemented the mandelbrot set for the 256 X 256 window size with values

  // ImageWidth = ImageHeight = 256;

  double MinRe = -2.0;
  double MaxRe = 1.0;
  double MinIm = -1.2;
  double MaxIm = 1.8;

  ComputeMandelbrot();

Next, I select a region of square and these are the coordinates for the upper left most tip (76,55), and rightmost bottom tip (116, 99) (square of side 44 is chosen)

so , I choose x2 = x1 + 44 ; y2 = y1 + 44;

How do I translate these new coordinates to the complex plane ? and how would the new real and imaginary values change in order to compute it for the new set of values ?

This is what I have tried so far..

double Re_factor = (MaxRe-MinRe)/(ImageWidth-1);
double Im_factor = (MaxIm-MinIm)/(ImageHeight-1);

double newMinRe = MinRe + (Re_factor* x1);
double newMaxRe = MaxRe + (Re_factor* x2);
double newMinIm = MinIm + (Im_factor* y1);
double newMaxIm = MaxIm + (Im_factor* y2);

// and then I compute c - real and c- imag values

  for(unsigned y=0; y<ImageHeight; ++y) 
{ 
  double c_im = newMaxIm - y*Im_factor;
  for(unsigned x=0; x<ImageWidth; ++x)
    {
      double c_re = newMinRe + x*Re_factor;

      // ComputeMandelbrot();

     }

 }

I am having a hard time figuring out the math, and also with regards to generating a 'zoom' view and any help is appreciated !!

解决方案

It's a linear scaling. Let's doing it in 1D. You have the screen space (screen coordinates), and the image space (the complex plane, in your case)

• screen space => [0, 255]
• image space => [-2, 1]

So to convert a coordinate X from screen space to image space X'

X' = (X / 255) * (1 - (-2)) + (-2)

To make it more generic

• screen space => [SMin, SMax]
• image space => [IMin, IMax]

X' = ((X - SMin) / (SMax - SMin)) * (IMax - IMin) + IMin

---

In your code, you do

double newMinRe = MinRe + (Re_factor* x1);

which is equivalent to what I show. But then you do

double newMaxRe = MaxRe + (Re_factor* x2);

which is not correct, and should be

double newMaxRe = MinRe + (Re_factor* x2);

Same problem in your loop, it should be

for(unsigned y=0; y<ImageHeight; ++y)  { 
  double c_im = MinIm + y*Im_factor;
  for(unsigned x=0; x<ImageWidth; ++x) {
    double c_re = MinRe + x*Re_factor;
    // ComputeMandelbrot();
  }
}

---

Additional detail for extra-goodness : to sample properly the image space , I suggest this

for(unsigned SX = SMin; x < SMax; ++x) {
  double k = (double(SX + 0.5) - SMin) / (SMax - SMin);
  double IX = (k * (IMax - IMin)) + IMin;
}

The +0.5 term is to sample right in the middle of the pixel...

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