Question

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 !!

Answer 1

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

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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();
  }
}

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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…