朱里叶集&曼得布罗特集

上学期选修了一门课《现代数学方法选讲》，包含两部分内容：模糊数学和分形几何。觉得分形几何中的论点新颖有趣，其中比较基础的就是朱里叶集和曼得布罗特集了。这两种集合的图形体现了无限可分的概念，往往带给人们意想不到的震撼和不同凡响的美感。恰好那时侯正在自学MFC，感觉该做些什么练练手，所以做了这个画这两种集合的程序。使用了内存缓冲绘图法，以提高绘图效率。可任意放大缩小，能够观察到分形图形无限可分的特性。

奇妙的分形几何图形

主要代码如下：

朱里叶集的绘制：

void CJuliaView::OnDraw(CDC* pDC/*pDC*/)
{
CJuliaDoc* pDoc = GetDocument();
ASSERT_VALID(pDoc);

// TODO: 在此处为本机数据添加绘制代码
int i,j,k;
CDC dcMem;
CBitmap* pOldBitmap = NULL;
if(b_Change==TRUE)
{
  double lx,ly,x,y,m,n,r,a,b;
  const int kmax=256,M=4,iColor[19]={0x800000,0x8000,0x808000,0x80,0x800080,0x8080,0xc0c0c0,0xc0dcc0,0xa6caf0,0xfffbf0,0xa0a0a4,0x808080,0xff0000,0xff00,0xffff00,0xff,0xff00ff,0xffff,0xffffff};
  CRect rect;
  GetClientRect(rect);
  cxClient=rect.right-rect.left;
  cyClient=rect.bottom-rect.top;
  lx=(m_Xmax-m_Xmin)/cxClient;
  ly=(m_Ymax-m_Ymin)/cyClient;
  a=pDoc->m_ParamA;
  b=pDoc->m_ParamB;
  dcMem.CreateCompatibleDC(pDC);
  pDoc->m_bmpBuffer.DeleteObject();
  pDoc->m_bmpBuffer.CreateCompatibleBitmap(pDC,cxClient,cyClient);
  pOldBitmap = dcMem.SelectObject(&pDoc->m_bmpBuffer);
  ::SetCursor(::LoadCursor(NULL,IDC_WAIT));
  for(i=0;i<cxClient;i++)
  {
   for(j=0;j<cyClient;j++)
   {
    x=m_Xmin+i*lx;
    y=m_Ymax-j*ly;
    k=0;
    while(TRUE)
    {
     m=x*x-y*y+a;
     n=2*x*y+b;
     k++;
     r=m*m+n*n;
     if(r>M)
     {
      dcMem.SetPixel(i,j,iColor[(k-1)%19]);
      break;
     }
     if(k==kmax)
     {
      dcMem.SetPixel(i,j,0);
      break;
     }
     x=m;
     y=n;
    }
   }
  }
  b_Change=FALSE;
  pDC->BitBlt(0,0,cxClient,cyClient,&dcMem,0,0,SRCCOPY);
}
else
{
  dcMem.CreateCompatibleDC(NULL);//这里我们就在内存中虚拟建造了DC
  pOldBitmap = dcMem.SelectObject(&pDoc->m_bmpBuffer);//我们调入了我们bitmap目标
  pDC->BitBlt(0,0,cxClient,cyClient,&dcMem,0,0,SRCCOPY);
}
dcMem.SelectObject(pOldBitmap);
dcMem.DeleteDC();
}

曼得布罗特集的绘制：

void CMandelbrotView::OnDraw(CDC* pDC/*pDC*/)
{
CMandelbrotDoc* pDoc = GetDocument();
ASSERT_VALID(pDoc);

// TODO: 在此处为本机数据添加绘制代码
int i,j,k;
CDC dcMem;
CBitmap* pOldBitmap = NULL;
if(b_Change==TRUE)
{
  double lx,ly,x,y,m,n,r,a,b;
  const int kmax=256,M=4,iColor[19]={0x800000,0x8000,0x808000,0x80,0x800080,0x8080,0xc0c0c0,0xc0dcc0,0xa6caf0,0xfffbf0,0xa0a0a4,0x808080,0xff0000,0xff00,0xffff00,0xff,0xff00ff,0xffff,0xffffff};
  CRect rect;
  GetClientRect(rect);
  cxClient=rect.right-rect.left;
  cyClient=rect.bottom-rect.top;
  lx=(m_Xmax-m_Xmin)/cxClient;
  ly=(m_Ymax-m_Ymin)/cyClient;
  dcMem.CreateCompatibleDC(pDC);
  pDoc->m_bmpBuffer.DeleteObject();
  pDoc->m_bmpBuffer.CreateCompatibleBitmap(pDC,cxClient,cyClient);
  pOldBitmap = dcMem.SelectObject(&pDoc->m_bmpBuffer);
  ::SetCursor(::LoadCursor(NULL,IDC_WAIT));
  for(i=0;i<cxClient;i++)
  {
   for(j=0;j<cyClient;j++)
   {
    a=m_Xmin+i*lx;
    b=m_Ymax-j*ly;
    k=0;
    x=0;
    y=0;
    while(TRUE)
    {
     m=x*x-y*y+a;
     n=2*x*y+b;
     k++;
     r=m*m+n*n;
     if(r>M)
     {
      dcMem.SetPixel(i,j,iColor[(k-1)%19]);
      break;
     }
     if(k==kmax)
     {
      dcMem.SetPixel(i,j,0);
      break;
     }
     x=m;
     y=n;
    }
   }
  }
  b_Change=FALSE;
  pDC->BitBlt(0,0,cxClient,cyClient,&dcMem,0,0,SRCCOPY);
}
else
{
  dcMem.CreateCompatibleDC(NULL);//这里我们就在内存中虚拟建造了DC
  pOldBitmap = dcMem.SelectObject(&pDoc->m_bmpBuffer);//我们调入了我们bitmap目标
  pDC->BitBlt(0,0,cxClient,cyClient,&dcMem,0,0,SRCCOPY);
}
dcMem.SelectObject(pOldBitmap);
dcMem.DeleteDC();
}

两者大同小异，仅在算法部分略有不同。