- 先了解一下曼德博集合
下面引用百科
曼德布洛特集合（Mandelbrot set）是在复平面上组成分形的点的集合，一种 案。
- 中文名
- 曼德勃罗集合 外文名
- Mandelbrot set
- 含 义
- 复平面上组成分形的点的集合 性 质
- 案
目录
- 1
- 2
曼德布洛特复数集合（Mandelbrot set，或译为曼德博集合）是一种在 上组成分形的点的集合，以数学家 的名字命名。曼德博集合与 有些相似的地方，例如使用相同的复二次多项式来进行 。
曼德布洛特集合可以用复二次多项式: f_c( z) = z^2 + c
来定义 其中 c是一个复参数。对于每一个 c，从开始对 f c( z)进行迭代。
序列 的值或者延伸到无限大，或者只停留在有限半径的圆盘内。
曼德布洛特集合就是使以上序列不延伸至无限大的所有 c点的集合。
从数学上来讲，曼德布洛特集合是一个 的集合。一个给定的复数 c或者属于曼德布洛特集合 M，或者不是。
计算的方法
曼德布洛特集合一般用 计算。对于大多数的分形软件，例如Ultra fractal，内部已经有了比较成熟的例子。下面的程序是一段 ，表达了曼德布洛特集合的计算思路[1]。
For Each z0 in Complex repeats = 0 z=z0 Do z=z^2+z0 repeate = repeats+1 Loop until abs(z)>Bailout or repeats >= MaxRepeats If repeats >= MaxRepeats Then Draw z0,Black Else Draw z0,f(z,z0,Repeats) 'f返回颜色 End IfNextf函数的一些例子直接利用循环终止时的Repeats 综合利用z和Repeats Orbit Traps也可以用Mathematica制作 DensityPlot[Block[{z, t = 0}, z = x + y*I; While[(Abs[z] < 2.0) && (t < 100), ++t; z = z^2 + x + y*I]; Return[t]],{x, -2, 0.8}, {y, -1.5, 1.5}, PlotPoints -> 500, Mesh -> False
效果
- 实现代码
// 曼德博集合.cpp : Defines the entry point for the application.//#include "stdafx.h"#include "resource.h"#include
#define MAX_LOADSTRING 100// Global Variables:HINSTANCE hInst; // current instanceTCHAR szTitle[MAX_LOADSTRING]; // The title bar textTCHAR szWindowClass[MAX_LOADSTRING]; // The title bar textconst int iXmax = 600;const int iYmax = 600;const double CxMin = -2.5;const double CxMax = 1.5;const double CyMin = -2.0;const double CyMax = 2.0;const int IterationMax = 200;const double EscapeRadius = 2;// Foward declarations of functions included in this code module:ATOM MyRegisterClass(HINSTANCE hInstance);BOOL InitInstance(HINSTANCE, int);LRESULT CALLBACK WndProc(HWND, UINT, WPARAM, LPARAM);LRESULT CALLBACK About(HWND, UINT, WPARAM, LPARAM);int APIENTRY WinMain(HINSTANCE hInstance, HINSTANCE hPrevInstance, LPSTR lpCmdLine, int nCmdShow){ // TODO: Place code here. MSG msg; HACCEL hAccelTable; // Initialize global strings LoadString(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING); LoadString(hInstance, IDC_MY, szWindowClass, MAX_LOADSTRING); MyRegisterClass(hInstance); // Perform application initialization: if (!InitInstance (hInstance, nCmdShow)) { return FALSE; } hAccelTable = LoadAccelerators(hInstance, (LPCTSTR)IDC_MY); // Main message loop: while (GetMessage(&msg, NULL, 0, 0)) { if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg)) { TranslateMessage(&msg); DispatchMessage(&msg); } } return msg.wParam;}//// FUNCTION: MyRegisterClass()//// PURPOSE: Registers the window class.//// COMMENTS://// This function and its usage is only necessary if you want this code// to be compatible with Win32 systems prior to the 'RegisterClassEx'// function that was added to Windows 95. It is important to call this function// so that the application will get 'well formed' small icons associated// with it.//ATOM MyRegisterClass(HINSTANCE hInstance){ WNDCLASSEX wcex; wcex.cbSize = sizeof(WNDCLASSEX); wcex.style = CS_HREDRAW | CS_VREDRAW; wcex.lpfnWndProc = (WNDPROC)WndProc; wcex.cbClsExtra = 0; wcex.cbWndExtra = 0; wcex.hInstance = hInstance; wcex.hIcon = LoadIcon(hInstance, (LPCTSTR)IDI_MY); wcex.hCursor = LoadCursor(NULL, IDC_ARROW); wcex.hbrBackground = (HBRUSH)(COLOR_WINDOW+1); wcex.lpszMenuName = (LPCSTR)IDC_MY; wcex.lpszClassName = szWindowClass; wcex.hIconSm = LoadIcon(wcex.hInstance, (LPCTSTR)IDI_SMALL); return RegisterClassEx(&wcex);}//// FUNCTION: InitInstance(HANDLE, int)//// PURPOSE: Saves instance handle and creates main window//// COMMENTS://// In this function, we save the instance handle in a global variable and// create and display the main program window.//BOOL InitInstance(HINSTANCE hInstance, int nCmdShow){ HWND hWnd; hInst = hInstance; // Store instance handle in our global variable hWnd = CreateWindow(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW, CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, NULL, NULL, hInstance, NULL); if (!hWnd) { return FALSE; } ShowWindow(hWnd, nCmdShow); UpdateWindow(hWnd); return TRUE;}//// FUNCTION: WndProc(HWND, unsigned, WORD, LONG)//// PURPOSE: Processes messages for the main window.//// WM_COMMAND - process the application menu// WM_PAINT - Paint the main window// WM_DESTROY - post a quit message and return////LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam){ int wmId, wmEvent; PAINTSTRUCT ps; HDC hdc; TCHAR szHello[MAX_LOADSTRING]; LoadString(hInst, IDS_HELLO, szHello, MAX_LOADSTRING); switch (message) { case WM_COMMAND: wmId = LOWORD(wParam); wmEvent = HIWORD(wParam); // Parse the menu selections: switch (wmId) { case IDM_ABOUT: DialogBox(hInst, (LPCTSTR)IDD_ABOUTBOX, hWnd, (DLGPROC)About); break; case IDM_EXIT: DestroyWindow(hWnd); break; default: return DefWindowProc(hWnd, message, wParam, lParam); } break; case WM_PAINT: { //添加大括号，使内部的变量初始化赋值编译通过 hdc = BeginPaint(hWnd, &ps); // TODO: Add any drawing code here... int iX,iY; double Cx,Cy; double PixelWidth = (CxMax - CxMin)/iXmax; double PixelHeight; PixelHeight = (CyMax - CyMin)/iYmax; COLORREF color; double Zx,Zy; double Zx2,Zy2; INT Iteration; double ER2; ER2 = EscapeRadius *EscapeRadius; //依据曼德博集合计算原理，获取每个像素的属性 for(iY = 0;iY < iYmax;iY++)//循环每一行 { Cy = CyMin + iY*PixelHeight; if(fabs(Cy) < PixelHeight/2) Cy = 0.0; for(iX=0;iX < iXmax;iX++)//循环每一列 { Cx = CxMin +iX* PixelHeight; Zx = Zy= Zy2=Zx2 = 0.0; Iteration = 0; //针对每个像素进行循环计算 while(Iteration < IterationMax && ((Zx2 + Zy2)