Blog Entry © Sunday, April 26, 2026, by James Pate Williams, Jr. Butterfly Curve

// ButterflyEquation.cpp : Defines the entry point for the application.
//

#include "framework.h"
#include "ButterflyEquation.h"
#include <vector>

#define MAX_LOADSTRING 100

typedef struct tagPoint3d
{
    double t, x, y;
} Point3d, * PPoint3d;

// Global Variables:
HINSTANCE hInst;                                // current instance
WCHAR szTitle[MAX_LOADSTRING];                  // The title bar text
WCHAR szWindowClass[MAX_LOADSTRING];            // the main window class name
std::vector<Point3d> points;

// Forward declarations of functions included in this code module:
ATOM                MyRegisterClass(HINSTANCE hInstance);
BOOL                InitInstance(HINSTANCE, int);
LRESULT CALLBACK    WndProc(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK    About(HWND, UINT, WPARAM, LPARAM);

int APIENTRY wWinMain(_In_ HINSTANCE hInstance,
                     _In_opt_ HINSTANCE hPrevInstance,
                     _In_ LPWSTR    lpCmdLine,
                     _In_ int       nCmdShow)
{
    UNREFERENCED_PARAMETER(hPrevInstance);
    UNREFERENCED_PARAMETER(lpCmdLine);

    // TODO: Place code here.

    // Initialize global strings
    LoadStringW(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
    LoadStringW(hInstance, IDC_BUTTERFLYEQUATION, szWindowClass, MAX_LOADSTRING);
    MyRegisterClass(hInstance);

    // Perform application initialization:
    if (!InitInstance (hInstance, nCmdShow))
    {
        return FALSE;
    }

    HACCEL hAccelTable = LoadAccelerators(hInstance, MAKEINTRESOURCE(IDC_BUTTERFLYEQUATION));

    MSG msg;

    // Main message loop:
    while (GetMessage(&msg, nullptr, 0, 0))
    {
        if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg))
        {
            TranslateMessage(&msg);
            DispatchMessage(&msg);
        }
    }

    return (int) msg.wParam;
}

//
//  FUNCTION: MyRegisterClass()
//
//  PURPOSE: Registers the window class.
//
ATOM MyRegisterClass(HINSTANCE hInstance)
{
    WNDCLASSEXW wcex = { 0 };

    wcex.cbSize = sizeof(WNDCLASSEX);

    wcex.style          = CS_HREDRAW | CS_VREDRAW;
    wcex.lpfnWndProc    = WndProc;
    wcex.cbClsExtra     = 0;
    wcex.cbWndExtra     = 0;
    wcex.hInstance      = hInstance;
    wcex.hIcon          = LoadIcon(hInstance, MAKEINTRESOURCE(IDI_BUTTERFLYEQUATION));
    wcex.hCursor        = LoadCursor(nullptr, IDC_ARROW);
    wcex.hbrBackground  = (HBRUSH)(COLOR_WINDOW+1);
    wcex.lpszMenuName   = MAKEINTRESOURCEW(IDC_BUTTERFLYEQUATION);
    wcex.lpszClassName  = szWindowClass;
    wcex.hIconSm        = LoadIcon(wcex.hInstance, MAKEINTRESOURCE(IDI_SMALL));

    return RegisterClassExW(&wcex);
}

//
//   FUNCTION: InitInstance(HINSTANCE, 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)
{
   hInst = hInstance; // Store instance handle in our global variable

   HWND hWnd = CreateWindowW(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
      CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, nullptr, nullptr, hInstance, nullptr);

   if (!hWnd)
   {
      return FALSE;
   }

   ShowWindow(hWnd, nCmdShow);
   UpdateWindow(hWnd);

   return TRUE;
}

void CreateButterflyGraphPoints()
{
    double p = 4.0 * atan(1.0);
    double h = 12.0 * p / 1024.0;
    double t = 0.0;

    for (int i = 0; i <= 1024; i++)
    {
        Point3d pt = { 0 };
        double c = 2.0 * cos(4.0 * t);
        double d = pow(sin(t / 12.0), 5.0);
        double x = sin(t) * (exp(cos(t)) - c - d);
        double y = cos(t) * (exp(cos(t)) - c - d);
        pt.t = t;
        pt.x = x;
        pt.y = y;
        points.push_back(pt);
        t += h;
    }
}

static void FindMinMax(
    double& xMin, double& xMax,
    double& yMin, double& yMax)
{
    // uses global 2D double points structure

    xMin = yMin = DBL_MAX;
    xMax = yMax = DBL_MIN;

    for (size_t i = 0; i < points.size(); i++)
    {
        Point3d pt = points[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }
}

static void DrawFormattedText(HDC hdc, char text[], RECT rect)
{
    // Draw the text with formatting options
    DrawTextA(hdc, text, -1, &rect, DT_SINGLELINE | DT_NOCLIP);
}

//
//  FUNCTION: WndProc(HWND, UINT, WPARAM, LPARAM)
//
//  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)
{
    switch (message)
    {
    case WM_CREATE:
        CreateButterflyGraphPoints();
        break;
    case WM_COMMAND:
        {
            int wmId = LOWORD(wParam);
            // Parse the menu selections:
            switch (wmId)
            {
            case IDM_ABOUT:
                DialogBox(hInst, MAKEINTRESOURCE(IDD_ABOUTBOX), hWnd, About);
                break;
            case IDM_EXIT:
                DestroyWindow(hWnd);
                break;
            default:
                return DefWindowProc(hWnd, message, wParam, lParam);
            }
        }
        break;
    case WM_PAINT:
        {
            PAINTSTRUCT ps;
            HDC hdc = BeginPaint(hWnd, &ps);
            double h = 0, pi = 0, plm = 0, theta = 0;
            double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
            FindMinMax(xMin, xMax, yMin, yMax);
            float xSpan = (float)(xMax - xMin);
            float ySpan = (float)(yMax - yMin);
            RECT rect = { };
            GetClientRect(hWnd, &rect);
            float width = (float)(rect.right - rect.left + 1);
            float height = (float)(rect.bottom - rect.top - 32 + 1);
            float sx0 = 2.0f * width / 16.0f;
            float sx1 = 14.0f * width / 16.0f;
            float sy0 = 2.0f * height / 16.0f;
            float sy1 = 14.0f * height / 16.0f;
            float deltaX = xSpan / 8.0f;
            float deltaY = ySpan / 8.0f;
            float xSlope = (sx1 - sx0) / xSpan;
            float xInter = (float)(sx0 - xSlope * xMin);
            float ySlope = (sy0 - sy1) / ySpan;
            float yInter = (float)(sy0 - ySlope * yMax);
            float px = 0, py = 0, sx = 0, sy = 0;
            POINT wPt = { };
            int i = 0;
            float x = (float)xMin;
            float y = (float)yMax;
            px = x;
            py = y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            char buffer[128] = { };

            while (i <= 8)
            {
                sx = xSlope * x + xInter;
                wPt.x = wPt.y = 0;
                MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
                LineTo(hdc, (int)sx, (int)sy1);

                sprintf_s(buffer, "%5.4lf", x);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    (int)strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx - size.cx / 2.0f);
                textRect.right = (long)(sx + size.cx / 2.0f);
                textRect.top = (long)sy1;
                textRect.bottom = (long)(sy1 + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
                x += deltaX;
                i++;
            }

            i = 0;
            y = (float)yMin;

            while (i <= 8)
            {
                sy = ySlope * y + yInter;
                wPt.x = wPt.y = 0;
                MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
                LineTo(hdc, (int)sx, (int)sy);

                if (i != 0)
                {
                    sprintf_s(buffer, "%+5.3lf", y);
                    SIZE size = { };
                    GetTextExtentPoint32A(
                        hdc,
                        buffer,
                        (int)strlen(buffer),
                        &size);
                    RECT textRect = { };
                    textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                    textRect.right = (long)(sx0 - size.cx / 2.0f);
                    textRect.top = (long)(sy - size.cy / 2.0f);
                    textRect.bottom = (long)(sy + size.cy / 2.0f);
                    DrawFormattedText(hdc, buffer, textRect);
                }

                y += deltaY;
                i++;
            }

            HGDIOBJ bPenNew = NULL;
            HGDIOBJ hPenOld = NULL;

            bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
            hPenOld = SelectObject(hdc, bPenNew);

            px = (float)points[0].x;
            py = (float)points[0].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy, &wPt);

            for (size_t j = 1; j < points.size(); j++)
            {
                px = (float)points[j].x;
                py = (float)points[j].y;
                sx = xSlope * px + xInter;
                sy = ySlope * py + yInter;
                LineTo(hdc, (int)sx, (int)sy);
            }

            SelectObject(hdc, hPenOld);
            DeleteObject(bPenNew);
            EndPaint(hWnd, &ps);
        }
        break;
    case WM_DESTROY:
        PostQuitMessage(0);
        break;
    default:
        return DefWindowProc(hWnd, message, wParam, lParam);
    }
    return 0;
}

// Message handler for about box.
INT_PTR CALLBACK About(HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    switch (message)
    {
    case WM_INITDIALOG:
        return (INT_PTR)TRUE;

    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    }
    return (INT_PTR)FALSE;
}