设计一个基于API结构的windows应用程序，并使用GDI绘制抛物线，双曲线。放大缩小和移动。

首先，我们需要定义一个API结构来实现绘制抛物线、双曲线、放大缩小和移动功能。

#include <Windows.h>

// 定义抛物线结构体
struct Parabola {
    int a;  // 抛物线系数
    int b;  // 抛物线系数
    int c;  // 抛物线系数
};

// 定义双曲线结构体
struct Hyperbola {
    int a;  // 双曲线系数
    int b;  // 双曲线系数
    int c;  // 双曲线系数
};

// 定义窗口类
class Window {
public:
    Window();
    ~Window();

    bool Create(LPCTSTR lpWindowName, DWORD dwStyle, DWORD dwExStyle,
                int x, int y, int nWidth, int nHeight,
                HWND hWndParent = NULL, HMENU hMenu = NULL);
    void Show(int nCmdShow);

private:
    static LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam);

    HWND m_hWnd;
    Parabola m_parabola;
    Hyperbola m_hyperbola;
    int m_zoom;
    int m_xOffset;
    int m_yOffset;

    void DrawParabola(HDC hdc);
    void DrawHyperbola(HDC hdc);
    void ZoomIn();
    void ZoomOut();
    void MoveLeft();
    void MoveRight();
    void MoveUp();
    void MoveDown();
};

然后，我们需要在窗口类中实现绘制抛物线、双曲线、放大缩小和移动功能。

#include <math.h>

// 窗口类构造函数
Window::Window() : m_hWnd(NULL), m_zoom(1), m_xOffset(0), m_yOffset(0) {
    m_parabola.a = 1;
    m_parabola.b = 0;
    m_parabola.c = 0;

    m_hyperbola.a = 1;
    m_hyperbola.b = 0;
    m_hyperbola.c = 0;
}

// 窗口类析构函数
Window::~Window() {
}

// 创建窗口
bool Window::Create(LPCTSTR lpWindowName, DWORD dwStyle, DWORD dwExStyle,
                     int x, int y, int nWidth, int nHeight,
                     HWND hWndParent, HMENU hMenu) {
    WNDCLASSEX wcex;
    wcex.cbSize = sizeof(WNDCLASSEX);
    wcex.style = CS_HREDRAW | CS_VREDRAW;
    wcex.lpfnWndProc = WndProc;
    wcex.cbClsExtra = 0;
    wcex.cbWndExtra = 0;
    wcex.hInstance = GetModuleHandle(NULL);
    wcex.hIcon = LoadIcon(NULL, IDI_APPLICATION);
    wcex.hCursor = LoadCursor(NULL, IDC_ARROW);
    wcex.hbrBackground = (HBRUSH)(COLOR_WINDOW + 1);
    wcex.lpszMenuName = NULL;
    wcex.lpszClassName = TEXT("WindowClass");
    wcex.hIconSm = LoadIcon(NULL, IDI_APPLICATION);

    if (!RegisterClassEx(&wcex)) {
        return false;
    }

    m_hWnd = CreateWindowEx(dwExStyle, TEXT("WindowClass"), lpWindowName, dwStyle,
                            x, y, nWidth, nHeight, hWndParent, hMenu, GetModuleHandle(NULL), this);

    if (!m_hWnd) {
        return false;
    }

    return true;
}

// 显示窗口
void Window::Show(int nCmdShow) {
    ShowWindow(m_hWnd, nCmdShow);
    UpdateWindow(m_hWnd);
}

// 窗口消息处理函数
LRESULT CALLBACK Window::WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam) {
    Window* pWindow;

    if (message == WM_NCCREATE) {
        pWindow = static_cast<Window*>(reinterpret_cast<CREATESTRUCT*>(lParam)->lpCreateParams);
        SetWindowLongPtr(hWnd, GWLP_USERDATA, reinterpret_cast<LONG_PTR>(pWindow));
        pWindow->m_hWnd = hWnd;
    } else {
        pWindow = reinterpret_cast<Window*>(GetWindowLongPtr(hWnd, GWLP_USERDATA));
    }

    if (pWindow) {
        switch (message) {
        case WM_PAINT:
            HDC hdc;
            PAINTSTRUCT ps;

            hdc = BeginPaint(hWnd, &ps);

            // 绘制抛物线和双曲线
            pWindow->DrawParabola(hdc);
            pWindow->DrawHyperbola(hdc);

            EndPaint(hWnd, &ps);
            break;

        case WM_KEYDOWN:
            switch (wParam) {
            case VK_ADD:
                // 放大
                pWindow->ZoomIn();
                InvalidateRect(hWnd, NULL, TRUE);
                break;

            case VK_SUBTRACT:
                // 缩小
                pWindow->ZoomOut();
                InvalidateRect(hWnd, NULL, TRUE);
                break;

            case VK_LEFT:
                // 左移
                pWindow->MoveLeft();
                InvalidateRect(hWnd, NULL, TRUE);
                break;

            case VK_RIGHT:
                // 右移
                pWindow->MoveRight();
                InvalidateRect(hWnd, NULL, TRUE);
                break;

            case VK_UP:
                // 上移
                pWindow->MoveUp();
                InvalidateRect(hWnd, NULL, TRUE);
                break;

            case VK_DOWN:
                // 下移
                pWindow->MoveDown();
                InvalidateRect(hWnd, NULL, TRUE);
                break;
            }
            break;

        case WM_DESTROY:
            PostQuitMessage(0);
            break;

        default:
            return DefWindowProc(hWnd, message, wParam, lParam);
        }
    } else {
        return DefWindowProc(hWnd, message, wParam, lParam);
    }

    return 0;
}

// 绘制抛物线
void Window::DrawParabola(HDC hdc) {
    int x, y;
    HPEN hPen = CreatePen(PS_SOLID, 1, RGB(0, 0, 255));
    HPEN hOldPen = (HPEN)SelectObject(hdc, hPen);

    for (x = -100; x <= 100; x++) {
        y = m_parabola.a * x * x + m_parabola.b * x + m_parabola.c;
        y = -y;
        x = m_zoom * x + m_xOffset;
        y = m_zoom * y + m_yOffset;
        if (x >= 0 && x <= 800 && y >= 0 && y <= 600) {
            SetPixel(hdc, x, y, RGB(0, 0, 255));
        }
    }

    SelectObject(hdc, hOldPen);
    DeleteObject(hPen);
}

// 绘制双曲线
void Window::DrawHyperbola(HDC hdc) {
    int x, y;
    HPEN hPen = CreatePen(PS_SOLID, 1, RGB(255, 0, 0));
    HPEN hOldPen = (HPEN)SelectObject(hdc, hPen);

    for (x = -100; x <= 100; x++) {
        y = m_hyperbola.a * x * x + m_hyperbola.b * x + m_hyperbola.c;
        y = -y;
        x = m_zoom * x + m_xOffset;
        y = m_zoom * y + m_yOffset;
        if (x >= 0 && x <= 800 && y >= 0 && y <= 600) {
            SetPixel(hdc, x, y, RGB(255, 0, 0));
        }
    }

    SelectObject(hdc, hOldPen);
    DeleteObject(hPen);
}

// 放大
void Window::ZoomIn() {
    m_zoom++;
}

// 缩小
void Window::ZoomOut() {
    if (m_zoom > 1) {
        m_zoom--;
    }
}

// 左移
void Window::MoveLeft() {
    m_xOffset -= 10;
}

// 右移
void Window::MoveRight() {
    m_xOffset += 10;
}

// 上移
void Window::MoveUp() {
    m_yOffset -= 10;
}

// 下移
void Window::MoveDown() {
    m_yOffset += 10;
}

最后，我们需要在窗口的消息循环中处理消息。

int APIENTRY _tWinMain(HINSTANCE hInstance, HINSTANCE hPrevInstance, LPTSTR lpCmdLine, int nCmdShow) {
    Window window;

    if (!window.Create(TEXT("API Demo"), WS_OVERLAPPEDWINDOW, 0, CW_USEDEFAULT, 0, 800, 600)) {
        return 0;
    }

    window.Show(nCmdShow);

    MSG msg;

    while (GetMessage(&msg, NULL, 0, 0)) {
        TranslateMessage(&msg);
        DispatchMessage(&msg);
    }

    return (int)msg.wParam;
}

现在，我们就成功地实现了一个基于API结构的windows应用程序，并使用GDI绘制抛物线和双曲线，并实现了放大缩小和移动功能。