矩阵中的数学旋转公式 转换到 C++中函数 替换DirectX 9.0中D3DXMatrixRotationAxis函数

还记得DirectX 9.0 C++ 教程 光照中这个旋转的水壶吗？当时我们用的是官方函数D3DXMatrixRotationY(D3DXMATRIX *pOut, FLOAT Angle)。刚刚我们已经学习了线性代数中的一些基础知识，Matrix 矩阵，单位矩阵，Transposition，矩阵与矩阵相乘，矩阵中的旋转(Rotation)，我已经迫不及待想用自己写的旋转函数替换官方旋转函数了，来更好的记住数学知识，你呢？

对上面这张gif图片是如何制作感兴趣？看截取视频，然后制成gif 教程

首先看下数学公式：

由上面的数学公式，我们转换的C++函数：

//-----------------------------------------------------------------------------
// Name: setupRotate()
// Desc: 绕x,y,z轴旋转,axis:1表示绕x轴，2表示y轴，3表示z轴
//-----------------------------------------------------------------------------
VOID setupRotate(D3DXMATRIXA16 *returnMatrix,int axis, float theta)
{
	float s, c;
	s = sin(theta),c=cos(theta);
	
	switch (axis) {
	case 1: // Rotate about the x-axis
		returnMatrix->_11 = 1.0f; returnMatrix->_12 = 0.0f; returnMatrix->_13 = 0.0f; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = 0.0f; returnMatrix->_22 = c; returnMatrix->_23 = s; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = 0.0f; returnMatrix->_32 = -s; returnMatrix->_33 = c; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	case 2: // Rotate about the y-axis
		returnMatrix->_11 = c; returnMatrix->_12 = 0.0f; returnMatrix->_13 = -s; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = 0.0f; returnMatrix->_22 = 1.0f; returnMatrix->_23 = 0.0f; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = s; returnMatrix->_32 = 0.0f; returnMatrix->_33 = c; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	case 3: // Rotate about the z-axis
		returnMatrix->_11 = c; returnMatrix->_12 = s; returnMatrix->_13 = 0.0f; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = -s; returnMatrix->_22 = c; returnMatrix->_23 = 0.0f; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = 0.0f; returnMatrix->_32 = 0.0f; returnMatrix->_33 = 1.0f; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	default:
		// bogus axis index
		assert(false);
	}
}
//-----------------------------------------------------------------------------
// Name: setupRotate()
// Desc: 绕自定义轴旋转
//-----------------------------------------------------------------------------
VOID setupRotate(const D3DXVECTOR3 &axis, D3DXMATRIXA16 *returnMatrix, float theta)
{
   //要旋转的向量轴必须是单位向量，不然物体会变形
   //assert(fabs(axis*axis - 1.0f) < .01f);
   float s, c;
	s = sin(theta),c=cos(theta);
	// 计算出一些重复利用的子表达式。
	float a = 1.0f - c;
	float ax = a * axis.x;
	float ay = a * axis.y;
	float az = a * axis.z;
	returnMatrix->_11 = ax*axis.x + c;
	returnMatrix->_12 = ax*axis.y + axis.z*s;
	returnMatrix->_13 = ax*axis.z - axis.y*s;
	returnMatrix->_14 = 0.0f;
	returnMatrix->_21 = ay*axis.x - axis.z*s;
	returnMatrix->_22 = ay*axis.y + c;
	returnMatrix->_23 = ay*axis.z + axis.x*s;
	returnMatrix->_24 = 0.0f;
	returnMatrix->_31 = az*axis.x + axis.y*s;
    returnMatrix->_32 = az*axis.y - axis.x*s;
    returnMatrix->_33 = az*axis.z + c;
	returnMatrix->_34 = 0.0f;
    returnMatrix->_41 = 0.0f;
	returnMatrix->_42 = 0.0f;
	returnMatrix->_43 = 0.0f;
	returnMatrix->_44 = 1.0f;
}

可能会有人对上面函数中的D3DXMATRIXA16和D3DXVECTOR3不理解，他们其实是Direct3D中表示4维矩阵和3维向量的结构体，可是我们数学公式里涉及到的是3维矩阵，可以看下下面的转换，因为要设计到矩阵相乘，故只能4维矩阵跟4维矩阵相乘，故把3维的转换成4维的了。

具体使用上面两个函数的例子：

    // 旋转
    D3DXMATRIXA16 matWorld;
	//自定义旋转的轴
	D3DXVECTOR3 axis(1.0f/sqrt(3), 1.0f/sqrt(3), 1.0f/sqrt(3));

    D3DXMatrixIdentity( &matWorld );

	/*DirectX 官方API*/
    //D3DXMatrixRotationY( &matWorld, timeGetTime()/500.0f );

	/*使用自定义旋转函数1*/
	//setupRotate(&matWorld,2,timeGetTime()/500.0f);

    /*使用自定义旋转函数2*/
	setupRotate(axis,&matWorld,timeGetTime()/500.0f);
    g_pd3dDevice->SetTransform( D3DTS_WORLD, &matWorld );

旋转效果：

我稍微修改了下自定义旋转的轴，改成一个非单位向量，(1.0f, 1.0f, 0.0f)，旋转效果(变形的飞起，哈哈)：

完整代码(看的麻烦的同学，可以拉到底下载整个项目来看)：

//-----------------------------------------------------------------------------
// File: Lights.cpp
// Copyright (c) Microsoft Corporation & Waitingfy.com. All rights reserved.
//-----------------------------------------------------------------------------
#include 
#include 
#include 
#include 
#include 
#include 



//-----------------------------------------------------------------------------
// Global variables
//-----------------------------------------------------------------------------
LPDIRECT3D9             g_pD3D       = NULL; // Used to create the D3DDevice
LPDIRECT3DDEVICE9       g_pd3dDevice = NULL; // Our rendering device
LPDIRECT3DVERTEXBUFFER9 g_pVB        = NULL; // Buffer to hold vertices
ID3DXMesh* Objects;//茶壶

// A structure for our custom vertex type. We added a normal, and omitted the
// color (which is provided by the material)
struct CUSTOMVERTEX
{
    D3DXVECTOR3 position; // The 3D position for the vertex
    D3DXVECTOR3 normal;   // The surface normal for the vertex
};

// Our custom FVF, which describes our custom vertex structure
#define D3DFVF_CUSTOMVERTEX (D3DFVF_XYZ|D3DFVF_NORMAL)




//-----------------------------------------------------------------------------
// Name: InitD3D()
// Desc: Initializes Direct3D
//-----------------------------------------------------------------------------
HRESULT InitD3D( HWND hWnd )
{
    // Create the D3D object.
    if( NULL == ( g_pD3D = Direct3DCreate9( D3D_SDK_VERSION ) ) )
        return E_FAIL;

    // Set up the structure used to create the D3DDevice. Since we are now
    // using more complex geometry, we will create a device with a zbuffer.
    D3DPRESENT_PARAMETERS d3dpp;
    ZeroMemory( &d3dpp, sizeof(d3dpp) );
    d3dpp.Windowed = TRUE;
    d3dpp.SwapEffect = D3DSWAPEFFECT_DISCARD;
    d3dpp.BackBufferFormat = D3DFMT_UNKNOWN;
    d3dpp.EnableAutoDepthStencil = TRUE;
    d3dpp.AutoDepthStencilFormat = D3DFMT_D16;

    // Create the D3DDevice
    if( FAILED( g_pD3D->CreateDevice( D3DADAPTER_DEFAULT, D3DDEVTYPE_HAL, hWnd,
                                      D3DCREATE_SOFTWARE_VERTEXPROCESSING,
                                      &d3dpp, &g_pd3dDevice ) ) )
    {
        return E_FAIL;
    }

    // Turn off culling
    g_pd3dDevice->SetRenderState( D3DRS_CULLMODE, D3DCULL_NONE );

    // Turn on the zbuffer
    g_pd3dDevice->SetRenderState( D3DRS_ZENABLE, TRUE );

    return S_OK;
}



//-----------------------------------------------------------------------------
// Name: Cleanup()
// Desc: Releases all previously initialized objects
//-----------------------------------------------------------------------------
VOID Cleanup()
{
    if( g_pVB != NULL )
        g_pVB->Release();

    if( g_pd3dDevice != NULL )
        g_pd3dDevice->Release();

    if( g_pD3D != NULL )
        g_pD3D->Release();
    if(Objects != NULL)
	   Objects->Release();
}
//-----------------------------------------------------------------------------
// Name: setupRotate()
// Desc: 绕x,y,z轴旋转,axis:1表示绕x轴，2表示y轴，3表示z轴
//-----------------------------------------------------------------------------
VOID setupRotate(D3DXMATRIXA16 *returnMatrix,int axis, float theta)
{
	float s, c;
	s = sin(theta),c=cos(theta);
	
	switch (axis) {
	case 1: // Rotate about the x-axis
		returnMatrix->_11 = 1.0f; returnMatrix->_12 = 0.0f; returnMatrix->_13 = 0.0f; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = 0.0f; returnMatrix->_22 = c; returnMatrix->_23 = s; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = 0.0f; returnMatrix->_32 = -s; returnMatrix->_33 = c; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	case 2: // Rotate about the y-axis
		returnMatrix->_11 = c; returnMatrix->_12 = 0.0f; returnMatrix->_13 = -s; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = 0.0f; returnMatrix->_22 = 1.0f; returnMatrix->_23 = 0.0f; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = s; returnMatrix->_32 = 0.0f; returnMatrix->_33 = c; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	case 3: // Rotate about the z-axis
		returnMatrix->_11 = c; returnMatrix->_12 = s; returnMatrix->_13 = 0.0f; returnMatrix->_14 = 0.0f;
		returnMatrix->_21 = -s; returnMatrix->_22 = c; returnMatrix->_23 = 0.0f; returnMatrix->_24 = 0.0f;
		returnMatrix->_31 = 0.0f; returnMatrix->_32 = 0.0f; returnMatrix->_33 = 1.0f; returnMatrix->_34 = 0.0f;
		returnMatrix->_41 = 0.0f; returnMatrix->_42 = 0.0f; returnMatrix->_43 = 0.0f; returnMatrix->_44 = 1.0f;
		break;
	default:
		// bogus axis index
		assert(false);
	}
}
//-----------------------------------------------------------------------------
// Name: setupRotate()
// Desc: 绕自定义轴旋转,类似DirectX中的函数D3DXMatrixRotationAxis
//-----------------------------------------------------------------------------
VOID setupRotate(const D3DXVECTOR3 &axis, D3DXMATRIXA16 *returnMatrix, float theta)
{
   //要旋转的向量轴必须是单位向量，不然物体会变形
   //assert(fabs(axis*axis - 1.0f) < .01f);
   float s, c;
	s = sin(theta),c=cos(theta);
	// 计算出一些重复利用的子表达式。
	float a = 1.0f - c;
	float ax = a * axis.x;
	float ay = a * axis.y;
	float az = a * axis.z;
	returnMatrix->_11 = ax*axis.x + c;
	returnMatrix->_12 = ax*axis.y + axis.z*s;
	returnMatrix->_13 = ax*axis.z - axis.y*s;
	returnMatrix->_14 = 0.0f;
	returnMatrix->_21 = ay*axis.x - axis.z*s;
	returnMatrix->_22 = ay*axis.y + c;
	returnMatrix->_23 = ay*axis.z + axis.x*s;
	returnMatrix->_24 = 0.0f;
	returnMatrix->_31 = az*axis.x + axis.y*s;
    returnMatrix->_32 = az*axis.y - axis.x*s;
    returnMatrix->_33 = az*axis.z + c;
	returnMatrix->_34 = 0.0f;
    returnMatrix->_41 = 0.0f;
	returnMatrix->_42 = 0.0f;
	returnMatrix->_43 = 0.0f;
	returnMatrix->_44 = 1.0f;
}

//-----------------------------------------------------------------------------
// Name: SetupMatrices()
// Desc: Sets up the world, view, and projection transform matrices.
//-----------------------------------------------------------------------------
VOID SetupMatrices()
{
    // 旋转
    D3DXMATRIXA16 matWorld;
	//自定义旋转的轴
	//D3DXVECTOR3 axis(1.0f/sqrt(3), 1.0f/sqrt(3), 1.0f/sqrt(3));
    D3DXVECTOR3 axis(1.0f, 1.0f, 0.0f);
    D3DXMatrixIdentity( &matWorld );

	/*DirectX 官方API*/
    //D3DXMatrixRotationY( &matWorld, timeGetTime()/500.0f );

	/*使用自定义旋转函数1*/
	//setupRotate(&matWorld,2,timeGetTime()/500.0f);

    /*使用自定义旋转函数2*/
	setupRotate(axis,&matWorld,timeGetTime()/500.0f);
    g_pd3dDevice->SetTransform( D3DTS_WORLD, &matWorld );

    // 摄像机的位置
    D3DXVECTOR3 vEyePt( 0.0f, 3.0f,-5.0f );
    D3DXVECTOR3 vLookatPt( 0.0f, 0.0f, 0.0f );
    D3DXVECTOR3 vUpVec( 0.0f, 1.0f, 0.0f );
    D3DXMATRIXA16 matView;
    D3DXMatrixLookAtLH( &matView, &vEyePt, &vLookatPt, &vUpVec );
    g_pd3dDevice->SetTransform( D3DTS_VIEW, &matView );

    // 设置视锥体大小
    D3DXMATRIXA16 matProj;
    D3DXMatrixPerspectiveFovLH( &matProj, D3DX_PI/4, 1.0f, 1.0f, 100.0f );
    g_pd3dDevice->SetTransform( D3DTS_PROJECTION, &matProj );
}




//-----------------------------------------------------------------------------
// Name: SetupLights()
// Desc: Sets up the Lights and materials for the scene.
//-----------------------------------------------------------------------------
VOID SetupLights()
{
    // 初始化一个材料
    D3DMATERIAL9 mtrl;
    ZeroMemory( &mtrl, sizeof(D3DMATERIAL9) );
    mtrl.Diffuse = mtrl.Ambient = D3DXCOLOR(1.0f,1.0f,0.0f,1.0f); //材质有漫射和环境光的设置，都为黄色
    g_pd3dDevice->SetMaterial( &mtrl );

    // 初始化一个白色的方向光
    D3DXVECTOR3 vecDir;
    D3DLIGHT9 light;
    ZeroMemory( &light, sizeof(D3DLIGHT9) );
    light.Type       = D3DLIGHT_DIRECTIONAL;//方向光
	light.Diffuse = D3DXCOLOR(1.0f,1.0f,1.0f,1.0f);//设置光源的漫射光颜色为白色
	light.Direction = D3DXVECTOR3(0.0f,0.0f,1.0f);//光的传播方向平行z轴
    light.Range       = 1000.0f;
    g_pd3dDevice->SetLight( 0, &light );
    g_pd3dDevice->LightEnable( 0, TRUE );
    g_pd3dDevice->SetRenderState( D3DRS_LIGHTING, TRUE );

    // Finally, turn on some ambient light.
    g_pd3dDevice->SetRenderState( D3DRS_AMBIENT, 0x00202020 );
}




//-----------------------------------------------------------------------------
// Name: Render()
// Desc: Draws the scene
//-----------------------------------------------------------------------------
VOID Render()
{
    // 背景为蓝色
    g_pd3dDevice->Clear( 0, NULL, D3DCLEAR_TARGET|D3DCLEAR_ZBUFFER,
                         D3DCOLOR_XRGB(0,0,255), 1.0f, 0 );

    // Begin the scene
    if( SUCCEEDED( g_pd3dDevice->BeginScene() ) )
    {
        // 茶壶的材料颜色也定义在这个函数中
        SetupLights();

        // Setup the world, view, and projection matrices
        SetupMatrices();

        // Render the vertex buffer contents
        g_pd3dDevice->SetStreamSource( 0, g_pVB, 0, sizeof(CUSTOMVERTEX) );
        g_pd3dDevice->SetFVF( D3DFVF_CUSTOMVERTEX );

		//创建一个茶壶
		Objects = 0;
		D3DXMATRIX  Worlds;
		D3DXCreateTeapot(g_pd3dDevice, &Objects, 0);
		Objects->DrawSubset(0);
                Objects->Release(); 
                Objects=0;
        // End the scene
        g_pd3dDevice->EndScene();
    }

    // Present the backbuffer contents to the display
    g_pd3dDevice->Present( NULL, NULL, NULL, NULL );
}




//-----------------------------------------------------------------------------
// Name: MsgProc()
// Desc: The window's message handler
//-----------------------------------------------------------------------------
LRESULT WINAPI MsgProc( HWND hWnd, UINT msg, WPARAM wParam, LPARAM lParam )
{
    switch( msg )
    {
        case WM_DESTROY:
            Cleanup();
            PostQuitMessage( 0 );
            return 0;
    }

    return DefWindowProc( hWnd, msg, wParam, lParam );
}




//-----------------------------------------------------------------------------
// Name: WinMain()
// Desc: The application's entry point
//-----------------------------------------------------------------------------
INT WINAPI WinMain( HINSTANCE hInst, HINSTANCE, LPSTR, INT )
{
    // Register the window class
    WNDCLASSEX wc = { sizeof(WNDCLASSEX), CS_CLASSDC, MsgProc, 0L, 0L,
                      GetModuleHandle(NULL), NULL, NULL, NULL, NULL,
                      "D3D Tutorial", NULL };
    RegisterClassEx( &wc );

    // Create the application's window
    HWND hWnd = CreateWindow( "D3D Tutorial", "D3D Tutorial 04: Lights",
                              WS_OVERLAPPEDWINDOW, 100, 100, 300, 300,
                              GetDesktopWindow(), NULL, wc.hInstance, NULL );

    // Initialize Direct3D
    if( SUCCEEDED( InitD3D( hWnd ) ) )
    {
            // Show the window
            ShowWindow( hWnd, SW_SHOWDEFAULT );
            UpdateWindow( hWnd );

            // Enter the message loop
            MSG msg;
            ZeroMemory( &msg, sizeof(msg) );
            while( msg.message!=WM_QUIT )
            {
                if( PeekMessage( &msg, NULL, 0U, 0U, PM_REMOVE ) )
                {
                    TranslateMessage( &msg );
                    DispatchMessage( &msg );
                }
                else
                    Render();
            }
    }

    UnregisterClass( "D3D Tutorial", wc.hInstance );
    return 0;
}

项目下载：
RotationArbitrary  (本站下载)
文章源地址：http://www.waitingfy.com/?p=363