世界坐標系到攝影坐標系的變換
攝影坐標系的原點為世界坐標系中的觀察位置eye點，z軸與觀察方向（從eye點出發到at點的向量）一致。
由于僅由一個坐標原點的位置和一個z坐標軸的方向不足以完全確定一個坐標系，因此，還需要指定一個up向量來定位攝影坐標系的y軸方向。這個在世界坐標系中給出的up向量是不必與觀察方向垂直的，只是指明攝影坐標系應該以向上的方向還是以向下的方向作為y軸。
Builds a left-handed, look-at matrix.
D3DXMATRIX * D3DXMatrixLookAtLH(
D3DXMATRIX * pOut,
CONST D3DXVECTOR3 * pEye,
CONST D3DXVECTOR3 * pAt,
CONST D3DXVECTOR3 * pUp
);
Parameters
- pOut
- [in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.
- pEye
- [in] Pointer to the D3DXVECTOR3 structure that defines the eye point. This value is used in translation.
- pAt
- [in] Pointer to the D3DXVECTOR3 structure that defines the camera look-at target.
- pUp
- [in] Pointer to the D3DXVECTOR3 structure that defines the current world's up, usually [0, 1, 0].
Return Values
Pointer to a D3DXMATRIX structure that is a left-handed, look-at matrix.
Remarks
The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixLookAtLH function can be used as a parameter for another function.
This function uses the following formula to compute the returned matrix.
zaxis = normal(At - Eye)
xaxis = normal(cross(Up, zaxis))
yaxis = cross(zaxis, xaxis)
xaxis.x yaxis.x zaxis.x 0
xaxis.y yaxis.y zaxis.y 0
xaxis.z yaxis.z zaxis.z 0
-dot(xaxis, eye) -dot(yaxis, eye) -dot(zaxis, eye) 1
要正確運行以下的示例程序，需要在工程中包含d3dx9.lib，或者在main函數前加入
#pragma comment(lib, "d3dx9.lib")
以指示編譯器鏈接d3dx9.lib。
代碼示例：
#include <d3dx9math.h>
#include <stdio.h>
#pragma warning(disable : 4305)
int main()
{
D3DXVECTOR3 p(3.0, 3.0, 3.0); // point with world coordinate
D3DXVECTOR3 eye(1.0, 0.0, 0.0); // camera position with world coordinate
D3DXVECTOR3 at(0.0, 0.0, 0.0); // look at target with world coordinate
D3DXVECTOR3 up(0.0, 1.0, 0.0); // define the current world's up
D3DXMATRIX m; // transform matrix from world coordinate to camera coordinate
// Builds a left-handed, look-at matrix.
D3DXMatrixLookAtLH(&m, &eye, &at, &up);
// print out matrix
printf(" |%f %f %f %f|\n", m._11, m._12, m._13, m._14);
printf(" |%f %f %f %f|\n", m._21, m._22, m._23, m._24);
printf("m =|%f %f %f %f|\n", m._31, m._32, m._33, m._34);
printf(" |%f %f %f %f|\n", m._41, m._42, m._43, m._44);
printf("\n\n\n");
D3DXVec3TransformCoord(&p, &p, &m);
printf("p(%f, %f, %f)", p.x, p.y, p.z);
printf("\n\n\n");
return 0;
}
輸出：
|0.000000 0.000000 -1.000000 0.000000|
|-0.000000 1.000000 0.000000 0.000000|
m = |1.000000 0.000000 0.000000 0.000000|
|-0.000000 -0.000000 1.000000 1.000000|
p(3.000000, 3.000000, -2.000000)
透視投影變換
為了顯示三維物體，必須將三維物體透視投影到平面上，以獲得一個二維的平面圖象，同時還要保存物體的深度信息（z軸上的位置信息）。
Builds a left-handed perspective projection matrix based on a field of view.
D3DXMATRIX * D3DXMatrixPerspectiveFovLH(
D3DXMATRIX * pOut,
FLOAT fovy,
FLOAT Aspect,
FLOAT zn,
FLOAT zf
);
Parameters
- pOut
- [in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.
- fovy
- [in] Field of view in the y direction, in radians.
- Aspect
- [in] Aspect ratio, defined as view space width divided by height.
- zn
- [in] Z-value of the near view-plane.
- zf
- [in] Z-value of the far view-plane.
Return Values
Pointer to a D3DXMATRIX structure that is a left-handed perspective projection matrix.
Remarks
The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveFovLH function can be used as a parameter for another function.
This function computes the returned matrix as shown:
xScale 0 0 0
0 yScale 0 0
0 0 zf/(zf-zn) 1
0 0 -zn*zf/(zf-zn) 0
where:
yScale = cot(fovY/2)
xScale = yScale / aspect ratio
視截體的平面計算
在世界坐標系中選擇觀察者位置，并確定視角的方向和視線的遠近距離，視截體的6個側面也就決定下來，由于三維物體一開始是在世界坐標系中給出的，因此視截體在世界坐標系中的6個平面的計算就顯得相當重要，因為可由這6個視截體的平面法向量計算出世界空間中的哪些三維物體應該顯示出來。
完全可以根據觀察者的坐標位置、觀察視角的大小和遠近平面的方位，直接計算出6個視截體平面在世界坐標系中的方程。不過，DirectX并沒有直接提供一個函數來計算這些平面的方程，但可以根據攝影變換和透視投影變換矩陣，反向計算出視截體的各個平面在世界坐標系中的方程。
世界空間中的視截體在攝影變換和透視投影變換后，變成攝影空間中的立方體[-1, 1] x [-1, 1] x [0, 1]，即6個視截體平面的方程分別為x=-1，x=1，y=-1，y=1，z=0和z=1。
代碼示例：
#include <d3dx9math.h>
#include <stdio.h>
#pragma warning(disable : 4305)
int main()
{
// build camera reference frame
D3DXVECTOR3 eye(0.0, 0.0, 3.0); // camera position with world coordinate
D3DXVECTOR3 at(0.0, 0.0, 30.0); // look at target with world coordinate
D3DXVECTOR3 up(0.0, 1.0, 3.0); // define the current world's up
D3DXMATRIX view_matrix; // transform matrix from world coordinate to camera coordinate
// Builds a left-handed, look-at matrix.
D3DXMatrixLookAtLH(&view_matrix, &eye, &at, &up);
D3DXMATRIX proj_matrix;
// Builds a left-handed perspective projection matrix based on a field of view.
D3DXMatrixPerspectiveFovLH(&proj_matrix, D3DX_PI/3, 2.0, 2.0, 8.0);
printf("calculate frustum plane equation in world reference frame:\n\n");
D3DXMATRIX M = view_matrix * proj_matrix;
D3DXPLANE plane[6];
plane[0].a = M._11 + M._14;
plane[0].b = M._21 + M._24;
plane[0].c = M._31 + M._34;
plane[0].d = M._41 + M._44;
D3DXPlaneNormalize(&plane[0], &plane[0]);
printf("left plane equation: a=%f, b=%f, c=%f, d=%f\n\n", plane[0].a, plane[0].b, plane[0].c, plane[0].d);
plane[1].a = M._14 - M._11;
plane[1].b = M._24 - M._21;
plane[1].c = M._34 - M._31;
plane[1].d = M._44 - M._41;
D3DXPlaneNormalize(&plane[1], &plane[1]);
printf("right plane equation: a=%f, b=%f, c=%f, d=%f\n\n", plane[1].a, plane[1].b, plane[1].c, plane[1].d);
plane[2].a = M._14 - M._12;
plane[2].b = M._24 - M._22;
plane[2].c = M._34 - M._32;
plane[2].d = M._44 - M._42;
D3DXPlaneNormalize(&plane[2], &plane[2]);
printf("top plane equation: a=%f, b=%f, c=%f, d=%f\n\n", plane[2].a, plane[2].b, plane[2].c, plane[2].d);
plane[3].a = M._12 + M._14;
plane[3].b = M._22 + M._24;
plane[3].c = M._32 + M._34;
plane[3].d = M._42 + M._44;
D3DXPlaneNormalize(&plane[3], &plane[3]);
printf("bottom plane equation: a=%f, b=%f, c=%f, d=%f\n\n", plane[3].a, plane[3].b, plane[3].c, plane[3].d);
plane[4].a = M._13;
plane[4].b = M._23;
plane[4].c = M._33;
plane[4].d = M._43;
D3DXPlaneNormalize(&plane[4], &plane[4]);
printf("near plane equation: a=%f, b=%f, c=%f, d=%f\n\n", plane[4].a, plane[4].b, plane[4].c, plane[4].d);
plane[5].a = M._14 - M._13;
plane[5].b = M._24 - M._23;
plane[5].c = M._34 - M._33;
plane[5].d = M._44 - M._43;
D3DXPlaneNormalize(&plane[5], &plane[5]);
printf("far plane equation: a=%f, b=%f, c=%f, d=%f\n\n\n\n", plane[5].a, plane[5].b, plane[5].c, plane[5].d);
return 0;
}
輸出：
calculate frustum plane equation in world reference frame:
left plane equation: a=0.654654, b=0.000000, c=0.755929, d=-2.267787
right plane equation: a=-0.654654, b=0.000000, c=0.755929, d=-2.267787
top plane equation: a=0.000000, b=-0.866025, c=0.500000, d=-1.500000
bottom plane equation: a=0.000000, b=0.866025, c=0.500000, d=-1.500000
near plane equation: a=0.000000, b=0.000000, c=1.000000, d=-5.000000
far plane equation: a=0.000000, b=0.000000, c=-1.000000, d=11.000000
透視投影空間到屏幕視口的變換
視截體透視投影到[-1, -1] x [-1, 1] x [0, 1]立方體后，就需要將xy平面中的二維圖像變換到計算機的屏幕視口中進行顯示，同時還將z軸[0, 1]變換為[min_z, max_z]，這就是透視投影空間到視口的坐標變換。對于DirectX來說，通常取min_z = 0和max_z = 1，即維持z軸的坐標值不變。
在三維游戲開發中，一般使用D3DVIEWPORT9結構體和 IDirect3DDevice9::SetViewport Method設置視口的區域。