投影變換

將攝影空間中的三維物體投影到二維膠片上，也就是Direct3D中的屏幕，這種三維到二維的變換過程就是投影變換，即從取景空間到攝影空間的變換。設三維物體在觀察空間中的坐標為P_view，投影矩陣為M_proj，則頂點在投影空間中的坐標為：

P_proj = P_view * M_proj

下面分別介紹兩種基本的投影變換：正交投影和透視投影，以及它們在Direct3D中的實現。

 

1、正交投影

正交投影中，投影向量和觀察平面垂直，物體坐標沿觀察坐標系的z軸平行投影到觀察平面上，觀察點和觀察平面間的距離不會影響物體的投影大小。

工程設計中的頂視圖、前視圖和側視圖就是典型的正交投影。與世界變換、取景變換類似，只需先生成一個投影矩陣mat_proj，然后調用下面的代碼就可以設置投影矩陣：

g_device->SetTransform(D3DTS_PROJECTION, &mat_proj);

下面來看看正交投影矩陣的生成。對于正交投影來說，它的取景范圍是一個長方體，只有在這個長方體中的景物才會被繪制出來。

Direct3D擴展實用庫提供了函數D3DXMatrixOrthoLH()，用于創建一個正交投影矩陣，函數D3DXMatrixOrthoLH()的聲明如下：

Builds a left-handed orthographic projection matrix.

D3DXMATRIX * D3DXMatrixOrthoLH(
  D3DXMATRIX * pOut,
  FLOAT w,
  FLOAT h,
  FLOAT zn,
  FLOAT zf
);

Parameters

pOut

[in, out] Pointer to the resulting D3DXMATRIX.

w

[in] Width of the view volume.

h

[in] Height of the view volume.

zn

[in] Minimum z-value of the view volume which is referred to as z-near.

zf

[in] Maximum z-value of the view volume which is referred to as z-far.

Return Values

Pointer to the resulting D3DXMATRIX.

Remarks

All the parameters of the D3DXMatrixOrthoLH function are distances in camera space. The parameters describe the dimensions of the view volume.

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixOrthoLH function can be used as a parameter for another function.

This function uses the following formula to compute the returned matrix.

2/w  0    0           0
0    2/h  0           0
0    0    1/(zf-zn)   0
0    0   -zn/(zf-zn)  1

2、透視投影

透視投影實現的是一個縮放、透視的投影。透視投影的特點是，距離攝像機越遠的物體在投影平面上的成像越小，透視投影的取景范圍是一個截頭體（四棱臺）。這個截頭體稱為取景截頭體（viewing frustum），攝像機位于四棱錐的頂點。這個四棱錐被截頭體的遠平面和近平面分割，遠近裁剪面中間的部分就是取景截頭體，只有這個空間里的對象才是可見的。

透視投影矩陣的作用就是將取景截頭體內的景物投影到攝像機的二維膠片上，可以利用Direct3D功能擴展庫提供的D3DXMatrixPerspectiveFovLH()，構建一個透視投影矩陣：

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

透視投影矩陣的作用是將一個取景截頭體轉換成一個立方體。因為截頭體的近端比遠端小，所以靠近攝像機的對象將被放大，而對象距離攝像機越遠，其成像越小，這就是場景的透視原理。透視變換把一個取景截頭體轉換成一個新的坐標空間，注意，該截頭體變成了一個立方體，同時，原點從場景的右上角移動到了立方體的中心。在透視變換中，x軸和z軸方向的極限都是-1和1，z軸方向對于前平面的極限是0，對后平面的極限是1。

另外，D3DX還提供了下列函數供程序員創建透視投影變換矩陣:

D3DXMatrixPerspectiveLH

Builds a left-handed perspective projection matrix

D3DXMATRIX * D3DXMatrixPerspectiveLH(
  D3DXMATRIX * pOut,
  FLOAT w,
  FLOAT h,
  FLOAT zn,
  FLOAT zf
);

Parameters

pOut

[in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.

w

[in] Width of the view volume at the near view-plane.

h

[in] Height of the view volume at the near view-plane.

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

All the parameters of the D3DXMatrixPerspectiveLH function are distances in camera space. The parameters describe the dimensions of the view volume.

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveLH function can be used as a parameter for another function.

This function uses the following formula to compute the returned matrix.

2*zn/w  0       0              0
0       2*zn/h  0              0
0       0       zf/(zf-zn)     1
0       0       zn*zf/(zn-zf)  0

D3DXMatrixPerspectiveRH

Builds a right-handed perspective projection matrix.

D3DXMATRIX * D3DXMatrixPerspectiveRH(
  D3DXMATRIX * pOut,
  FLOAT w,
  FLOAT h,
  FLOAT zn,
  FLOAT zf
);

Parameters

pOut

[in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.

w

[in] Width of the view volume at the near view-plane.

h

[in] Height of the view volume at the near view-plane.

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 right-handed perspective projection matrix.

Remarks

All the parameters of the D3DXMatrixPerspectiveRH function are distances in camera space. The parameters describe the dimensions of the view volume.

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveRH function can be used as a parameter for another function.

This function uses the following formula to compute the returned matrix.

2*zn/w  0       0              0
0       2*zn/h  0              0
0       0       zf/(zn-zf)    -1
0       0       zn*zf/(zn-zf)  0

 

D3DXMatrixPerspectiveFovRH

Builds a right-handed perspective projection matrix based on a field of view.

D3DXMATRIX * D3DXMatrixPerspectiveFovRH(
  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 right-handed perspective projection matrix.

Remarks

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveFovRH 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/(zn-zf)        -1
0        0        zn*zf/(zn-zf)      0
where:
yScale = cot(fovY/2)
    
xScale = yScale / aspect ratio

 

D3DXMatrixPerspectiveOffCenterLH

Builds a customized, left-handed perspective projection matrix.

D3DXMATRIX * D3DXMatrixPerspectiveOffCenterLH(
  D3DXMATRIX * pOut,
  FLOAT l,
  FLOAT r,
  FLOAT b,
  FLOAT t,
  FLOAT zn,
  FLOAT zf
);

Parameters

pOut

[in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.

l

[in] Minimum x-value of the view volume.

r

[in] Maximum x-value of the view volume.

b

[in] Minimum y-value of the view volume.

t

[in] Maximum y-value of the view volume.

zn

[in] Minimum z-value of the view volume.

zf

[in] Maximum z-value of the view volume.

Return Values

Pointer to a D3DXMATRIX structure that is a customized, left-handed perspective projection matrix.

Remarks

All the parameters of the D3DXMatrixPerspectiveOffCenterLH function are distances in camera space. The parameters describe the dimensions of the view volume.

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveOffCenterLH function can be used as a parameter for another function.

This function uses the following formula to compute the returned matrix.

2*zn/(r-l)   0            0              0
0            2*zn/(t-b)   0              0
(l+r)/(l-r)  (t+b)/(b-t)  zf/(zf-zn)     1
0            0            zn*zf/(zn-zf)  0

D3DXMatrixPerspectiveOffCenterRH

Builds a customized, right-handed perspective projection matrix.

D3DXMATRIX * D3DXMatrixPerspectiveOffCenterRH(
  D3DXMATRIX * pOut,
  FLOAT l,
  FLOAT r,
  FLOAT b,
  FLOAT t,
  FLOAT zn,
  FLOAT zf
);

Parameters

pOut

[in, out] Pointer to the D3DXMATRIX structure that is the result of the operation.

l

[in] Minimum x-value of the view volume.

r

[in] Maximum x-value of the view volume.

b

[in] Minimum y-value of the view volume.

t

[in] Maximum y-value of the view volume.

zn

[in] Minimum z-value of the view volume.

zf

[in] Maximum z-value of the view volume.

Return Values

Pointer to a D3DXMATRIX structure that is a customized, right-handed perspective projection matrix.

Remarks

All the parameters of the D3DXMatrixPerspectiveOffCenterRH function are distances in camera space. The parameters describe the dimensions of the view volume.

The return value for this function is the same value returned in the pOut parameter. In this way, the D3DXMatrixPerspectiveOffCenterRH function can be used as a parameter for another function.

This function uses the following formula to compute the returned matrix.

2*zn/(r-l)   0            0                0
0            2*zn/(t-b)   0                0
(l+r)/(r-l)  (t+b)/(t-b)  zf/(zn-zf)      -1
0            0            zn*zf/(zn-zf)    0

3、w友好投影矩陣

經過頂點坐標變換后，每個頂點坐標將具有4個元素（x, y, z, w）。Direct3D使用這個w坐標在深度緩沖區和霧化效果中執行一些深度相關的運算。為了能夠使用這個w坐標進行深度相關運算，要求投影矩陣必須是w友好投影矩陣（w-friendly projection matrix，也稱作兼容矩陣），即投影矩陣第三行第四列的元素必須是1，以使w坐標與世界空間中頂點的z坐標相當。如果投影變換矩陣第三行第四列的元素不是1，必須將所有的矩陣元素除以投影矩陣第三行第四列元素的值，將投影矩陣變換為w友好投影矩陣。如果沒有提供一個w友好投影矩陣，基于深度的霧化效果和深度緩沖就不能正確實現。

下面給出的就是從一個非w友好投影矩陣到w友好投影矩陣的轉換。

Direct3D在進行以w為基礎的深度計算中，需要使用w友好投影矩陣，因此即使應用程序不需要進行頂點坐標變換，也需要設置一個w友好投影矩陣。通過實用庫函數D3DXMatrixPerspectiveFovLH()得到的投影矩陣通常都是w友好投影矩陣，所以通常不需要關心這個問題。