OpenGL Transformation 几何变换的顺序概要(MVP,NDC,Window坐标变换过程)

OpenGL Transformation 几何变换的顺序概要(MVP,NDC,Window坐标变换过程)

Geometry transforming or culling sequence:

• application passing the vertex pos(object space pos) to the shader program。
• transforming of clip space pos from object space pos by mulipling the MVP matrix。
  • to world space pos from object space pos by multipling the model/world matrix。
    $$\left(\begin{array}{c}{x}_{world}\\ y_{world}\\ z_{world}\end{array}\right)=M_{model/world}\left(\begin{array}{c}{x}_{obj}\\ y_{obj}\\ z_{obj}\end{array}\right)$$
  • to view space pos from world space pos by multipling the view/camera/eye matrix。
    • right-hand coordinates
      $$\left(\begin{array}{c}{x}_{eye}\\ y_{eye}\\ z_{eye}\end{array}\right)=M_{modelView}\left(\begin{array}{c}{x}_{obj}\\ y_{obj}\\ z_{obj}\end{array}\right)=M_{view}\cdot M_{model/world}\left(\begin{array}{c}{x}_{obj}\\ y_{obj}\\ z_{obj}\end{array}\right)$$
  • to projection/clip space pos from view space pos by multipling the projective/clip matrix。
    $$\left(\begin{array}{c}{x}_{clip}\\ y_{clip}\\ z_{clip}\end{array}\right)=M_{project}\left(\begin{array}{c}{x}_{eye}\\ y_{eye}\\ z_{eye}\end{array}\right)$$
    • culling the vertex which $-w_{clip}>xy{z}_{clip}>w_{clip}$。
      • add edge/vertex when culling occurs。
    • only reserve the vertex which $-w_{clip}<xy{z}_{clip}<w_{clip}$。
    • projection/clip space pos still are Homogeneouse coordinates。
      • to perspective effect, transforming to NDC space (Euclidean/Cartesian space) from projection/clip space pos, then dividing by the $w_{clip}$，$xy{z}_{ndc}=xy{z}_{clip}/w_{clip}$。
• to NDC(Normalized Device Coordinates) space, just using clip space pos divide by $w_{clip}$。
  $$\begin{gathered} \because eyespaceisright-handcoordinates \\ \therefore w_{clip}=-z_{clip} \\ \therefore \left(\begin{array}{c}{x}_{clip}\\ y_{clip}\\ z_{clip}\\ w_{clip}\to (-z_{clip})\end{array}\right)=\left[\begin{array}{cccc}\cdot & \cdots & \cdots & \cdot \\ ⋮& \ddots & \ddots & ⋮\\ \cdot & \cdots & \cdots & \cdot \\ 0& 0& -1& 0\end{array}\right]\left(\begin{array}{c}{x}_{eye}\\ y_{eye}\\ z_{eye}\\ w_{eye}\end{array}\right) \\ \left(\begin{array}{c}{x}_{ndc}\\ y_{ndc}\\ z_{ndc}\end{array}\right)=\left(\begin{array}{c}{x}_{clip}/w_{clip}\\ y_{clip}/w_{clip}\\ z_{clip}/w_{clip}\end{array}\right)\{\begin{array}{l}-w_{clip}<xy{z}_{ndc}<w_{clip}\\ xy{z}_{ndc}=xy{z}_{clip}/w_{clip}\\ -1<xy{z}_{ndc}<1\end{array} \end{gathered}$$
  • $w_{clip}=-z_{clip}$ see OpenGL Projection Matrix。
  • NDC like a cube, [l,r] range [-1,1], [b,t] range [-1,1], [-n,-f] range [-1,1]，see Perspective Projection。
• to mapping Window Coordinates (Screen Coordinates), using glViewport(x,y,w,h), glDepthRange(n,f) functions set the parameters of mapping, see following formula：

$$\left(\begin{array}{c}{x}_{window/screen}\\ y_{window/screen}\\ z_{window/screen}\end{array}\right)=\left(\begin{array}{c}\frac{w}{2}{x}_{ndc}+\left(x+\frac{w}{2}\right)\\ \frac{h}{2}{y}_{ndc}+\left(y+\frac{h}{2}\right)\\ \frac{f-n}{2}{z}_{ndc}+\frac{\left(f+n\right)}{2}\end{array}\right)\{\begin{array}{l}{x}_{window}=\{\begin{array}{l}-1\Rightarrow x\\ 1\Rightarrow x+w\end{array}\\ y_{window}=\{\begin{array}{l}-1\Rightarrow y\\ 1\Rightarrow y+h\end{array}\\ z_{window}=\{\begin{array}{l}-1\Rightarrow n\\ 1\Rightarrow f\end{array}\end{array}$$

参考前辈一文：纹理投影，里面有 GPU-GEMS 一图

References

• LaTeX 各种命令，符号
• OpenGL Transformation
  • Object Space Coordinates
  • Eye Space Coordinates
  • Clip Space Coordinates
  • NDC(Normalized Device Coordinates)
    • Perspective Projection
  • Window Coordinates (Screen Coordinates)
  • OpenGL Projection Matrix
  • Homogeneous Coordinates