计算机图形学（四）Viewing 视见变换

1. Projection Transformation

• Perspective 透视投影

  • 视见体 view volume：四棱台

  • General Perspective Matrix

    • $h=z_{prp}-z$

    • $p_h=M_{pers}\cdot p$

    • $$
      \begin{pmatrix}
      x_h\y_h\z_h\h
      \end{pmatrix}
      =
      \begin{bmatrix}
      z_{prp}-z_{vp}&0&-x_{prp}&x_{prp}z_{vp}\
      0&z_{prp}-z_{vp}&-y_{prp}&y_{prp}z_{vp}\
      0&0&s_z&t_z\
      0&0&-1&z_{prp}
      \end{bmatrix}
      \cdot
      \begin{pmatrix}
      x\y\z\1
      \end{pmatrix}
      $$

      • $s_z=-\frac{near+far}{near-far}$
      • $t_z=\frac{2nearfar}{near-far}$

    • $$
      \begin{pmatrix}
      x_p\y_p\z_{vp}\1
      \end{pmatrix}
      =
      \begin{pmatrix}
      x_p\y_p\d\1
      \end{pmatrix}
      =
      \begin{pmatrix}
      x_h/h\y_h/h\d\1
      \end{pmatrix}
      $$

  • Perspective in View Volume

    • perspective(fovy, aspect, near, far)
    • $aspect=w/h$;
    • $top=h/2=near*\tan(fovy/2)$;
    • $right=w=aspect*h$
    • Six planes of view volume
      • $x=\pm \frac{right-left}{-2*near}$
      • $y=\pm\frac{top-down}{-2*near}$
      • $z_{min}=-near$
      • $z_{max}=-far$

  • Perspective Normalization Matrix

    • 错切变换
      • $((left+right)/2, (top+bottom)/2,-near)$变到$(0,0,-near)$
      • $H(\theta,\phi)=H(arccot\frac{left+right}{-2near},arccot(\frac{top+bottom}{-2near}))$
    • scaling from view volume to normalization
      • 变为 $x=\pm z$; $y=\pm z$
      • $s_x=1/x=-2*near/(right-left)$
      • $s_y=1/y=-2*near/(top-bottom)$
      • $s_z=1$
    • 投影规范化矩阵
      • $N=\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&\alpha&\beta\0&0&-1&0\end{bmatrix}$

    $$
    M_{persp}=NSH=
    \begin{bmatrix}
    \frac{2near}{right-left}&0&\frac{right+left}{right-left}&0\
    0&\frac{2near}{top-bottom}&\frac{top+bottom}{top-bottom}&0\
    0&0&-\frac{far+near}{far-near}&\frac{-2farnear}{far-near}\
    0&0&-1&0
    \end{bmatrix}
    $$

• Orthographic 正则投影

  • 视见体：长方体

  • Orthographic Matrix

  • Orthographic in View Volume

    • Ortho(left, right, bottom, top, near, far)

  • Orthographic Normalization Matrix

    • 先移动到原点
    • Scaling from view volume to normalization
      • $s_x=1/x=2/(right-left)$
      • $s_y=1/y=2/(top-bottom)$
      • $s_z=1/z=2/(near-far)$
    • So,

    $$
    \begin{align}
    M_{ortho}&=ST\&=
    \begin{bmatrix}
    \frac{2}{right-left}&0&0&0\
    0&\frac{2}{top-bottom}&0&0\
    0&0&\frac{2}{far-near}&0\
    0&0&0&1
    \end{bmatrix}
    \cdot
    \begin{bmatrix}
    1&0&0&-\frac{right+left}{2}\
    0&1&0&-\frac{top+bottom}{2}\
    0&0&1&-\frac{far+near}{2}\
    0&0&0&1
    \end{bmatrix}
    \&=\begin{bmatrix}
    \frac{2}{right-left}&0&0&-\frac{left+right}{right-left}\
    0&\frac{2}{top-bottom}&0&-\frac{top+bottom}{top-bottom}\
    0&0&-\frac{2}{far-near}&-\frac{far+near}{far-near}\
    0&0&0&1
    \end{bmatrix}\
    \end{align}
    $$

2. The Modelview Duality

• Inverses Matrices（此部分查看03）

• Viewing and Modeling transformations

  • Viewing：LookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz)
  • Modeling：
    • Translatef(tx, ty, tz)
    • Rotatef(angle, x, y, z)
    • Scalef(sx, sy, sz)

• Viewing Transformation

  • eye as origin;

  • view-plane normal $n$ as z-axis;

  • view-up vector (VUP) $v$ as y-axis;

  • x-axis: $u=v\times n$

  • LookAt(eye, at, up) code

    mat4 LookAt( const vec4& eye, const vec4& at, const vec4& up){
      vec4 n = normalize(eye - at); 
      vec4 u = normalize(cross(up,n)); 
      vec4 v = normalize(cross(n,u)); 
      vec4 t = vec4(0.0, 0.0, 0.0, 1.0); 
      mat4 c = mat4(u, v, n, t);
    
      return c * Translate( -eye ); 
    }

3. Definition of Viewport

• Viewport is a rectangular region in the window to which you can draw.
• Viewport Matrix
  • $P’=T_2ST_1*P$

4. Sequence of OpenGL Transformation

• Sequence Picture

• Current Transformation Matrix CTM
  • Conceptually there is a 4 x 4 homogeneous coordinate matrix, the current transformation matrix (CTM) that is part of the state and is applied to all vertices that pass down the pipeline
  • The CTM is defined in the user program and loaded into a transformation unit
  • Remember that last matrix specified in the program is the first applied
  • CTM in OpenGL: Model-view & Projection

5. Shadows

• shadow areas are:

  • to be seen from the view position
  • not to be seen from light source position

• Shadows from a directional light
  $$
  \begin{pmatrix}
  x’\y’\z’\w’
  \end{pmatrix}
  =
  \begin{bmatrix}
  b\times dy+c\times dz&-b\times dx&-c\times dx & -d\times dx\
  -a\times dy& a\times dx+c\times dz&-c\times dy&-d\times dy\
  -a\times dz& -b\times dz&a\times dx+b\times dy&-d\times dz\
  0&0&0&a\times dx+b\times dy+c\times dz
  \end{bmatrix}
  \cdot
  \begin{pmatrix}
  x\y\z\1
  \end{pmatrix}
  $$

• shadows from a point light
  $$
  \begin{pmatrix}
  x’\y’\z’\w’
  \end{pmatrix}
  =
  \begin{bmatrix}
  1&0&0&0\
  0&1&0&0\
  0&0&1&0\
  \frac{a}{-d}&\frac{b}{-d}&\frac{c}{-d}&0
  \end{bmatrix}
  \begin{pmatrix}
  x\y\z\1
  \end{pmatrix}
  $$