視野変換の覚書

視野変換

視野変換のための行列を得る

必要となる座標・空間ベクトル
  • 目の位置 Eye Point = (e_{x}, e_{y}, e_{z})
  • 注視している位置 Target Point = ( t_{x}, t_{y}, t_{z} )
  • 視線ベクトル

\vec{EV} =\left( \begin{array}{c}E_{x}\\E_{y}\\E_{z}\end{array}\right) =\left( \begin{array}{c}e_{x}-t_{x}\\e_{y}-t_{y}\\e_{z}-t_{z}\end{array}\right)

  • 視線ベクトルの頭上方向のベクトル

\vec{UV} =\left( \begin{array}{c}U_{x}\\U_{y}\\U_{z}\end{array}\right) = \left(\begin{array}{c}u_{x}\\u_{y}\\u_{z}\end{array}\right) , 通常は \left(\begin{array}{c}0\\1\\0\end{array}\right)

Z軸について
  • 視線ベクトルの長さ

|\vec{EV}| =\sqrt{{E_{x}}^{2}+{E_{y}}^{2}+{E_{z}}^{2}} =\sqrt{(e_{x}-t_{x})^{2}+(e_{y}-t_{y})^{2}+(e_{z}-t_{z})^{2}}

  • 視線ベクトルのノルム（単位ベクトル）をZ軸周りのベクトルとする

\vec{Z} = \dfrac{\vec{EV}}{|\vec{EV}|} =\left(\begin{array}{c} \dfrac{E_{x}}{\sqrt{{E_{x}}^{2}+{E_{y}}^{2}+{E_{z}}^{2}}}\\ \dfrac{E_{y}}{\sqrt{{E_{x}}^{2}+{E_{y}}^{2}+{E_{z}}^{2}}}\\ \dfrac{E_{z}}{\sqrt{{E_{x}}^{2}+{E_{y}}^{2}+{E_{z}}^{2}}}\\ \end{array}\right) =\left(\begin{array}{c} \dfrac{e_{x}-t_{x}}{\sqrt{(e_{x}-t_{x})^{2}+(e_{y}-t_{y})^{2}+(e_{z}-t_{z})^{2}}}\\ \dfrac{e_{y}-t_{y}}{\sqrt{(e_{x}-t_{x})^{2}+(e_{y}-t_{y})^{2}+(e_{z}-t_{z})^{2}}}\\ \dfrac{e_{z}-t_{z}}{\sqrt{(e_{x}-t_{x})^{2}+(e_{y}-t_{y})^{2}+(e_{z}-t_{z})^{2}}} \end{array}\right) =\left(\begin{array}{c}Z_{x}\\Z_{y}\\Z_{z}\end{array}\right)


X軸について
  • 頭上方向のベクトルとZ軸のベクトルとのクロス積

\vec{UZ}=\vec{UV}\times\vec{Z} =\left(\begin{array}{c}U_{x}\\U_{y}\\U_{z}\end{array}\right)\times\left(\begin{array}{c}Z_{x}\\Z_{y}\\Z_{z}\end{array}\right) =\left(\begin{array}{c}U_{z}Z_{y}-U_{y}Z_{z}\\U_{x}Z_{z}-U_{z}Z_{x}\\U_{y}Z_{x}-U_{x}Z_{y}\end{array}\right) =\left(\begin{array}{c}UZ_{x}\\UZ_{y}\\UZ_{z}\end{array}\right)

  • クロス積の長さを求める

|\vec{UZ}|=\sqrt{{UZ_{x}}^{2}+{UZ_{y}}^{2}+{UZ_{z}}^{2}} =\sqrt{(U_{z}Z_{y}-U_{y}Z_{z})^{2}+(U_{x}Z_{z}-U_{z}Z_{x})^{2}+(U_{y}Z_{x}-U_{x}Z_{y})^{2}}

  • クロス積を単位ベクトルとしたものをX軸周りのベクトルとする

\vec{X} =\dfrac{\vec{UZ}}{|\vec{UZ}|} =\left(\begin{array}{c} \dfrac{U_{z}Z_{y}-U_{y}Z_{z}}{\sqrt{(U_{z}Z_{y}-U_{y}Z_{z})^{2}+(U_{x}Z_{z}-U_{z}Z_{x})^{2}+(U_{y}Z_{x}-U_{x}Z_{y})^{2}}}\\ \dfrac{U_{x}Z_{z}-U_{z}Z_{x}}{\sqrt{(U_{z}Z_{y}-U_{y}Z_{z})^{2}+(U_{x}Z_{z}-U_{z}Z_{x})^{2}+(U_{y}Z_{x}-U_{x}Z_{y})^{2}}}\\ \dfrac{U_{y}Z_{x}-U_{x}Z_{y}}{\sqrt{(U_{z}Z_{y}-U_{y}Z_{z})^{2}+(U_{x}Z_{z}-U_{z}Z_{x})^{2}+(U_{y}Z_{x}-U_{x}Z_{y})^{2}}} \end{array}\right) =\left(\begin{array}{c}X_{x}\\X_{y}\\X_{z}\end{array}\right)

Y軸について
  • X軸周りのベクトルとZ軸周りのベクトルのクロス積をY軸周りのベクトルとする

\vec{Y}=\vec{X}\times\vec{Z} =\left(\begin{array}{c}X_{x}\\X_{y}\\X_{z}\end{array}\right)\times\left(\begin{array}{c}Z_{x}\\Z_{y}\\Z_{z}\end{array}\right) =\left(\begin{array}{c}X_{z}Z_{y}-X_{y}Z_{z}\\X_{x}Z_{z}-X_{z}Z_{x}\\X_{y}Z_{x}-X_{x}Z_{y}\end{array}\right) =\left(\begin{array}{c}Y_{x}\\Y_{y}\\Y_{z}\end{array}\right)


３軸のベクトルから行列を構成する

M=\left(\begin{array}{cccc} X_{x} & X_{y} & X_{z} & -(\vec{EV}\cdot\vec{X})\\ Y_{x} & Y_{y} & Y_{z} & -(\vec{EV}\cdot\vec{Y})\\ Z_{x} & Z_{y} & Z_{z} & -(\vec{EV}\cdot\vec{Z})\\ 0 & 0 & 0 & 1 \end{array}\right) =\left(\begin{array}{cccc} X_{x} & X_{y} & X_{z} & -(E_{x}X_{x}+E_{y}X_{y}+E_{z}X_{z})\\ Y_{x} & Y_{y} & Y_{z} & -(E_{x}Y_{x}+E_{y}Y_{y}+E_{z}Y_{z})\\ Z_{x} & Z_{y} & Z_{z} & -(E_{x}Z_{x}+E_{y}Z_{y}+E_{z}Z_{z})\\ 0 & 0 & 0 & 1 \end{array}\right)


点 p(px,py,pz)で視野変換を計算する

p\cdot M=\left(\begin{array}{c}p_{x}\\p_{y}\\p_{z}\\1\end{array}\right)\cdot M =\left(\begin{array}{c} p_{x}X_{x}+p_{y}X_{y}+p_{z}X_{z}-E_{x}X_{x}-E_{y}X_{y}-E_{z}X_{z}\\ p_{x}Y_{x}+p_{y}Y_{y}+p_{z}Y_{z}-E_{x}Y_{x}-E_{y}Y_{y}-E_{z}Y_{z}\\ p_{x}Z_{x}+p_{y}Z_{y}+p_{z}Z_{z}-E_{x}Z_{x}-E_{y}Z_{y}-E_{z}Z_{z}\\ 1 \end{array}\right) =\left(\begin{array}{c}P_{x}\\P_{y}\\P_{z}\\1\end{array}\right)


視野錐台による透視投影変換行列

正規化

M_{a} = \left(\begin{array}{cccc} \dfrac{2}{right-left} & 0 & 0 & 0 \\ 0 & \dfrac{2}{top-bottom} & 0 & 0 \\ 0 & 0 & -\dfrac{2}{far-near} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)

透視変換

M_{b} = \left(\begin{array}{cccc} near & 0 & 0 & 0 \\ 0 & near & 0 & 0 \\ 0 & 0 & \dfrac{far+near}{2} & far\cdot near \\ 0 & 0 & -1 & 0 \end{array}\right)

剪断（せんだん）

M_{c} = \left(\begin{array}{cccc} 1 & 0 & \dfrac{right+left}{2\cdot near} & 0 \\ 0 & 1 & \dfrac{top+bottom}{2\cdot near} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)

変換行列

M_{p} = M_{a}\cdot M_{b}\cdot M_{c} = \left(\begin{array}{cccc} \dfrac{2\cdot near}{right-left} & 0 & \dfrac{right+left}{right-left} & 0 \\ 0 & \dfrac{2\cdot near}{top-bottom} & \dfrac{top+bottom}{top-bottom} & 0 \\ 0 & 0 & -\dfrac{far+near}{far-near} & -\dfrac{2\cdot far \cdot near}{far-near} \\ 0 & 0 & -1 & 0 \end{array}\right)