Z軸(じく)で回転(かいてん)する

見(み)た目(め)が変(か)わらないのはここまで。ポリゴンを回転(かいてん)させて見(み)ましょう。

Matrixどうしの掛(か)け算(ざん)

以下(いか)の用(よう)に定義(ていぎ)されています

$$ \left( \begin{array}{ccc} a{11} & a{12} & a{13} & a{14}\ a{21} & a{22} & a{23} & a{24}\ a{31} & a{32} & a{33} & a{34}\ a{41} & a{42} & a{43} & a{44}\ \end{array} \right) \left( \begin{array}{ccc} b{11} & b{12} & b{13} & b{14}\ b{21} & b{22} & b{23} & b{24}\ b{31} & b{32} & b{33} & b{34}\ b{41} & b{42} & b{43} & b{44}\ \end{array} \right) = \left( \begin{array}{ccc} M{11} & M{12} & M{13} & M{14}\ M{21} & M{22} & M{23} & M{24}\ M{31} & M{32} & M{33} & M{34}\ M{41} & M{42} & M{43} & M{44}\ \end{array} \right)

$$\ M{11} = a{11} \times b{11} + a{12} \times b{21} + a{13} \times b{31} + a{14} \times b{41}\ M{21} = a{21} \times b{11} + a{22} \times b{21} + a{23} \times b{31} + a{24} \times b{41}\ M{31} = a{31} \times b{11} + a{32} \times b{21} + a{33} \times b{31} + a{34} \times b{41}\ M{41} = a{41} \times b{11} + a{42} \times b{21} + a{43} \times b{31} + a{44} \times b{41}\

$$\ M{12} = a{11} \times b{12} + a{12} \times b{22} + a{13} \times b{32} + a{14} \times b{42}\ M{22} = a{21} \times b{12} + a{22} \times b{22} + a{23} \times b{32} + a{24} \times b{42}\ M{32} = a{31} \times b{12} + a{32} \times b{22} + a{33} \times b{32} + a{34} \times b{42}\ M{42} = a{41} \times b{12} + a{42} \times b{22} + a{43} \times b{32} + a{44} \times b{42}\

$$\ M{13} = a{11} \times b{13} + a{12} \times b{23} + a{13} \times b{33} + a{14} \times b{43}\ M{23} = a{21} \times b{13} + a{22} \times b{23} + a{23} \times b{33} + a{24} \times b{43}\ M{33} = a{31} \times b{13} + a{32} \times b{23} + a{33} \times b{33} + a{34} \times b{43}\ M{43} = a{41} \times b{13} + a{42} \times b{23} + a{43} \times b{33} + a{44} \times b{43}\

$$\ M{14} = a{11} \times b{14} + a{12} \times b{24} + a{13} \times b{34} + a{14} \times b{44}\ M{24} = a{21} \times b{14} + a{22} \times b{24} + a{23} \times b{34} + a{24} \times b{44}\ M{34} = a{31} \times b{14} + a{32} \times b{24} + a{33} \times b{34} + a{34} \times b{44}\ M{44} = a{41} \times b{14} + a{42} \times b{24} + a{43} \times b{34} + a{44} \times b{44}\

Z軸(じく)で回転(かいてん)した後(あと)の値(あたい)

計算(けいさん)してみます。

$$ \left( \begin{array}{ccc} a{11} & a{12} & a{13} & a{14}\ a{21} & a{22} & a{23} & a{24}\ a{31} & a{32} & a{33} & a{34}\ a{41} & a{42} & a{43} & a{44}\ \end{array} \right) \left(\begin{array}{ccc} \cos\theta & -\sin\theta & 0 & 0\ \sin\theta & \cos\theta & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1\ \end{array} \right) = \left( \begin{array}{ccc} M{11} & M{12} & a{13} & a{14}\ M{21} & M{22} & a{23} & a{24}\ M{31} & M{32} & a{33} & a{34}\ M{41} & M{42} & a{43} & a{44}\ \end{array} \right)

$$\ M{11} = a{11} \times \cos\theta + a{12} \times \sin\theta \ M{21} = a{21} \times \cos\theta + a{22} \times \sin\theta \ M{31} = a{31} \times \cos\theta + a{32} \times \sin\theta \ M{41} = a{41} \times \cos\theta + a{42} \times \sin\theta \

$$\ M{12} = a{11} \times -\sin\theta + a{12} \times \cos\theta \ M{22} = a{21} \times -\sin\theta + a{22} \times \cos\theta \ M{32} = a{31} \times -\sin\theta + a{32} \times \cos\theta \ M{42} = a{41} \times -\sin\theta + a{42} \times \cos\theta \