式展開: 6月 2020

四元数の行列表現において，可逆元とその逆元を左右から作用させる

\begin{array}{rcl} q = w_{q} + x_{q} i + y_{q} j + z_{q} k & \leftrightarrow & [\begin{array}{c} w_{q} + x_{q} i & y_{q} + z_{q} i \\ - (y_{q} - z_{q} i) & w_{q} - x_{q} i \end{array}] \\ p = w_{p} + x_{p} i + y_{p} j + z_{p} k & \leftrightarrow & [\begin{array}{c} w_{p} + x_{p} i & y_{p} + z_{p} i \\ - (y_{p} - z_{p} i) & w_{p} - x_{p} i \end{array}] \end{array}

\begin{array}{rcl} q^{- 1} = \frac{\bar{q}}{| q |^{2}} & \leftrightarrow & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \end{array}

\begin{array}{rcl} q p q^{- 1} & \leftrightarrow & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} w_{q} + x_{q} i & y_{q} + z_{q} i \\ - (y_{q} - z_{q} i) & w_{q} - x_{q} i \end{array}] [\begin{array}{c} w_{p} + x_{p} i & y_{p} + z_{p} i \\ - (y_{p} - z_{p} i) & w_{p} - x_{p} i \end{array}] [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} + ((w_{q}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i & 2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} + (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i \\ - (2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} - (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i) & (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} - ((w_{p}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i \end{array}] \\ \dots 式 展 開 は 最 後 に 添 付 \end{array}

\begin{array}{rcl} w_{q p q^{- 1}} & = & \frac{(w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p}}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} = w_{p} \\ x_{q p q^{- 1}} & = & \frac{(w_{q}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} \\ y_{q p q^{- 1}} & = & \frac{2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p}}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} \\ z_{q p q^{- 1}} & = & \frac{2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} \end{array}

特殊なp, qを考える

w_{q} = \cos (\frac{θ}{2}), x_{q}^{2} + y_{q}^{2} + z_{q}^{2} = \sin^{2} (\frac{θ}{2})

となる

q

及び

w_{p} = 0

となる

p

を考える．
また，この時

v_{x} = \frac{x_{q}}{\sin (\frac{θ}{2})}, v_{y} = \frac{y_{q}}{\sin (\frac{θ}{2})}, v_{z} = \frac{z_{q}}{\sin (\frac{θ}{2})}, v_{x}^{2} + v_{y}^{2} + v_{z}^{2} = 1

となるような

v_{x}, v_{y}, v_{z}

を用意する．

\begin{array}{rcl} w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2} & = & (\cos^{2} (\frac{θ}{2}) + {(\sin (\frac{θ}{2}) v_{x})}^{2} + {(\sin (\frac{θ}{2}) v_{y})}^{2} + {(\sin (\frac{θ}{2}) v_{z})}^{2}) \\ = & (\cos^{2} (\frac{θ}{2}) + \sin^{2} (\frac{θ}{2}) (v_{x}^{2} + v_{y}^{2} + v_{z}^{2})) \\ = & (\cos^{2} (\frac{θ}{2}) + \sin^{2} (\frac{θ}{2})) \\ = & 1 \\ w_{q p q^{- 1}} & = & w_{p} \\ = & 0 \\ x_{q p q^{- 1}} & = & (w_{q}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p} \\ = & (\cos^{2} (\frac{θ}{2}) + {(\sin (\frac{θ}{2}) v_{x})}^{2} - {(\sin (\frac{θ}{2}) v_{y})}^{2} - {(\sin (\frac{θ}{2}) v_{z})}^{2} - 2 {(\sin (\frac{θ}{2}) v_{x})}^{2} + 2 {(\sin (\frac{θ}{2}) v_{x})}^{2}) x_{p} + 2 (\sin (\frac{θ}{2}) v_{x} \sin (\frac{θ}{2}) v_{y} - \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) v_{z}) y_{p} + 2 (\cos (\frac{θ}{2}) \sin (\frac{θ}{2}) v_{y} + \sin (\frac{θ}{2}) v_{x} \sin (\frac{θ}{2}) v_{z}) z_{p} \\ = & (\cos^{2} (\frac{θ}{2}) - \sin^{2} (\frac{θ}{2}) (v_{x}^{2} + v_{y}^{2} + v_{z}^{2}) + 2 \sin^{2} (\frac{θ}{2}) v_{x}^{2}) x_{p} + 2 \sin^{2} (\frac{θ}{2}) v_{x} v_{y} y_{p} - 2 \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) v_{z} y_{p} + 2 \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) v_{y} z_{p} + 2 \sin^{2} (\frac{θ}{2}) v_{x} v_{z} z_{p} \\ = & (\cos^{2} (\frac{θ}{2}) - \sin^{2} (\frac{θ}{2}) \cdot 1) x_{p} + 2 \sin^{2} (\frac{θ}{2}) v_{x}^{2} x_{p} + 2 \sin^{2} (\frac{θ}{2}) (v_{x} v_{y} y_{p} + v_{x} v_{z} z_{p}) + 2 \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) (v_{y} z_{p} - v_{z} y_{p}) \\ = & (\cos^{2} (\frac{θ}{2}) - \sin^{2} (\frac{θ}{2})) x_{p} + 2 \sin^{2} (\frac{θ}{2}) (v_{x}^{2} x_{p} + v_{x} v_{y} y_{p} + v_{x} v_{z} z_{p}) + 2 \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) (v_{y} z_{p} - v_{z} y_{p}) \\ = & \cos (θ) x_{p} + (1 - \cos (θ)) ((v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{x}) + \sin (θ) (v_{y} z_{p} - v_{z} y_{p}) \\ \dots \cos^{2} (\frac{θ}{2}) - \sin^{2} (\frac{θ}{2}) = \cos (\frac{θ}{2} + \frac{θ}{2}) = \cos (θ) \\ \dots 2 \sin^{2} (\frac{θ}{2}) = \sin^{2} (\frac{θ}{2}) + \sin^{2} (\frac{θ}{2}) = (1 - \cos^{2} (\frac{θ}{2})) + \sin^{2} (\frac{θ}{2}) = 1 - (\cos^{2} (\frac{θ}{2}) - \sin^{2} (\frac{θ}{2})) = 1 - \cos (θ) \\ \dots 2 \cos (\frac{θ}{2}) \sin (\frac{θ}{2}) = \sin (\frac{θ}{2} + \frac{θ}{2}) = \sin (θ) \\ = & (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{x} + \cos (θ) (x_{p} - (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{x}) + \sin (θ) (v_{y} z_{p} - v_{z} y_{p}) \\ = & (V \cdot V_{p}) v_{x} + \cos (θ) (x_{p} - (V \cdot V_{p}) v_{x}) + \sin (θ) (v_{y} z_{p} - v_{z} y_{p}) \dots V = (v_{x}, v_{y}, v_{z}), V_{p} = (x_{p}, y_{p}, z_{p}) \\ y_{q p q^{- 1}} & = & 2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} \\ = & (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{y} + \cos (θ) (y_{p} - (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{y}) + \sin (θ) (v_{z} x_{p} - v_{x} z_{p}) \\ = & (V \cdot V_{p}) v_{y} + \cos (θ) (y_{p} - (V \cdot V_{p}) v_{y}) + \sin (θ) (v_{z} x_{p} - v_{x} z_{p}) \\ z_{q p q^{- 1}} & = & 2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p} \\ = & (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{z} + \cos (θ) (z_{p} - (v_{x} x_{p} + v_{y} y_{p} + v_{z} z_{p}) v_{z}) + \sin (θ) (v_{x} y_{p} - v_{y} x_{p}) \\ = & (V \cdot V_{p}) v_{z} + \cos (θ) (z_{p} - (V \cdot V_{p}) v_{z}) + \sin (θ) (v_{x} y_{p} - v_{y} x_{p}) \end{array}

\begin{array}{rcl} P_{q p q^{- 1}} & = & w_{q p q^{- 1}} + x_{q p q^{- 1}} i + y_{q p q^{- 1}} j + z_{q p q^{- 1}} k \\ = & 0 \\ + ((V \cdot V_{p}) v_{x} + \cos (θ) (x_{p} - (V \cdot V_{p}) v_{x}) + \sin (θ) (v_{y} z_{p} - v_{z} y_{p})) i \\ + ((V \cdot V_{p}) v_{y} + \cos (θ) (y_{p} - (V \cdot V_{p}) v_{y}) + \sin (θ) (v_{z} x_{p} - v_{x} z_{p})) j \\ + ((V \cdot V_{p}) v_{z} + \cos (θ) (z_{p} - (V \cdot V_{p}) v_{z}) + \sin (θ) (v_{x} y_{p} - v_{y} x_{p})) k \\ = & (V \cdot V_{p}) (v_{x} i + v_{y} j + v_{z} k) + \cos (θ) (x_{p} i + y_{p} j + z_{p} k) - \cos (θ) (V \cdot V_{p}) (v_{x} i + v_{y} j + v_{z} k) + \sin (θ) ((v_{y} z_{p} - v_{z} y_{p}) i + (v_{z} x_{p} - v_{x} z_{p}) j (v_{x} y_{p} - v_{y} x_{p}) k) \\ = & (V \cdot V_{p}) V + \cos (θ) V_{p} - \cos (θ) (V \cdot V_{p}) V + \sin (θ) (V \times V_{p}) \\ = & (V \cdot V_{p}) V + \cos (θ) (V_{p} - (V \cdot V_{p})) V + \sin (θ) (V \times V_{p}) \end{array}

これはベクトル

V_{p} = (x_{p}, y_{p}, z_{p})

を，回転軸

V = (v_{x}, v_{y}, v_{z})

周りに

θ

だけ回す変換となる．

\begin{array}{rcl} q p q^{- 1} & \leftrightarrow & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} w_{q} + x_{q} i & y_{q} + z_{q} i \\ - (y_{q} - z_{q} i) & w_{q} - x_{q} i \end{array}] [\begin{array}{c} w_{p} + x_{p} i & y_{p} + z_{p} i \\ - (y_{p} - z_{p} i) & w_{p} - x_{p} i \end{array}] [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} (w_{q} + x_{q} i) (w_{p} + x_{p} i) + (y_{q} + z_{q} i) (- (y_{p} - z_{p} i)) & (w_{q} + x_{q} i) (y_{p} + z_{p} i) + (y_{q} + z_{q} i) (w_{p} - x_{p} i) \\ (- (y_{q} - z_{q} i)) (w_{p} + x_{p} i) + (w_{q} - x_{q} i) (- (y_{p} - z_{p} i)) & (- (y_{q} - z_{q} i)) (y_{p} + z_{p} i) + (w_{q} - x_{q} i) (w_{p} - x_{p} i) \end{array}] [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} w_{q} w_{p} + w_{q} x_{p} i + x_{q} i w_{p} + x_{q} i x_{p} i + y_{q} (- y_{p}) + y_{q} z_{p} i + z_{q} i (- y_{p}) + z_{q} i z_{p} i & w_{q} y_{p} + w_{q} z_{p} i + x_{q} i y_{p} + x_{q} i z_{p} i + y_{q} w_{p} + y_{q} (- x_{p} i) + z_{q} i w_{p} + z_{q} i (- x_{p} i) \\ (- y_{q}) w_{p} + (- y_{q}) x_{p} i + z_{q} i w_{p} + z_{q} i x_{p} i + w_{q} (- y_{p}) + w_{q} z_{p} i + (- x_{q} i) (- y_{p}) + (- x_{q} i) z_{p} i & (- y_{q}) y_{p} + (- y_{q}) z_{p} i + z_{q} i y_{p} + z_{q} i z_{p} i + w_{q} w_{p} + w_{q} (- x_{p} i) + (- x_{q} i) w_{p} + (- x_{q} i) (- x_{p} i) \end{array}] [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i & w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i \\ - (w_{q} y_{p} - x_{q} z_{p} + y_{q} w_{p} + z_{q} x_{p} - (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i & w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i \end{array}] [\begin{array}{c} w_{q} - x_{q} i & - y_{q} - z_{q} i \\ - (- y_{q} + z_{q} i) & w_{q} + x_{q} i \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} - x_{q} i) + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (- (- y_{q} + z_{q} i)) & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- y_{q} - z_{q} i) + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (w_{q} + x_{q} i) \\ (- (w_{q} y_{p} - x_{q} z_{p} + y_{q} w_{p} + z_{q} x_{p} - (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (w_{q} - x_{q} i) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- (- y_{q} + z_{q} i)) & (- (w_{q} y_{p} - x_{q} z_{p} + y_{q} w_{p} + z_{q} x_{p} - (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- y_{q} - z_{q} i) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} + x_{q} i) \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} - x_{q} i) + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (y_{q} - z_{q} i)) & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- y_{q} - z_{q} i) + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (w_{q} + x_{q} i) \\ (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (w_{q} - x_{q} i) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (y_{q} - z_{q} i) & (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- y_{q} - z_{q} i) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} + x_{q} i) \end{array}] \\ = & \frac{1}{w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}} [\begin{array}{c} (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} + ((w_{q}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i & 2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} + (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i \\ - (2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} - (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i) & (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} - ((w_{p}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i \end{array}] \\ \dots 各 要 素 の 式 展 開 は 以 降 に 添 付 \end{array}

(1,1)要素

(1,1)要素の第一項

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} - x_{q} i) & = & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q}) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- x_{q} i) \\ = & w_{p} w_{q} w_{q} - w_{q} x_{p} x_{q} - w_{q} y_{p} y_{q} - w_{q} z_{p} z_{q} + w_{q} x_{p} x_{q} + w_{p} x_{q} x_{q} + x_{q} y_{q} z_{p} - x_{q} y_{p} z_{q} + (w_{q} w_{q} x_{p} + w_{p} w_{q} x_{q} + w_{q} y_{q} z_{p} - w_{q} y_{p} z_{q} - w_{p} w_{q} x_{q} + x_{p} x_{q} x_{q} + x_{q} y_{p} y_{q} + x_{q} z_{p} z_{q}) i \end{array}

(1,1)要素の第二項

\begin{array}{rcl} (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (- (- y_{q} + z_{q} i)) & = & (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (y_{q}) + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (- z_{q} i) \\ = & w_{q} y_{p} y_{q} + w_{p} y_{q} y_{q} + x_{p} y_{q} z_{q} - x_{q} y_{q} z_{p} + w_{q} z_{p} z_{q} + w_{p} z_{q} z_{q} + x_{q} y_{p} z_{q} - x_{p} y_{q} z_{q} + (w_{q} y_{q} z_{p} + w_{p} y_{q} z_{q} + x_{q} y_{p} y_{q} - x_{p} y_{q} y_{q} - w_{q} y_{p} z_{q} - w_{p} y_{q} z_{q} - x_{p} z_{q} z_{q} + x_{q} z_{p} z_{q}) i \end{array}

(1,1)要素の第一，二項の和

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} - x_{q} i) \\ + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (- (- y_{q} + z_{q} i)) & = & w_{p} w_{q} w_{q} - w_{q} x_{p} x_{q} - w_{q} y_{p} y_{q} - w_{q} z_{p} z_{q} + w_{q} x_{p} x_{q} + w_{p} x_{q} x_{q} + x_{q} y_{q} z_{p} - x_{q} y_{p} z_{q} + w_{q} y_{p} y_{q} + w_{p} y_{q} y_{q} + x_{p} y_{q} z_{q} - x_{q} y_{q} z_{p} + w_{q} z_{p} z_{q} + w_{p} z_{q} z_{q} + x_{q} y_{p} z_{q} - x_{p} y_{q} z_{q} + (w_{q} w_{q} x_{p} + w_{p} w_{q} x_{q} + w_{q} y_{q} z_{p} - w_{q} y_{p} z_{q} - w_{p} w_{q} x_{q} + x_{p} x_{q} x_{q} + x_{q} y_{p} y_{q} + x_{q} z_{p} z_{q} + w_{q} y_{q} z_{p} + w_{p} y_{q} z_{q} + x_{q} y_{p} y_{q} - x_{p} y_{q} y_{q} - w_{q} y_{p} z_{q} - w_{p} y_{q} z_{q} - x_{p} z_{q} z_{q} + x_{q} z_{p} z_{q}) i \\ = & (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} + ((w_{q}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i \end{array}

(1,2)要素

(1,2)要素の第一項

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- y_{q} - z_{q} i) & = & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- y_{q}) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- z_{q} i) \\ = & - w_{p} w_{q} y_{q} + x_{p} x_{q} y_{q} + y_{p} y_{q} y_{q} + z_{p} z_{q} y_{q} + w_{q} x_{p} z_{q} + w_{p} x_{q} z_{q} + y_{q} z_{p} z_{q} - y_{p} z_{q} z_{q} + (- w_{q} x_{p} y_{q} - w_{p} x_{q} y_{q} - y_{q} y_{q} z_{p} + y_{p} y_{q} z_{q} - w_{p} w_{q} z_{q} + x_{p} x_{q} z_{q} + y_{p} y_{q} z_{q} + z_{p} z_{q} z_{q}) i \end{array}

(1,2)要素の第二項

\begin{array}{rcl} (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (w_{q} + x_{q} i) & = & (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) w_{q} + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) x_{q} i \\ = & w_{q} w_{q} y_{p} + w_{p} w_{q} y_{q} + w_{q} x_{p} z_{q} - w_{q} x_{q} z_{p} - w_{q} x_{q} z_{p} - w_{p} x_{q} z_{q} - x_{q} x_{q} y_{p} + x_{p} x_{q} y_{q} + (w_{q} w_{q} z_{p} + w_{p} w_{q} z_{q} + w_{q} x_{q} y_{p} - w_{q} x_{p} y_{q} + w_{q} x_{q} y_{p} + w_{p} x_{q} y_{q} + x_{p} x_{q} z_{q} - x_{q} x_{q} z_{p}) i \end{array}

(1,2)要素の第一，二項の和

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} + (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- y_{q} - z_{q} i) \\ + (w_{q} y_{p} + y_{q} w_{p} + z_{q} x_{p} - x_{q} z_{p} + (w_{q} z_{p} + z_{q} w_{p} + x_{q} y_{p} - y_{q} x_{p}) i) (w_{q} + x_{q} i) & = & - w_{p} w_{q} y_{q} + x_{p} x_{q} y_{q} + y_{p} y_{q} y_{q} + z_{p} z_{q} y_{q} + w_{q} x_{p} z_{q} + w_{p} x_{q} z_{q} + y_{q} z_{p} z_{q} - y_{p} z_{q} z_{q} + w_{q} w_{q} y_{p} + w_{p} w_{q} y_{q} + w_{q} x_{p} z_{q} - w_{q} x_{q} z_{p} - w_{q} x_{q} z_{p} - w_{p} x_{q} z_{q} - x_{q} x_{q} y_{p} + x_{p} x_{q} y_{q} + (- w_{q} x_{p} y_{q} - w_{p} x_{q} y_{q} - y_{q} y_{q} z_{p} + y_{p} y_{q} z_{q} - w_{p} w_{q} z_{q} + x_{p} x_{q} z_{q} + y_{p} y_{q} z_{q} + z_{p} z_{q} z_{q} + w_{q} w_{q} z_{p} + w_{p} w_{q} z_{q} + w_{q} x_{q} y_{p} - w_{q} x_{p} y_{q} + w_{q} x_{q} y_{p} + w_{p} x_{q} y_{q} + x_{p} x_{q} z_{q} - x_{q} x_{q} z_{p}) i \\ = & 2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} + (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i \end{array}

(2,1)要素

(2,1)要素の第一項

\begin{array}{rcl} (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (w_{q} - x_{q} i) & = & - w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (w_{q}) - w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- x_{q} i) \\ = & - w_{q} y_{p} w_{q} + x_{q} z_{p} w_{q} - y_{q} w_{p} w_{q} - z_{q} x_{p} w_{q} + w_{q} z_{p} x_{q} + x_{q} y_{p} x_{q} - y_{q} x_{p} x_{q} + z_{q} w_{p} x_{q} + (w_{q} z_{p} w_{q} + x_{q} y_{p} w_{q} - y_{q} x_{p} w_{q} + z_{q} w_{p} w_{q} + w_{q} y_{p} x_{q} - x_{q} z_{p} x_{q} + y_{q} w_{p} x_{q} + z_{q} x_{p} x_{q}) i \end{array}

(2,1)要素の第二項

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (y_{q} - z_{q} i) & = & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (y_{q}) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (- z_{q} i) \\ = & w_{q} w_{p} y_{q} - x_{q} x_{p} y_{q} - y_{q} y_{p} y_{q} - z_{q} z_{p} y_{q} - w_{q} x_{p} z_{q} - x_{q} w_{p} z_{q} - y_{q} z_{p} z_{q} + z_{q} y_{p} z_{q} + (- w_{q} x_{p} y_{q} - x_{q} w_{p} y_{q} - y_{q} z_{p} y_{q} + z_{q} y_{p} y_{q} - w_{q} w_{p} z_{q} + x_{q} x_{p} z_{q} + y_{q} y_{p} z_{q} + z_{q} z_{p} z_{q}) i \end{array}

(2,1)要素の第一，二項の和

\begin{array}{rcl} (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (w_{q} - x_{q} i) \\ + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (y_{q} - z_{q} i) & = & - w_{q} y_{p} w_{q} + x_{q} z_{p} w_{q} - y_{q} w_{p} w_{q} - z_{q} x_{p} w_{q} + w_{q} z_{p} x_{q} + x_{q} y_{p} x_{q} - y_{q} x_{p} x_{q} + z_{q} w_{p} x_{q} + (w_{q} z_{p} w_{q} + x_{q} y_{p} w_{q} - y_{q} x_{p} w_{q} + z_{q} w_{p} w_{q} + w_{q} y_{p} x_{q} - x_{q} z_{p} x_{q} + y_{q} w_{p} x_{q} + z_{q} x_{p} x_{q}) i + w_{q} w_{p} y_{q} - x_{q} x_{p} y_{q} - y_{q} y_{p} y_{q} - z_{q} z_{p} y_{q} - w_{q} x_{p} z_{q} - x_{q} w_{p} z_{q} - y_{q} z_{p} z_{q} + z_{q} y_{p} z_{q} + (- w_{q} x_{p} y_{q} - x_{q} w_{p} y_{q} - y_{q} z_{p} y_{q} + z_{q} y_{p} y_{q} - w_{q} w_{p} z_{q} + x_{q} x_{p} z_{q} + y_{q} y_{p} z_{q} + z_{q} z_{p} z_{q}) i \\ = & - w_{q} y_{p} w_{q} + x_{q} z_{p} w_{q} - y_{q} w_{p} w_{q} - z_{q} x_{p} w_{q} + w_{q} z_{p} x_{q} + x_{q} y_{p} x_{q} - y_{q} x_{p} x_{q} + z_{q} w_{p} x_{q} + w_{q} w_{p} y_{q} - x_{q} x_{p} y_{q} - y_{q} y_{p} y_{q} - z_{q} z_{p} y_{q} - w_{q} x_{p} z_{q} - x_{q} w_{p} z_{q} - y_{q} z_{p} z_{q} + z_{q} y_{p} z_{q} + (w_{q} z_{p} w_{q} + x_{q} y_{p} w_{q} - y_{q} x_{p} w_{q} + z_{q} w_{p} w_{q} + w_{q} y_{p} x_{q} - x_{q} z_{p} x_{q} + y_{q} w_{p} x_{q} + z_{q} x_{p} x_{q} - w_{q} x_{p} y_{q} - x_{q} w_{p} y_{q} - y_{q} z_{p} y_{q} + z_{q} y_{p} y_{q} - w_{q} w_{p} z_{q} + x_{q} x_{p} z_{q} + y_{q} y_{p} z_{q} + z_{q} z_{p} z_{q}) i \\ = & - 2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} (- w_{q}^{2} + x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) y_{p} + 2 (w_{q} x_{q} - y_{q} z_{q}) z_{p} + ((w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + 2 (x_{q} z_{q} - w_{q} y_{q}) x_{p}) i \\ = & - (2 (w_{q} z_{q} + x_{q} y_{q}) x_{p} + (w_{q}^{2} - x_{q}^{2} + y_{q}^{2} - z_{q}^{2}) y_{p} + 2 (y_{q} z_{q} - w_{q} x_{q}) z_{p} - (2 (x_{q} z_{q} - w_{q} y_{q}) x_{p} + 2 (w_{q} x_{q} + y_{q} z_{q}) y_{p} + (w_{q}^{2} - x_{q}^{2} - y_{q}^{2} + z_{q}^{2}) z_{p}) i) \end{array}

(2,2)要素

(2,2)要素の第一項

\begin{array}{rcl} (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- y_{q} - z_{q} i) & = & (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- y_{q}) + (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- z_{q} i) \\ = & w_{q} y_{p} y_{q} - x_{q} z_{p} y_{q} + y_{q} w_{p} y_{q} + z_{q} x_{p} y_{q} + w_{q} z_{p} z_{q} + x_{q} y_{p} z_{q} - y_{q} x_{p} z_{q} + z_{q} w_{p} z_{q} + (- w_{q} z_{p} y_{q} - x_{q} y_{p} y_{q} + y_{q} x_{p} y_{q} - z_{q} w_{p} y_{q} + w_{q} y_{p} z_{q} - x_{q} z_{p} z_{q} + y_{q} w_{p} z_{q} + z_{q} x_{p} z_{q}) i \end{array}

(2,2)要素の第二項

\begin{array}{rcl} (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} + x_{q} i) & = & (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q}) + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (x_{q} i) \\ = & w_{q} w_{p} w_{q} - x_{q} x_{p} w_{q} - y_{q} y_{p} w_{q} - z_{q} z_{p} w_{q} + w_{q} x_{p} x_{q} + x_{q} w_{p} x_{q} + y_{q} z_{p} x_{q} - z_{q} y_{p} x_{q} + (- w_{q} x_{p} w_{q} - x_{q} w_{p} w_{q} - y_{q} z_{p} w_{q} + z_{q} y_{p} w_{q} + w_{q} w_{p} x_{q} - x_{q} x_{p} x_{q} - y_{q} y_{p} x_{q} - z_{q} z_{p} x_{q}) i \end{array}

(2,1)要素の第一，二項の和

\begin{array}{rcl} (- w_{q} y_{p} + x_{q} z_{p} - y_{q} w_{p} - z_{q} x_{p} + (w_{q} z_{p} + x_{q} y_{p} - y_{q} x_{p} + z_{q} w_{p}) i) (- y_{q} - z_{q} i) \\ + (w_{q} w_{p} - x_{q} x_{p} - y_{q} y_{p} - z_{q} z_{p} - (w_{q} x_{p} + x_{q} w_{p} + y_{q} z_{p} - z_{q} y_{p}) i) (w_{q} + x_{q} i) & = & (w_{q} y_{p} y_{q} - x_{q} z_{p} y_{q} + y_{q} w_{p} y_{q} + z_{q} x_{p} y_{q} + w_{q} z_{p} z_{q} + x_{q} y_{p} z_{q} - y_{q} x_{p} z_{q} + z_{q} w_{p} z_{q} + (- w_{q} z_{p} y_{q} - x_{q} y_{p} y_{q} + y_{q} x_{p} y_{q} - z_{q} w_{p} y_{q} + w_{q} y_{p} z_{q} - x_{q} z_{p} z_{q} + y_{q} w_{p} z_{q} + z_{q} x_{p} z_{q}) i) + (w_{q} w_{p} w_{q} - x_{q} x_{p} w_{q} - y_{q} y_{p} w_{q} - z_{q} z_{p} w_{q} + w_{q} x_{p} x_{q} + x_{q} w_{p} x_{q} + y_{q} z_{p} x_{q} - z_{q} y_{p} x_{q} + (- w_{q} x_{p} w_{q} - x_{q} w_{p} w_{q} - y_{q} z_{p} w_{q} + z_{q} y_{p} w_{q} + w_{q} w_{p} x_{q} - x_{q} x_{p} x_{q} - y_{q} y_{p} x_{q} - z_{q} z_{p} x_{q}) i) \\ = & w_{q} y_{p} y_{q} - x_{q} z_{p} y_{q} + y_{q} w_{p} y_{q} + z_{q} x_{p} y_{q} + w_{q} z_{p} z_{q} + x_{q} y_{p} z_{q} - y_{q} x_{p} z_{q} + z_{q} w_{p} z_{q} + w_{q} w_{p} w_{q} - x_{q} x_{p} w_{q} - y_{q} y_{p} w_{q} - z_{q} z_{p} w_{q} + w_{q} x_{p} x_{q} + x_{q} w_{p} x_{q} + y_{q} z_{p} x_{q} - z_{q} y_{p} x_{q} + (- w_{q} z_{p} y_{q} - x_{q} y_{p} y_{q} + y_{q} x_{p} y_{q} - z_{q} w_{p} y_{q} + w_{q} y_{p} z_{q} - x_{q} z_{p} z_{q} + y_{q} w_{p} z_{q} + z_{q} x_{p} z_{q} - w_{q} x_{p} w_{q} - x_{q} w_{p} w_{q} - y_{q} z_{p} w_{q} + z_{q} y_{p} w_{q} + w_{q} w_{p} x_{q} - x_{q} x_{p} x_{q} - y_{q} y_{p} x_{q} - z_{q} z_{p} x_{q}) i) \\ = & (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} + ((- w_{p}^{2} - x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) x_{p} + 2 (w_{q} z_{q} - x_{q} y_{q}) y_{p} - 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i \\ = & (w_{q}^{2} + x_{q}^{2} + y_{q}^{2} + z_{q}^{2}) w_{p} - ((w_{p}^{2} + x_{q}^{2} - y_{q}^{2} - z_{q}^{2}) x_{p} + 2 (x_{q} y_{q} - w_{q} z_{q}) y_{p} + 2 (w_{q} y_{q} + x_{q} z_{q}) z_{p}) i \end{array}

四元数の行列表現での逆元

\begin{array}{rcl} q = w + x i + y j + z k & \leftrightarrow & [\begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array}] \end{array}

\begin{array}{rcl} | q |^{2} = | w + x i + y j + z k |^{2} & \leftrightarrow & | \begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array} | \\ = & (w + x i) (w - x i) - (y + z i) (- (y - z i)) \\ = & (w + x i) (w - x i) + (y + z i) (y - z i) \\ = & w^{2} + x^{2} + y^{2} + z^{2} \end{array}

\begin{array}{rcl} \bar{q} = w - x i - y j - z k & \leftrightarrow & [\begin{array}{c} w + (- x) i & (- y) + (- z) i \\ - ((- y) - (- z) i) & w - (- x) i \end{array}] \\ = & [\begin{array}{c} w - x i & - y - z i \\ - (- y + z i) & w + x i \end{array}] \end{array}

\begin{array}{rcl} q \bar{q} & = & (w + x i + y j + z k) (w - x i - y j - z k) \\ \leftrightarrow & [\begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array}] [\begin{array}{c} w - x i & - y - z i \\ - (- y + z i) & w + x i \end{array}] \\ = & [\begin{array}{c} (w + x i) (w - x i) + (y + z i) (- (- y + z i)) & (w + x i) (- y - z i) + (y + z i) (w + x i) \\ (- (y - z i)) (w - x i) + (w - x i) (- (- y + z i)) & (- (y - z i)) (- y - z i) + (w - x i) (w + x i) \end{array}] \\ = & [\begin{array}{c} (w + x i) (w - x i) + (y + z i) (y - z i) & (w + x i) (- y - z i) + (y + z i) (w + x i) \\ (- y + z i) (w - x i) + (w - x i) (y - z i) & (- y + z i) (- y - z i) + (w - x i) (w + x i) \end{array}] \\ = & [\begin{array}{c} w^{2} - w x i + w x i + x^{2} + y^{2} - y z i + y z i + z^{2} & - w y - w z i - x y i + x z + w y + x y i + w z i - x z \\ - w y + x y i + w z i + x z + w y - w z i - x y i - x z & y^{2} + y z i - y z i + z^{2} + w^{2} + w x i - w x i + x^{2} \end{array}] \\ = & [\begin{array}{c} w^{2} + x^{2} + y^{2} + z^{2} & 0 \\ 0 & w^{2} + x^{2} + y^{2} + z^{2} \end{array}] \end{array}

\begin{array}{rcl} \bar{q} q & = & (w - x i - y j - z k) (w + x i + y j + z k) \\ \leftrightarrow & [\begin{array}{c} w - x i & - y - z i \\ - (- y + z i) & w + x i \end{array}] [\begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array}] \\ = & [\begin{array}{c} (w - x i) (w + x i) + (- y - z i) (- (y - z i)) & (w - x i) (y + z i) + (- y - z i) (w - x i) \\ (- (- y + z i)) (w + x i) + (w + x i) (- (y - z i)) & (- (- y + z i)) (y + z i) + (w + x i) (w - x i) \end{array}] \\ = & [\begin{array}{c} (w - x i) (w + x i) + (- y - z i) (- y + z i) & (w - x i) (y + z i) + (- y - z i) (w - x i) \\ (y - z i) (w + x i) + (w + x i) (- y + z i) & (y - z i) (y + z i) + (w + x i) (w - x i) \end{array}] \\ = & [\begin{array}{c} w^{2} + w x i - w x i + x^{2} + y^{2} + y z i - y z i + z^{2} & w y + w z i - x y i + x z - w y + x y i - w z i - x z \\ w y + x y i - w z i + x z - w y + w z i - x y i - x z & y^{2} + y z i - y z i + z^{2} + w^{2} - w x i + w x i + x^{2} \end{array}] \\ = & [\begin{array}{c} w^{2} + x^{2} + y^{2} + z^{2} & 0 \\ 0 & w^{2} + x^{2} + y^{2} + z^{2} \end{array}] \dots 行 列 の 積 は 一 般 に 非 可 換 だ が ， こ の 形 の 行 列 で は 可 換 で あ る \end{array}

\begin{array}{rcl} \frac{q \bar{q}}{| q |^{2}} & = & \frac{\bar{q} q}{| q |^{2}} = q \frac{\bar{q}}{| q |^{2}} = \frac{\bar{q}}{| q |^{2}} q = \frac{1}{| q |^{2}} q \bar{q} = \frac{1}{| q |^{2}} \bar{q} q \\ \leftrightarrow & \frac{1}{w^{2} + x^{2} + y^{2} + z^{2}} [\begin{array}{c} w^{2} + x^{2} + y^{2} + z^{2} & 0 \\ 0 & w^{2} + x^{2} + y^{2} + z^{2} \end{array}] \\ = & [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] \\ = & E \end{array}

\begin{array}{rcl} q^{- 1} & = & \frac{\bar{q}}{| q |^{2}} \\ \leftrightarrow & \frac{1}{w^{2} + x^{2} + y^{2} + z^{2}} [\begin{array}{c} w - x i & - y - z i \\ - (- y + z i) & w + x i \end{array}] \end{array}

四元数の行列表現での積

\begin{array}{rcl} w + x i + y j + z k & \leftrightarrow & [\begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array}] \\ (w_{1} + x_{1} i + y_{1} j + z_{1} k) (w_{2} + x_{2} i + y_{2} j + z_{2} k) & \leftrightarrow & [\begin{array}{c} w_{1} + x_{1} i & y_{1} + z_{1} i \\ - (y_{1} - z_{1} i) & w_{1} - x_{1} i \end{array}] [\begin{array}{c} w_{2} + x_{2} i & y_{2} + z_{2} i \\ - (y_{2} - z_{2} i) & w_{2} - x_{2} i \end{array}] \\ = & [\begin{array}{c} (w_{1} + x_{1} i) (w_{2} + x_{2} i) + (y_{1} + z_{1} i) (- (y_{2} - z_{2} i)) & (w_{1} + x_{1} i) (y_{2} + z_{2} i) + (y_{1} + z_{1} i) (w_{2} - x_{2} i) \\ (- (y_{1} - z_{1} i)) (w_{2} + x_{2} i) + (w_{1} - x_{1} i) (- (y_{2} - z_{2} i)) & (- (y_{1} - z_{1} i)) (y_{2} + z_{2} i) + (w_{1} - x_{1} i) (w_{2} - x_{2} i) \end{array}] \\ = & [\begin{array}{c} w_{1} w_{2} + w_{1} x_{2} i + x_{1} i w_{2} + x_{1} i x_{2} i + y_{1} (- y_{2}) + y_{1} z_{2} i + z_{1} i (- y_{2}) + z_{1} i z_{2} i & w_{1} y_{2} + w_{1} z_{2} i + x_{1} i y_{2} + x_{1} i z_{2} i + y_{1} w_{2} + y_{1} (- x_{2} i) + z_{1} i w_{2} + z_{1} i (- x_{2} i) \\ (- y_{1}) w_{2} + (- y_{1}) x_{2} i + z_{1} i w_{2} + z_{1} i x_{2} i + w_{1} (- y_{2}) + w_{1} z_{2} i + (- x_{1} i) (- y_{2}) + (- x_{1} i) z_{2} i & (- y_{1}) y_{2} + (- y_{1}) z_{2} i + z_{1} i y_{2} + z_{1} i z_{2} i + w_{1} w_{2} + w_{1} (- x_{2} i) + (- x_{1} i) w_{2} + (- x_{1} i) (- x_{2} i) \end{array}] \\ = & [\begin{array}{c} (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) + (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i & (w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) + (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) i \\ - ((w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) - (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) i) & (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) - (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i \end{array}] \\ \leftrightarrow & (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) \\ + (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i \\ + (w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) j \\ + (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) k \end{array}

\begin{array}{rcl} (w_{2} + x_{2} i + y_{2} j + z_{2} k) (w_{1} + x_{1} i + y_{1} j + z_{1} k) & \leftrightarrow & [\begin{array}{c} w_{2} + x_{2} i & y_{2} + z_{2} i \\ - (y_{2} - z_{2} i) & w_{2} - x_{2} i \end{array}] [\begin{array}{c} w_{1} + x_{1} i & y_{1} + z_{1} i \\ - (y_{1} - z_{1} i) & w_{1} - x_{1} i \end{array}] \\ = & [\begin{array}{c} (w_{2} + x_{2} i) (w_{1} + x_{1} i) + (y_{2} + z_{2} i) (- (y_{1} - z_{1} i)) & (w_{2} + x_{2} i) (y_{1} + z_{1} i) + (y_{2} + z_{2} i) (w_{1} - x_{1} i) \\ (- (y_{2} - z_{2} i)) (w_{1} + x_{1} i) + (w_{2} - x_{2} i) (- (y_{1} - z_{1} i)) & (- (y_{2} - z_{2} i)) (y_{1} + z_{1} i) + (w_{2} - x_{2} i) (w_{1} - x_{1} i) \end{array}] \\ = & [\begin{array}{c} w_{2} w_{1} + w_{1} x_{2} i + x_{1} i w_{2} + x_{1} i x_{2} i + y_{2} (- y_{1}) + y_{2} z_{1} i + z_{2} i (- y_{1}) + z_{2} i z_{1} i & w_{2} y_{1} + w_{2} z_{1} i + x_{2} i y_{1} + x_{2} i z_{1} i + y_{2} w_{1} + y_{2} (- x_{1} i) + z_{2} i w_{1} + z_{2} i (- x_{1} i) \\ (- y_{2}) w_{1} + (- y_{2}) x_{1} i + z_{2} i w_{1} + z_{2} i x_{1} i + w_{2} (- y_{1}) + w_{2} z_{1} i + (- x_{2} i) (- y_{1}) + (- x_{2} i) z_{1} i & (- y_{2}) y_{1} + (- y_{2}) z_{1} i + z_{2} i y_{1} + z_{2} i z_{1} i + w_{2} w_{1} + w_{2} (- x_{1} i) + (- x_{2} i) w_{1} + (- x_{2} i) (- x_{1} i) \end{array}] \\ = & [\begin{array}{c} (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) + (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i & (w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) + (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) i \\ - ((w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) - (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) i) & (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) - (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i \end{array}] \\ \dots 行 列 の 積 は 一 般 に 非 可 換 だ が ， こ の 形 の 行 列 で は 可 換 で あ る \\ \leftrightarrow & (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) \\ + (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) i \\ + (w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) j \\ + (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) k \end{array}

四元数の積としては以下のようにまとまる．

\begin{array}{rcl} w & = & (w_{1} w_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}) \\ x & = & (w_{1} x_{2} + w_{2} x_{1} + y_{1} z_{2} - z_{1} y_{2}) \\ y & = & (w_{1} y_{2} + w_{2} y_{1} + z_{1} x_{2} - x_{1} z_{2}) \\ z & = & (w_{1} z_{2} + w_{2} z_{1} + x_{1} y_{2} - y_{1} x_{2}) \end{array}

これは以前の計算結果と一致する．

四元数の行列表現

\begin{array}{rcl} w + x i + y j + z k & \leftrightarrow & [\begin{array}{c} w + x i & y + z i \\ - (y - z i) & w - x i \end{array}] \\ = & w [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] + x [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] + y [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] + z [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \\ = & w E + x I + y J + z K \end{array}

四元数の単位同士の積の確認

\begin{array}{rcl} i \cdot i \leftrightarrow I \cdot I & = & [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \cdot [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \\ = & [\begin{array}{c} i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot (- i) \\ 0 \cdot i + (- i) \cdot 0 & 0 \cdot 0 + (- i) \cdot (- i) \end{array}] \\ = & [\begin{array}{c} - 1 & 0 \\ 0 & - 1 \end{array}] \\ = & - E \\ j \cdot j \leftrightarrow J \cdot J & = & [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \cdot [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \\ = & [\begin{array}{c} 0 \cdot 0 + 1 \cdot (- 1) & 0 \cdot 1 + 1 \cdot 0 \\ (- 1) \cdot 0 + 0 \cdot (- 1) & (- 1) \cdot 1 + 0 \cdot 0 \end{array}] \\ = & [\begin{array}{c} - 1 & 0 \\ 0 & - 1 \end{array}] \\ = & - E \\ k \cdot k \leftrightarrow K \cdot K & = & [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \cdot [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \\ = & [\begin{array}{c} 0 \cdot 0 + i \cdot i & 0 \cdot i + i \cdot 0 \\ i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \end{array}] \\ = & [\begin{array}{c} - 1 & 0 \\ 0 & - 1 \end{array}] \\ = & - E \\ i \cdot j \leftrightarrow I \cdot J & = & [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \cdot [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \\ = & [\begin{array}{c} i \cdot 0 + 0 \cdot (- 1) & i \cdot 1 + 0 \cdot 0 \\ 0 \cdot 0 + (- i) \cdot (- 1) & 0 \cdot 1 + (- i) \cdot 0 \end{array}] \\ = & [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \\ = & K \\ j \cdot i \leftrightarrow J \cdot I & = & [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \cdot [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \\ = & [\begin{array}{c} 0 \cdot i + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot (- i) \\ (- 1) \cdot i + 0 \cdot 0 & (- 1) \cdot 0 + 0 \cdot (- i) \end{array}] \\ = & [\begin{array}{c} 0 & - i \\ - i & 0 \end{array}] \\ = & - K \\ i \cdot k \leftrightarrow I \cdot K & = & [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \cdot [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \\ = & [\begin{array}{c} i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \\ 0 \cdot 0 + (- i) \cdot i & 0 \cdot i + (- i) \cdot 0 \end{array}] \\ = & [\begin{array}{c} 0 & - 1 \\ 1 & 0 \end{array}] \\ = & - J \\ k \cdot i \leftrightarrow K \cdot I & = & [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \cdot [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \\ = & [\begin{array}{c} 0 \cdot i + i \cdot 0 & 0 \cdot 0 + i \cdot (- i) \\ i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot (- i) \end{array}] \\ = & [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \\ = & J \\ j \cdot k \leftrightarrow J \cdot K & = & [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \cdot [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \\ = & [\begin{array}{c} 0 \cdot 0 + 1 \cdot i & 0 \cdot i + 1 \cdot 0 \\ (- 1) \cdot 0 + 0 \cdot i & (- 1) \cdot i + 0 \cdot 0 \end{array}] \\ = & [\begin{array}{c} i & 0 \\ 0 & - i \end{array}] \\ = & I \\ k \cdot j \leftrightarrow K \cdot J & = & [\begin{array}{c} 0 & i \\ i & 0 \end{array}] \cdot [\begin{array}{c} 0 & 1 \\ - 1 & 0 \end{array}] \\ = & [\begin{array}{c} 0 \cdot 0 + i \cdot (- 1) & 0 \cdot 1 + i \cdot 0 \\ i \cdot 0 + 0 \cdot (- 1) & i \cdot 1 + 0 \cdot 0 \end{array}] \\ = & [\begin{array}{c} - i & 0 \\ 0 & i \end{array}] \\ = & - I \end{array}

複素数の行列表現

\begin{array}{rcl} a + b i & \leftrightarrow & [\begin{array}{c} a & - b \\ b & a \end{array}] \\ 積 \\ (a + b i) \cdot (c + d i) & = & a c + a d i + b i c + b i d i \\ = & a c - b d + a d i + b c i \\ = & (a c - b d) + (a d + b c) i \\ [\begin{array}{c} a & - b \\ b & a \end{array}] \cdot [\begin{array}{c} c & - d \\ d & c \end{array}] & = & [\begin{array}{c} a c - b d & - (a d + b c) \\ a d + b c & a c - b d \end{array}] \\ 積 (交 換) \\ (c + d i) \cdot (a + b i) & = & (a c - b d) + (a d + b c) i \\ [\begin{array}{c} c & - d \\ d & c \end{array}] \cdot [\begin{array}{c} a & - b \\ b & a \end{array}] & = & [\begin{array}{c} a c - b d & - (a d + b c) \\ a d + b c & a c - b d \end{array}] \dots 行 列 の 積 は 一 般 に 非 可 換 だ が ， こ の 形 の 行 列 で は 可 換 で あ る \\ 和 \\ (a + i b) + (c + i d) & = & (a + c) + (b + d) i \\ [\begin{array}{c} a & - b \\ b & a \end{array}] + [\begin{array}{c} c & - d \\ d & c \end{array}] & = & [\begin{array}{c} a + c & - (b + d) \\ b + d & a + c \end{array}] \\ 差 \\ (a + b i) - (c + d i) & = & (a - c) + (b - d) i \\ [\begin{array}{c} a & - b \\ b & a \end{array}] - [\begin{array}{c} c & - d \\ d & c \end{array}] & = & [\begin{array}{c} a - c & - (b - d) \\ b - d & a - c \end{array}] \\ 逆 数 \\ \frac{1}{c + d i} & = & \frac{1}{c + d i} \frac{c - d i}{c - d i} \\ = & \frac{c - d i}{c^{2} + d^{2}} \\ \frac{1}{[\begin{array}{c} c & - d \\ d & c \end{array}]} & = & {[\begin{array}{c} c & - d \\ d & c \end{array}]}^{- 1} \dots \frac{1}{A} = A^{- 1} \\ = & \frac{1}{| \begin{array}{c} c & d \\ - d & c \end{array} |} [\begin{array}{c} c & d \\ - d & c \end{array}] \dots {[\begin{array}{c} a & b \\ c & d \end{array}]}^{- 1} = \frac{1}{| \begin{array}{c} a & b \\ c & d \end{array} |} [\begin{array}{c} d & - b \\ - c & a \end{array}] \\ = & \frac{1}{c^{2} + d^{2}} [\begin{array}{c} c & d \\ - d & c \end{array}] \\ 商 \\ \frac{a + b i}{c + d i} & = & \frac{(a + b i) (c - d i)}{(c + d i) (c - d i)} \\ = & \frac{(a c + b d) + (b c - a d) i}{c^{2} + d^{2}} \\ \frac{[\begin{array}{c} a & - b \\ b & a \end{array}]}{[\begin{array}{c} c & - d \\ d & c \end{array}]} & = & [\begin{array}{c} a & - b \\ b & a \end{array}] \cdot \frac{1}{[\begin{array}{c} c & - d \\ d & c \end{array}]} \\ = & [\begin{array}{c} a & - b \\ b & a \end{array}] \cdot (\frac{1}{c^{2} + d^{2}} [\begin{array}{c} c & d \\ - d & c \end{array}]) \dots \frac{1}{[\begin{array}{c} c & - d \\ d & c \end{array}]} = \frac{1}{c^{2} + d^{2}} [\begin{array}{c} c & d \\ - d & c \end{array}] \\ = & \frac{1}{c^{2} + d^{2}} [\begin{array}{c} a & - b \\ b & a \end{array}] \cdot [\begin{array}{c} c & d \\ - d & c \end{array}] \\ = & \frac{1}{c^{2} + d^{2}} [\begin{array}{c} a c + b d & - (b c - a d) \\ b c - a d & a c + b d \end{array}] \end{array}

外積に外積を組み合わされた式　( ベクトル三重積 )

外積に外積を組み合わされた式

\begin{array}{rcl} (A \times B) \times C & = & | \begin{array}{c} i & j & k \\ x_{a} & y_{a} & z_{a} \\ x_{b} & y_{b} & z_{b} \end{array} | \cdot (x_{c} i + y_{c} j + z_{c} k) \\ = & ((y_{a} z_{b} - y_{b} z_{a}) i + (x_{b} z_{a} - x_{a} z_{b}) j + (x_{a} y_{b} - x_{b} y_{a}) k) \times (x_{c} i + y_{c} j + z_{c} k) \\ = & | \begin{array}{c} i & j & k \\ (y_{a} z_{b} - y_{b} z_{a}) i & (x_{b} z_{a} - x_{a} z_{b}) j & (x_{a} y_{b} - x_{b} y_{a}) k \\ x_{c} & y_{c} & z_{c} \end{array} | \\ = & {(x_{b} z_{a} - x_{a} z_{b}) z_{c} - (x_{a} y_{b} - x_{b} y_{a}) y_{c}} i \\ + {(x_{a} y_{b} - x_{b} y_{a}) x_{c} - (y_{a} z_{b} - y_{b} z_{a}) z_{c}} j \\ + {(y_{a} z_{b} - y_{b} z_{a}) y_{c} - (x_{b} z_{a} - x_{a} z_{b}) x_{c}} k \\ = & {(x_{b} z_{a} z_{c} - x_{a} z_{b} z_{c}) - (x_{a} y_{b} y_{c} - x_{b} y_{a} y_{c})} i \\ + {(y_{b} x_{a} x_{c} - y_{a} x_{b} x_{c}) - (y_{a} z_{b} z_{c} - y_{b} z_{a} z_{c})} j \\ + {(z_{b} y_{a} y_{c} - z_{a} y_{b} y_{c}) - (z_{a} x_{b} x_{c} - z_{b} x_{a} x_{c})} k \\ = & {(x_{b} z_{a} z_{c} - x_{a} z_{b} z_{c}) - (x_{a} y_{b} y_{c} - x_{b} y_{a} y_{c}) + x_{b} x_{a} x_{c} - x_{a} x_{b} x_{c}} i \\ + {(y_{b} x_{a} x_{c} - y_{a} x_{b} x_{c}) - (y_{a} z_{b} z_{c} - y_{b} z_{a} z_{c}) + y_{b} y_{a} y_{c} - y_{a} y_{b} y_{c}} j \\ + {(z_{b} y_{a} y_{c} - z_{a} y_{b} y_{c}) - (z_{a} x_{b} x_{c} - z_{b} x_{a} x_{c}) + z_{b} z_{a} z_{c} - z_{a} z_{b} z_{c}} k \\ = & {(x_{b} x_{a} x_{c} + x_{b} y_{a} y_{c} + x_{b} z_{a} z_{c}) i + (y_{b} x_{a} x_{c} + y_{b} y_{a} y_{c} + y_{b} z_{a} z_{c}) j + (z_{b} x_{a} x_{c} + z_{b} y_{a} y_{c} + z_{b} z_{a} z_{c}) k} \\ - {(x_{a} x_{b} x_{c} + x_{a} y_{b} y_{c} + x_{a} z_{b} z_{c}) i + (y_{a} x_{b} x_{c} + y_{a} y_{b} y_{c} + y_{a} z_{b} z_{c}) j + (z_{a} x_{b} x_{c} + z_{a} y_{b} y_{c} + z_{a} z_{b} z_{c}) k} \\ = & {(x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) x_{b} i + (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) y_{b} j + (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) z_{b} k} \\ - {(x_{b} x_{c} + y_{b} y_{c} + z_{b} z_{c}) x_{a} i + (x_{b} x_{c} + y_{b} y_{c} + z_{b} z_{c}) y_{a} j + (x_{b} x_{c} + y_{b} y_{c} + z_{b} z_{c}) z_{a} k} \\ = & (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) (x_{b} i + y_{b} j + z_{b} k) - (x_{b} x_{c} + y_{b} y_{c} + z_{b} z_{c}) (x_{a} i + y_{a} j + z_{a} k) \\ = & (A \cdot C) B - (B \cdot C) A \end{array}

\begin{array}{rcl} A \times (B \times C) & = & (x_{a} i + y_{a} j + z_{a} k) \times | \begin{array}{c} i & j & k \\ x_{b} & y_{b} & z_{b} \\ x_{c} & y_{c} & z_{c} \end{array} | \\ = & (x_{a} i + y_{a} j + z_{a} k) \times ((y_{b} z_{c} - y_{c} z_{b}) i + (x_{c} z_{b} - x_{b} z_{c}) j + (x_{b} y_{c} - x_{c} y_{b}) k) \\ = & | \begin{array}{c} i & j & k \\ x_{a} & y_{a} & z_{a} \\ (y_{b} z_{c} - y_{c} z_{b}) & (x_{c} z_{b} - x_{b} z_{c}) & (x_{b} y_{c} - x_{c} y_{b}) \end{array} | \\ = & {y_{a} (x_{b} y_{c} - x_{c} y_{b}) - z_{a} (x_{c} z_{b} - x_{b} z_{c})} i \\ + {z_{a} (y_{b} z_{c} - y_{c} z_{b}) - x_{a} (x_{b} y_{c} - x_{c} y_{b})} j \\ + {x_{a} (x_{c} z_{b} - x_{b} z_{c}) - y_{a} (y_{b} z_{c} - y_{c} z_{b})} k \\ = & {(x_{b} y_{a} y_{c} - x_{c} y_{a} y_{b}) - (x_{c} z_{a} z_{b} - x_{b} z_{a} z_{c})} i \\ + {(y_{b} z_{a} z_{c} - y_{c} z_{a} z_{b}) - (y_{c} x_{a} x_{b} - y_{b} x_{a} x_{c})} j \\ + {(z_{b} x_{a} x_{c} - z_{c} x_{a} x_{b}) - (z_{c} y_{a} y_{b} - z_{b} y_{a} y_{c})} k \\ = & {(x_{b} y_{a} y_{c} - x_{c} y_{a} y_{b}) - (x_{c} z_{a} z_{b} - x_{b} z_{a} z_{c}) + x_{b} x_{a} x_{c} - x_{c} x_{a} x_{b}} i \\ + {(y_{b} z_{a} z_{c} - y_{c} z_{a} z_{b}) - (y_{c} x_{a} x_{b} - y_{b} x_{a} x_{c}) + y_{b} y_{a} y_{c} - y_{c} y_{a} y_{b}} j \\ + {(z_{b} x_{a} x_{c} - z_{c} x_{a} x_{b}) - (z_{c} y_{a} y_{b} - z_{b} y_{a} y_{c}) + z_{b} z_{a} z_{c} - z_{c} z_{a} z_{b}} k \\ = & {(x_{b} x_{a} x_{c} + x_{b} y_{a} y_{c} + x_{b} z_{a} z_{c}) i + (y_{b} x_{a} x_{c} + y_{b} y_{a} y_{c} + y_{b} z_{a} z_{c}) j + (z_{b} x_{a} x_{c} + z_{b} y_{a} y_{c} + z_{b} z_{a} z_{c}) k} \\ - {(x_{c} x_{a} x_{b} + x_{c} y_{a} y_{b} + x_{c} z_{a} z_{b}) i + (y_{c} x_{a} x_{b} + y_{c} y_{a} y_{b} + y_{c} z_{a} z_{b}) j + (z_{c} x_{a} x_{b} + z_{c} y_{a} y_{b} + z_{c} z_{a} z_{b}) k} \\ = & {(x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) x_{b} i + (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) y_{b} j + (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) z_{b} k} \\ - {(x_{a} x_{b} + y_{a} y_{b} + z_{a} z_{b}) x_{c} i + (x_{a} x_{b} + y_{a} y_{b} + z_{a} z_{b}) y_{c} j + (x_{a} x_{b} + y_{a} y_{b} + z_{a} z_{b}) z_{c} k} \\ = & (x_{a} x_{c} + y_{a} y_{c} + z_{a} z_{c}) (x_{b} i + y_{b} j + z_{b} k) - (x_{a} x_{b} + y_{a} y_{b} + z_{a} z_{b}) (x_{c} i + y_{c} j + z_{c} k) \\ = & (A \cdot C) B - (A \cdot B) C \\ (A \times B) \times C & = & - C \times (A \times B) \dots A \times B = - B \times A \\ = & ((- C) \cdot B) A - ((- C) \cdot A) B \dots A \times (B \times C) = (A \cdot C) B - (A \cdot B) C \\ = & (A \cdot C) B - (B \cdot C) A \dots A \cdot B = B \cdot A, ((- A) \cdot B) = - (A \cdot B) \end{array}

内積と外積が組み合わされた式 ( スカラー三重積 )

内積と外積が組み合わされた式

外積したものと内積をとった場合

\begin{array}{rcl} A \cdot (B \times C) & = & (x_{a} i + y_{a} j + z_{a} k) \cdot | \begin{array}{c} i & j & k \\ x_{b} & y_{b} & z_{b} \\ x_{c} & y_{c} & z_{c} \end{array} | \\ = & (x_{a} i + y_{a} j + z_{a} k) \cdot ((y_{b} z_{c} - y_{c} z_{b}) i + (x_{c} z_{b} - x_{b} z_{c}) j + (x_{b} y_{c} - x_{c} y_{b}) k) \\ = & x_{a} (y_{b} z_{c} - y_{c} z_{b}) + y_{a} (x_{c} z_{b} - x_{b} z_{c}) + z_{a} (x_{b} y_{c} - x_{c} y_{b}) \\ = & x_{a} y_{b} z_{c} + x_{b} y_{c} z_{a} + x_{c} y_{a} z_{b} - x_{a} y_{c} z_{b} - x_{b} y_{a} z_{c} - x_{c} y_{b} z_{a} \end{array}

\begin{array}{rcl} (A \times B) \cdot C & = & | \begin{array}{c} i & j & k \\ x_{a} & y_{a} & z_{a} \\ x_{b} & y_{b} & z_{b} \end{array} | \cdot (x_{c} i + y_{c} j + z_{c} k) \\ = & ((y_{a} z_{b} - y_{b} z_{a}) i + (x_{b} z_{a} - x_{a} z_{b}) j + (x_{a} y_{b} - x_{b} y_{a}) k) \cdot (x_{c} i + y_{c} j + z_{c} k) \\ = & (y_{a} z_{b} - y_{b} z_{a}) x_{c} + (x_{b} z_{a} - x_{a} z_{b}) y_{c} + (x_{a} y_{b} - x_{b} y_{a}) z_{c} \\ = & x_{c} y_{a} z_{b} - x_{c} y_{b} z_{a} + x_{b} y_{c} z_{a} - x_{a} y_{c} z_{b} + x_{a} y_{b} z_{c} - x_{b} y_{a} z_{c} \\ = & x_{a} y_{b} z_{c} + x_{b} y_{c} z_{a} + x_{c} y_{a} z_{b} - x_{a} y_{c} z_{b} - x_{b} y_{a} z_{c} - x_{c} y_{b} z_{a} \end{array}

以上より内積と外積の位置が入れ替わった上記2式が等しいことがわかった．この計算をスカラー三重積と呼ぶ．

\begin{array}{rcl} A \cdot (B \times C) & = & (A \times B) \cdot C \end{array}

スカラー三重積と循環シフト

前述の等式の各ベクトルの表す文字を循環シフトさせて文字を入れ替え，更に2つの等式を用意する．

\begin{array}{rcl} A \cdot (B \times C) & = & (A \times B) \cdot C \dots 前 述 の 等 式 \\ B \cdot (C \times A) & = & (B \times C) \cdot A = A \cdot (B \times C) \dots A \cdot B = B \cdot A \\ C \cdot (A \times B) & = & (C \times A) \cdot B = B \cdot (C \times A) \dots A \cdot B = B \cdot A \end{array}

以上より6つの“内積と外積が組み合わされた式”は等しいことがわかった．

\begin{array}{rcl} A \cdot (B \times C) & = & B \cdot (C \times A) \\ = & C \cdot (A \times B) \\ = & (A \times B) \cdot C \\ = & (B \times C) \cdot A \\ = & (C \times A) \cdot B \end{array}

(これらは

A

B

C

を循環シフトした並びであり，

A

C

B

のような入替えを含めた並びにはならない点に注意)

点と直線の距離 (直角三角形の面積を経由して求める)

直角三角形を作成し面積の公式より求める方法

点 P (x_{p}, y_{p})

から

x 軸

と水平に

直 線 a x + b y + c = 0

に向かって線分を引き

交 点 A (x_{a}, y_{a} = y_{p})

と，

点 P

から

x 軸

と垂直に

直 線 a x + b y + c = 0

に向かって線分を引き

交 点 B (x_{b} = x_{p}, y_{b})

を用意する．

△ A B P

は

∠ P

を直角とした直角三角形のため，

△ A B P

の面積は

\frac{1}{2} \cdot \overset{―}{A P} \cdot \overset{―}{B P}

となる．一方で，

\overset{―}{A B}

を底辺として考えると

点 P

への

高 さ d

を用いることで，

△ A B P

の面積は

\frac{1}{2} \cdot \overset{―}{A B} \cdot d

で表せる．この2式が同一の面積を示すので等式にして，式変形で

d

を求める(

d

は

点 P (x_{p}, y_{p})

と

直 線 a x + b y + c = 0

との距離となる)．

\begin{array}{rcl} \frac{1}{2} \cdot \overset{―}{A P} \cdot \overset{―}{B P} & = & \frac{1}{2} \cdot \overset{―}{A B} \cdot d \\ \overset{―}{A P} \cdot \overset{―}{B P} & = & \overset{―}{A B} \cdot d \\ d & = & \frac{\overset{―}{A P} \cdot \overset{―}{B P}}{\overset{―}{A B}} \end{array}

\overset{―}{A P}

を求める．

\begin{array}{rcl} a x_{a} + b y_{p} + c & = & 0 \dots y_{a} = y_{p} な の で y = y_{p} と し て x_{a} を 求 め る ． \\ x_{a} & = & - \frac{b}{a} y_{p} - \frac{c}{a} \\ \overset{―}{A P} & = & | x_{a} - x_{p} | \\ = & | - \frac{b}{a} y_{p} - \frac{c}{a} - x_{p} | \\ = & | - \frac{1}{a} (a x_{p} + b y_{p} + c) | \\ = & | \frac{1}{a} (a x_{p} + b y_{p} + c) | \end{array}

\overset{―}{B P}

を求める．

\begin{array}{rcl} a x_{p} + b y_{b} + c & = & 0 \dots x_{b} = x_{p} な の で x = x_{p} と し て y_{b} を 求 め る ． \\ y_{b} & = & - \frac{a}{b} x_{p} - \frac{c}{b} \\ \overset{―}{B P} & = & | y_{b} - y_{p} | \\ = & | - \frac{a}{b} x_{p} - \frac{c}{b} - y_{p} | \\ = & | - \frac{1}{b} (a x_{p} + b y_{p} + c) | \\ = & | \frac{1}{b} (a x_{p} + b y_{p} + c) | \end{array}

\overset{―}{A B}

を求める．

\begin{array}{rcl} \overset{―}{A B} & = & \sqrt{{\overset{―}{A P}}^{2} + {\overset{―}{B P}}^{2}} \\ = & \sqrt{{| \frac{1}{a} (a x_{p} + b y_{p} + c) |}^{2} + {| \frac{1}{b} (a x_{p} + b y_{p} + c) |}^{2}} \\ = & \sqrt{\frac{1}{a^{2}} {| a x_{p} + b y_{p} + c |}^{2} + \frac{1}{b^{2}} {| a x_{p} + b y_{p} + c |}^{2}} \\ = & \sqrt{(\frac{1}{a^{2}} + \frac{1}{b^{2}}) {| a x_{p} + b y_{p} + c |}^{2}} \\ = & \sqrt{(\frac{a^{2} + b^{2}}{a^{2} b^{2}}) {| a x_{p} + b y_{p} + c |}^{2}} \\ = & \frac{\sqrt{a^{2} + b^{2}}}{| a b |} | a x_{p} + b y_{p} + c | \end{array}

d

を求める．

\begin{array}{rcl} d & = & \frac{\overset{―}{A P} \cdot \overset{―}{B P}}{\overset{―}{A B}} \\ = & \frac{\overset{―}{A P} \cdot \overset{―}{B P}}{\sqrt{{\overset{―}{A P}}^{2} + {\overset{―}{B P}}^{2}}} \\ = & \frac{| \frac{1}{a} (a x_{p} + b y_{p} + c) | \cdot | \frac{1}{b} (a x_{p} + b y_{p} + c) |}{\frac{\sqrt{a^{2} + b^{2}}}{| a b |} | a x_{p} + b y_{p} + c |} \\ = & \frac{\frac{1}{| a b |} {| a x_{p} + b y_{p} + c |}^{2}}{\frac{\sqrt{a^{2} + b^{2}}}{| a b |} | a x_{p} + b y_{p} + c |} \\ = & \frac{| a x_{p} + b y_{p} + c |}{\sqrt{a^{2} + b^{2}}} \end{array}

点と直線の距離 (直交する直線の式を経由して求める)

直交する直線の式より交点を求め，距離を求める方法

点 P (x_{p}, y_{p})

と

直 線 a x + b y + c = 0

の距離の公式を導く．

\begin{array}{rcl} a x + b y + c & = & 0 \\ y & = & - \frac{a}{b} x - \frac{c}{b} \end{array}

傾き

α

の直線に直交する直線の傾きは

- \frac{1}{α}

となるので上記の直線の場合，直交する直線の傾きは以下のようになる．

\begin{array}{rcl} - \frac{1}{α} & = & - \frac{1}{- \frac{a}{b}} \\ = & \frac{b}{a} \end{array}

上記傾きの直線が

点 P (x_{p}, y_{p})

を通るのでこの直線の式は以下のようになる．

\begin{array}{rcl} y - y_{p} & = & \frac{b}{a} (x - x_{p}) \\ y & = & \frac{b}{a} (x - x_{p}) + y_{p} \\ = & \frac{b}{a} x - \frac{b}{a} x_{p} + y_{p} \end{array}

もとの

直 線 y = - \frac{a}{b} x - \frac{c}{b}

と上記の直交する

直 線 y = \frac{b}{a} x - \frac{b}{a} x_{p} + y_{p}

との

交 点 Q (x_{q}, y_{q})

を求める．

\begin{array}{rcl} - \frac{a}{b} x_{q} - \frac{c}{b} & = & \frac{b}{a} x_{q} - \frac{b}{a} x_{p} + y_{p} \\ - \frac{a}{b} x_{q} - \frac{b}{a} x_{q} & = & - \frac{b}{a} x_{p} + y_{p} + \frac{c}{b} \\ (- \frac{a}{b} - \frac{b}{a}) x_{q} & = & - \frac{b}{a} x_{p} + y_{p} + \frac{c}{b} \\ - \frac{a^{2} + b^{2}}{a b} x_{q} & = & - \frac{b}{a} x_{p} + y_{p} + \frac{c}{b} \\ x_{q} & = & \frac{- a b}{a^{2} + b^{2}} (- \frac{b}{a} x_{p} + y_{p} + \frac{c}{b}) \\ = & \frac{1}{a^{2} + b^{2}} (b^{2} x_{p} - a b y_{p} - a c) \\ y_{q} & = & - \frac{a}{b} x_{q} - \frac{c}{b} \\ = & - \frac{a}{b} {\frac{1}{a^{2} + b^{2}} (b^{2} x_{p} - a b y_{p} - a c)} - \frac{c}{b} \\ = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} + a^{2} y_{p} + \frac{a^{2}}{b} c) - \frac{c}{b} \\ = & \frac{1}{a^{2} + b^{2}} {- a b x_{p} + a^{2} y_{p} + \frac{a^{2}}{b} c - (a^{2} + b^{2}) \frac{c}{b}} \\ = & \frac{1}{a^{2} + b^{2}} {- a b x_{p} + a^{2} y_{p} + \frac{c}{b} (a^{2} - a^{2} - b^{2})} \\ = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} + a^{2} y_{p} - \frac{b^{2} c}{b}) \\ = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} + a^{2} y_{p} - b c) \end{array}

以上より

点 P (x_{p}, y_{p})

と

点 Q (x_{q}, y_{q})

の距離を求める．

\begin{array}{rcl} x_{q} - x_{p} & = & \frac{1}{a^{2} + b^{2}} (b^{2} x_{p} - a b y_{p} - a c) - x_{p} \\ = & \frac{1}{a^{2} + b^{2}} {b^{2} x_{p} - a b y_{p} - a c - (a^{2} + b^{2}) x_{p}} \\ = & \frac{1}{a^{2} + b^{2}} (b^{2} x_{p} - a b y_{p} - a c - a^{2} x_{p} - b^{2} x_{p}) \\ = & \frac{1}{a^{2} + b^{2}} (- a^{2} x_{p} - a b y_{p} - a c) \\ = & \frac{- a}{a^{2} + b^{2}} (a x_{p} + b y_{p} + c) \\ {(x_{q} - x_{p})}^{2} & = & {\frac{- a}{a^{2} + b^{2}} (a x_{p} + b y_{p} + c)}^{2} \\ = & {(\frac{- a}{a^{2} + b^{2}})}^{2} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{{(- a)}^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{a^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ y_{q} - y_{p} & = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} + a^{2} y_{p} - b c) - y_{p} \\ = & \frac{1}{a^{2} + b^{2}} {- a b x_{p} + a^{2} y_{p} - b c - (a^{2} + b^{2}) y_{p}} \\ = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} + a^{2} y_{p} - b c - a^{2} y_{p} - b^{2} y_{p}) \\ = & \frac{1}{a^{2} + b^{2}} (- a b x_{p} - b^{2} y_{p} - b c) \\ = & \frac{- b}{a^{2} + b^{2}} (a x_{p} + b y_{p} + c) \\ {(y_{q} - y_{p})}^{2} & = & {\frac{- b}{a^{2} + b^{2}} (a x_{p} + b y_{p} + c)}^{2} \\ = & {(\frac{- b}{a^{2} + b^{2}})}^{2} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{{(- b)}^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{b^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ {(x_{q} - x_{p})}^{2} + {(y_{q} - y_{p})}^{2} & = & \frac{a^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} + \frac{b^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & {\frac{a^{2}}{{(a^{2} + b^{2})}^{2}} + \frac{b^{2}}{{(a^{2} + b^{2})}^{2}}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{a^{2} + b^{2}}{{(a^{2} + b^{2})}^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{1}{a^{2} + b^{2}} {(a x_{p} + b y_{p} + c)}^{2} \\ = & \frac{{(a x_{p} + b y_{p} + c)}^{2}}{a^{2} + b^{2}} \\ \sqrt{{(x_{q} - x_{p})}^{2} + {(y_{q} - y_{p})}^{2}} & = & \sqrt{\frac{{(a x_{p} + b y_{p} + c)}^{2}}{a^{2} + b^{2}}} \\ = & \frac{| a x_{p} + b y_{p} + c |}{\sqrt{a^{2} + b^{2}}} \end{array}

与えられる

直 線

が

直 線 y = a x + b

の形の場合，上記公式の

b = 1

で

c

を

b

と表すことになる．よって公式は以下のように変わることになる．

\begin{array}{rcl} \sqrt{{(x_{q} - x_{p})}^{2} + {(y_{q} - y_{p})}^{2}} & = & \frac{| a x_{p} + y_{p} + b |}{\sqrt{a^{2} + 1^{2}}} \end{array}

2点を補間した位置への回転操作のための四元数

位置ベクトル

p

への四元数

q_{1}

q_{2}

による回転操作を行った結果を

p_{1}

，

p_{2}

とする．また

p_{1}

と

p_{2}

との間を補間した位置を

p_{t}

とした時，

p

に対する回転操作で

p_{t}

となる四元数

q_{t}

を考える．

\begin{array}{rcl} q & = & w + x i + y j + z k \\ = & (w, x i + y j + z k) \dots (実 部 (ス カ ラ ー), i j k 部 (ベ ク ト ル)) \\ = & (w, V) \dots V = x i + y j + z k \\ \overset{―}{q} & = & w - x i - y j - z k \\ = & (w, - x i - y j - z k) \\ = & (w, - V) \\ q_{回 転} & = & (\cos (\frac{θ}{2}), \sin (\frac{θ}{2}) V) \\ \dots 特 に 位 置 ベ ク ト ル を 回 転 軸 V (た だ し V は 単 位 ベ ク ト ル . | V | = 1), 回 転 角 θ で 回 す 際 の 四 元 数 \\ q_{回 転} と q_{回 転}^{- 1} で 位 置 ベ ク ト ル p へ 左 右 か ら 作 用 さ せ q_{回 転} p q_{回 転}^{- 1} と し て 用 い る . \\ q_{1} & = & (\cos (\frac{θ_{1}}{2}), \sin (\frac{θ_{1}}{2}) V_{1}) \\ \dots | q_{1} | = 1, | V_{1} | = 1, 位 置 ベ ク ト ル p を p_{1} に 向 け て 回 転 さ せ る 際 の 軸 \\ = & (C_{1}, S_{1} V_{1}) \\ q_{2} & = & (\cos (\frac{θ_{2}}{2}), \sin (\frac{θ_{2}}{2}) V_{2}) \\ \dots | q_{2} | = 1, | V_{2} | = 1 位 置 ベ ク ト ル p を p_{2} に 向 け て 回 転 さ せ る 際 の 軸 \\ = & (C_{2}, S_{2} V_{2}) \\ q_{1 \to 2} & = & (\cos (\frac{θ_{1 \to 2}}{2}), \sin (\frac{θ_{1 \to 2}}{2}) V_{1 \to 2}) \\ \dots | q_{1 \to 2} | = 1, | V_{1 \to 2} | = 1, 位 置 ベ ク ト ル p を p_{1} か ら p_{2} に 向 け て 回 転 さ せ る 際 の 軸 \\ = & (C_{1 \to 2}, S_{1 \to 2} V_{1 \to 2}) \\ q_{1 \to t} & = & (\cos (t \frac{θ_{1 \to 2}}{2}), \sin (t \frac{θ_{1 \to 2}}{2}) V_{1 \to 2}) \\ \dots 0 \leq t \leq 1, 位 置 ベ ク ト ル p を p_{1} か ら p_{2} に 向 け て 回 転 さ せ る 途 中 の p_{t} ま で の 際 の 軸 な の で V_{1 \to 2} で 変 わ ら な い . 回 転 量 だ け t で 調 整 さ れ る . \\ = & (C_{1 \to t}, S_{1 \to t} V_{1 \to 2}) \\ q_{t} & = & (\cos (\frac{θ_{t}}{2}), \sin (\frac{θ_{t}}{2}) V_{t}) \\ \dots | q_{t} | = 1, | V_{t} | = 1, 位 置 ベ ク ト ル p を p_{t} に 向 け て 回 転 さ せ る 際 の 軸 \\ = & (C_{t}, S_{t} V_{t}) \end{array}

p

，

p_{1}

，

p_{2}

，

q_{1}

，

q_{2}

の関係性から

q_{1 \to 2} = (C_{1 \to 2}, S_{1 \to 2} V_{1 \to 2})

を求める．

\begin{array}{rcl} p_{1} & = & q_{1} p q_{1}^{- 1} \\ p_{2} & = & q_{2} p q_{2}^{- 1} \\ p_{2} & = & q_{1 \to 2} p_{1} q_{1 \to 2}^{- 1} \\ = & q_{1 \to 2} (q_{1} p q_{1}^{- 1}) q_{1 \to 2}^{- 1} \dots p_{1} = q_{1} p q_{1}^{- 1} \\ = & (q_{1 \to 2} q_{1}) p (q_{1}^{- 1} q_{1 \to 2}^{- 1}) \\ q_{2} & = & q_{1 \to 2} q_{1} \\ q_{1 \to 2} & = & q_{2} q_{1}^{- 1} \\ = & q_{2} \frac{\overset{―}{q_{1}}}{| q_{1} |^{2}} \dots q^{- 1} = \frac{\overset{―}{q}}{| q |^{2}} \\ = & q_{2} \overset{―}{q_{1}} \dots | q_{1} |^{2} = 1^{2} = 1 \\ = & (C_{2}, S_{2} V_{2}) (C_{1}, - S_{1} V_{1}) \\ = & (C_{1} C_{2} - {- S_{1}} S_{2} V_{2} \cdot V_{1}, - C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + {- S_{1}} S_{2} V_{2} \times V_{1}) \\ \dots q_{1} q_{2} = (w_{1}, V_{1}) (w_{2}, V_{2}) = (w_{1} w_{2} - V_{1} \cdot V_{2}, w_{1} V_{2} + w_{2} V_{1} + V_{1} \times V_{2}) \\ = & (C_{1} C_{2} + S_{1} S_{2} V_{1} \cdot V_{2}, - C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2}) \dots A \cdot B = B \cdot A, A \times B = - B \times A \\ C_{1 \to 2} & = & C_{1} C_{2} + S_{1} S_{2} V_{1} \cdot V_{2} \\ S_{1 \to 2} V_{1 \to 2} & = & - C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2} \end{array}

p

，

p_{1}

，

p_{t}

，

q_{1}

，

q_{1 \to t}

の関係性から

q_{t} = (C_{t}, S_{t} V_{t})

を求める．

\begin{array}{rcl} p_{t} & = & q_{t} p q_{t}^{- 1} \\ p_{t} & = & q_{1 \to t} p_{1} q_{1 \to t}^{- 1} \\ = & q_{1 \to t} (q_{1} p q_{1}^{- 1}) q_{1 \to t}^{- 1} \dots p_{1} = q_{1} p q_{1}^{- 1} \\ = & (q_{1 \to t} q_{1}) p (q_{1}^{- 1} q_{1 \to t}^{- 1}) \\ q_{t} & = & q_{1 \to t} q_{1} \\ = & (C_{1 \to t}, S_{1 \to t} V_{1 \to 2}) (C_{1}, S_{1} V_{1}) \\ = & (C_{1} C_{1 \to t} - S_{1} S_{1 \to t} V_{1 \to 2} \cdot V_{1}, C_{1 \to t} S_{1} V_{1} + C_{1} S_{1 \to t} V_{1 \to 2} + S_{1} S_{1 \to t} V_{1 \to 2} \times V_{1}) \\ = & (C_{1} C_{1 \to t} - S_{1} S_{1 \to t} V_{1} \cdot V_{1 \to 2}, C_{1 \to t} S_{1} V_{1} + C_{1} S_{1 \to t} V_{1 \to 2} + S_{1} S_{1 \to t} V_{1 \to 2} \times V_{1}) \dots A \cdot B = B \cdot A \\ C_{t} & = & C_{1} C_{1 \to t} - S_{1} S_{1 \to t} V_{1} \cdot V_{1 \to 2} \\ S_{t} V_{t} & = & C_{1 \to t} S_{1} V_{1} + C_{1} S_{1 \to t} V_{1 \to 2} + S_{1} S_{1 \to t} V_{1 \to 2} \times V_{1} \end{array}

上記で求めた

S_{1 \to 2} V_{1 \to 2}

を用いて，

V_{1} \cdot V_{1 \to 2}

を求めておく．

\begin{array}{rcl} S_{1 \to 2} V_{1 \to 2} & = & - C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2} \\ V_{1 \to 2} & = & \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2}} \\ V_{1} \cdot V_{1 \to 2} & = & V_{1} \cdot [\frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} (V_{1} \times V_{2})}] \\ = & \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} \cdot V_{1} + C_{1} S_{2} V_{1} \cdot V_{2} + S_{1} S_{2} V_{1} \cdot (V_{1} \times V_{2})} \\ = & \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} + C_{1} S_{2} V_{1} \cdot V_{2} + S_{1} S_{2} V_{2} \cdot (V_{1} \times V_{1})} \dots V_{1} \cdot V_{1} = {| V_{1} |}^{2} = 1^{2} = 1, A \cdot (B \times C) = C \cdot (A \times B) (ス カ ラ ー 三 重 積) \\ = & \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} + C_{1} S_{2} V_{1} \cdot V_{2}} \dots A \times A = 0 \end{array}

上記で求めた

V_{1} \cdot V_{1 \to 2}

を用いて，

C_{t}

を変形する．

\begin{array}{rcl} C_{t} & = & C_{1} C_{1 \to t} - S_{1} S_{1 \to t} V_{1} \cdot V_{1 \to 2} \\ = & C_{1} C_{1 \to t} - S_{1} S_{1 \to t} [\frac{1}{S_{1 \to 2}} {- C_{2} S_{1} + C_{1} S_{2} V_{1} \cdot V_{2}}] \\ \dots V_{1} \cdot V_{1 \to 2} = \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} + C_{1} S_{2} V_{1} \cdot V_{2}} \\ = & C_{1} C_{1 \to t} - \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} S_{1}^{2} + C_{1} S_{1} S_{2} V_{1} \cdot V_{2}} \\ = & C_{1} C_{1 \to t} - \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} (1 - C_{1}^{2}) + C_{1} S_{1} S_{2} V_{1} \cdot V_{2}} \dots \sin^{2} (θ) = 1 - \cos^{2} (θ) \\ = & C_{1} C_{1 \to t} - \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} + C_{1}^{2} C_{2} + C_{1} S_{1} S_{2} V_{1} \cdot V_{2}} \\ = & C_{1} C_{1 \to t} - \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} + C_{1} (C_{1} C_{2} + S_{1} S_{2} V_{1} \cdot V_{2})} \\ = & C_{1} C_{1 \to t} - \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} + C_{1} C_{1 \to 2}} \dots C_{1 \to 2} = C_{1} C_{2} + S_{1} S_{2} V_{1} \cdot V_{2} \\ = & C_{1} C_{1 \to t} + \frac{S_{1 \to t}}{S_{1 \to 2}} C_{2} - \frac{S_{1 \to t}}{S_{1 \to 2}} C_{1} C_{1 \to 2} \\ = & \frac{S_{1 \to 2} C_{1 \to t} - C_{1 \to 2} S_{1 \to t}}{S_{1 \to 2}} C_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} C_{2} \\ = & \frac{\sin (\frac{θ_{1 \to 2}}{2}) \cos (t \frac{θ_{1 \to 2}}{2}) - \cos (\frac{θ_{1 \to 2}}{2}) \sin (t \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} C_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} C_{2} \\ = & \frac{\sin (\frac{θ_{1 \to 2}}{2} - t \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} C_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} C_{2} \dots \sin (α) \cos (β) - \cos (α) \sin (β) = \sin (α - β) \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} C_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} C_{2} \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{2} \end{array}

上記で求めた

V_{1 \to 2}

を用いて，

S_{t} V_{t}

を変形する．

\begin{array}{rcl} S_{t} V_{t} & = & C_{1 \to t} S_{1} V_{1} + C_{1} S_{1 \to t} V_{1 \to 2} + S_{1} S_{1 \to t} V_{1 \to 2} \times V_{1} \\ = & C_{1 \to t} S_{1} V_{1} + C_{1} S_{1 \to t} [\frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2}}] + S_{1} S_{1 \to t} [\frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2}}] \times V_{1} \\ \dots V_{1 \to 2} = \frac{1}{S_{1 \to 2}} {- C_{2} S_{1} V_{1} + C_{1} S_{2} V_{2} + S_{1} S_{2} V_{1} \times V_{2}} \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{1} C_{2} S_{1} V_{1} + C_{1}^{2} S_{2} V_{2} + C_{1} S_{1} S_{2} V_{1} \times V_{2}} + \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{2} S_{1}^{2} V_{1} \times V_{1} + C_{1} S_{1} S_{2} V_{2} \times V_{1} + S_{1}^{2} S_{2} V_{1} \times V_{2} \times V_{1}} \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{1} C_{2} S_{1} V_{1} + C_{1}^{2} S_{2} V_{2} + C_{1} S_{1} S_{2} V_{1} \times V_{2} - C_{2} S_{1}^{2} V_{1} \times V_{1} + C_{1} S_{1} S_{2} V_{1} \times V_{2} + S_{1}^{2} S_{2} V_{1} \times V_{2} \times V_{1}} \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} [- C_{1} C_{2} S_{1} V_{1} + C_{1}^{2} S_{2} V_{2} + S_{1}^{2} S_{2} {(V_{1} \cdot V_{1}) V_{2} - (V_{2} \cdot V_{1}) V_{1}}] \dots A \times A = 0, A \times B \times C = (A \cdot C) B - (B \cdot C) A \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} [- C_{1} C_{2} S_{1} V_{1} + C_{1}^{2} S_{2} V_{2} + S_{1}^{2} S_{2} {{| V_{1} |}^{2} V_{2} - | V_{2} | | V_{1} | \cos (θ_{1 \to 2}) V_{1}}] \dots A \cdot A = {| A |}^{2}, A \cdot B = | A | | B | \cos (θ) \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} {- C_{1} C_{2} S_{1} V_{1} + C_{1}^{2} S_{2} V_{2} + S_{1}^{2} S_{2} V_{2} - S_{1}^{2} S_{2} \cos (θ_{1 \to 2}) V_{1}} \dots | V_{1} | = 1, | V_{2} | = 1 \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} [{- C_{1} C_{2} S_{1} - S_{1}^{2} S_{2} \cos (θ_{1 \to 2})} V_{1} + {C_{1}^{2} S_{2} + S_{1}^{2} S_{2}} V_{2}] \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} [- S_{1} {C_{1} C_{2} + S_{1} S_{2} \cos (θ_{1 \to 2})} V_{1} + S_{2} {C_{1}^{2} + S_{1}^{2}} V_{2}] \\ = & C_{1 \to t} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} {- S_{1} C_{1 \to 2} V_{1} + S_{2} V_{2}} \dots C_{1 \to 2} = C_{1} C_{2} + S_{1} S_{2} \cos (θ_{1 \to 2}), \cos^{2} (θ) + \sin^{2} (θ) = 1 \\ = & \frac{S_{1 \to 2} C_{1 \to t} - C_{1 \to 2} S_{1 \to t}}{S_{1 \to 2}} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} S_{2} V_{2} \\ = & \frac{\sin (\frac{θ_{1 \to 2}}{2}) \cos (t \frac{θ_{1 \to 2}}{2}) - \cos (\frac{θ_{1 \to 2}}{2}) \sin (t \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} S_{2} V_{2} \\ = & \frac{\sin (\frac{θ_{1 \to 2}}{2} - t \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} S_{2} V_{2} \dots \sin (α) \cos (β) - \cos (α) \sin (β) = \sin (α - β) \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{S_{1 \to 2}} S_{1} V_{1} + \frac{S_{1 \to t}}{S_{1 \to 2}} S_{2} V_{2} \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{1} V_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{2} V_{2} \end{array}

以上により

p_{t}

へ

p

を回転させる四元数

q_{t}

が

q_{1}

，

q_{2}

(及び

θ_{1 \to 2}

と

t

)によって表すことができた．

\begin{array}{rcl} q_{t} & = & (C_{t}, S_{t} V_{t}) \\ = & (\frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{2}, \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{1} V_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{2} V_{2}) \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{2} + \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{1} V_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{2} V_{2} \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{1} + \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{1} V_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} C_{2} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} S_{2} V_{2} \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} (C_{1}, S_{1} V_{1}) + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} (C_{2}, S_{2} V_{2}) \\ = & \frac{\sin ((1 - t) \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} q_{1} + \frac{\sin (t \frac{θ_{1 \to 2}}{2})}{\sin (\frac{θ_{1 \to 2}}{2})} q_{2} \end{array}

これは“ 2点を補間した位置への回転操作のための三角凾数”と同様の式の形をしている．

2点を補間した位置への回転操作のための三角凾数

原 点 O を 中 心 と し た 等 距 離 R 上 の 点 P の 点 P_{1} か ら 点 P_{2} へ の 移 動 (半 径 R の 円 周 上) を 考 え る . \vec{O P_{1}} と \vec{O P_{2}} の な す 角 を θ と す る .

\begin{array}{rcl} | \vec{O P} | & = & | \vec{O P_{1}} | = | \vec{O P_{2}} | = R \\ \vec{O P_{1}} \cdot \vec{O P_{2}} & = & | \vec{O P_{1}} | | \vec{O P_{2}} | \cos (θ) \end{array}

パラメータ

t

(ただし

t

は

0 \leq t \leq 1

の実数)を用いて

\vec{O P}

との関係(内積)を求めると以下のようになる

\begin{array}{rcl} \vec{O P} \cdot \vec{O P_{1}} & = & | \vec{O P} | | \vec{O P_{1}} | \cos (t θ) \\ = & R^{2} \cos (t θ) \cdot た だ し t は 0 \leq t \leq 1 の 実 数 \\ \vec{O P} \cdot \vec{O P_{2}} & = & | \vec{O P} | | \vec{O P_{2}} | \cos ((1 - t) θ) \\ = & R^{2} \cos ((1 - t) θ) \end{array}

\vec{O P} = α \vec{O P_{1}} + β \vec{O P_{2}}

で表すと仮定する．

\begin{array}{rcl} \vec{O P} \cdot \vec{O P_{1}} & = & (α \vec{O P_{1}} + β \vec{O P_{2}}) \cdot \vec{O P_{1}} \\ = & α \vec{O P_{1}} \cdot \vec{O P_{1}} + β \vec{O P_{2}} \cdot \vec{O P_{1}} \\ = & α | \vec{O P_{1}} | | \vec{O P_{1}} | \cos (0) + β | \vec{O P_{1}} | | \vec{O P_{2}} | \cos (θ) \\ = & α R^{2} \cdot 1 + β R^{2} \cos (θ) \\ = & R^{2} (α + β \cos (θ)) \\ \vec{O P} \cdot \vec{O P_{2}} & = & (α \vec{O P_{1}} + β \vec{O P_{2}}) \cdot \vec{O P_{2}} \\ = & α \vec{O P_{1}} \cdot \vec{O P_{2}} + β \vec{O P_{2}} \cdot \vec{O P_{2}} \\ = & α | \vec{O P_{1}} | | \vec{O P_{2}} | \cos (θ) + β | \vec{O P_{2}} | | \vec{O P_{2}} | \cos (0) \\ = & α R^{2} \cos (θ) + β R^{2} \cdot 1 \\ = & R^{2} (α \cos (θ) + β) \end{array}

以上より以下の連立方程式が作れる．

{\begin{aligned} α + β \cos (θ) & = & \cos (t θ) \\ α \cos (θ) + β & = & \cos ((1 - t) θ) \end{aligned}

連立方程式を解く．

\begin{array}{rcl} (α + β \cos (θ)) \cos (θ) & = & \cos (t θ) \cos (θ) \dots 連 立 方 程 式 上 側 の 式 の 両 辺 に \cos (θ) を 掛 け る ． \\ α \cos (θ) + β \cos^{2} (θ) & = & \cos (t θ) \cos (θ) \\ (α \cos (θ) + β \cos^{2} (θ)) - (α \cos (θ) + β) & = & (\cos (t θ) \cos (θ)) - (\cos ((1 - t) θ)) \dots 連 立 方 程 式 上 側 の 式 の 変 形 か ら 連 立 方 程 式 下 側 の 式 を 引 く ． \\ β \cos^{2} (θ) - β & = & \cos (t θ) \cos (θ) - (\cos (θ - t θ)) \\ β (\cos^{2} (θ) - 1) & = & \cos (t θ) \cos (θ) - (\cos (θ) \cos (t θ) + \sin (θ) \sin (t θ)) \dots \cos (α - β) = \cos (α) \cos (β) + \sin (α) \sin (β) \\ β (\cos^{2} (θ) - 1) & = & - \sin (θ) \sin (t θ) \\ β (1 - \cos^{2} (θ)) & = & \sin (θ) \sin (t θ) \\ β \sin^{2} (θ) & = & \sin (θ) \sin (t θ) \\ β & = & \frac{\sin (θ) \sin (t θ)}{\sin^{2} (θ)} \\ β & = & \frac{\sin (t θ)}{\sin (θ)} \\ α + (\frac{\sin (t θ)}{\sin (θ)}) \cos (θ) & = & \cos (t θ) \dots 連 立 方 程 式 上 側 の 式 に 求 め た β を 代 入 す る ． \\ α & = & \cos (t θ) - (\frac{\sin (t θ)}{\sin (θ)}) \cos (θ) \\ α & = & \frac{\sin (θ) \cos (t θ) - \cos (θ) \sin (t θ)}{\sin (θ)} \\ α & = & \frac{\sin (θ - t θ)}{\sin (θ)} \dots \sin (α) \cos (β) - \cos (α) \sin (β) = \sin (α - β) \\ = & \frac{\sin ((1 - t) θ)}{\sin (θ)} \\ α, β & = & \frac{\sin ((1 - t) θ)}{\sin (θ)}, \frac{\sin (t θ)}{\sin (θ)} \end{array}

解より点

P_{1}

から点

P_{2}

への移動の間の点

P

を変数

t

を用いて以下のように表現できることが示せた．

\begin{array}{rcl} \vec{O P} & = & α \vec{O P_{1}} + β \vec{O P_{2}} \\ = & \frac{\sin ((1 - t) θ)}{\sin (θ)} \vec{O P_{1}} + \frac{\sin (t θ)}{\sin (θ)} \vec{O P_{2}} \end{array}

三次元でも四元数を用いると同様の形の式となることを示した記事が“ 2点を補間した位置への回転操作のための四元数”です．

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