2点を補間した位置への回転操作のための四元数

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$\mathbf{p}$への四元数${\mathbf{q}}_{\mathbf{1}}$, ${\mathbf{q}}_{\mathbf{2}}$による回転操作を行った結果を${\mathbf{p}}_{\mathbf{1}}$，${\mathbf{p}}_{\mathbf{2}}$とする． また${\mathbf{p}}_{\mathbf{1}}$と${\mathbf{p}}_{\mathbf{2}}$との間を補間した位置を${\mathbf{p}}_{\mathbf{t}}$とした時，$\mathbf{p}$に対する回転操作で${\mathbf{p}}_{\mathbf{t}}$となる四元数${\mathbf{q}}_{\mathbf{t}}$を考える． $$\begin{array}{rcl}\mathbf{q}& =& w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ & =& (w,x\mathbf{i}+y\mathbf{j}+z\mathbf{k})\cdots (\mathrm{実}\mathrm{部}(\mathrm{ス}\mathrm{カ}\mathrm{ラ}\mathrm{ー}),ijk\mathrm{部}(\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}))\\ & =& (w,\mathbf{V})\cdots \mathbf{V}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ \bar{\mathbf{q}}& =& w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}\\ & =& (w,-x\mathbf{i}-y\mathbf{j}-z\mathbf{k})\\ & =& (w,-\mathbf{V})\\ {\mathbf{q}}_{\mathrm{回}\mathrm{転}}& =& (\cos \left(\frac{\theta }{2}\right),\sin \left(\frac{\theta }{2}\right)\mathbf{V})\\ & & \cdots \mathrm{特}\mathrm{に}\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{を}\mathrm{回}\mathrm{転}\mathrm{軸}\mathbf{V}(\mathrm{た}\mathrm{だ}\mathrm{し}\mathbf{V}\mathrm{は}\mathrm{単}\mathrm{位}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}.\left|\mathbf{V}\right|=1),\mathrm{回}\mathrm{転}\mathrm{角}\theta \mathrm{で}\mathrm{回}\mathrm{す}\mathrm{際}\mathrm{の}\mathrm{四}\mathrm{元}\mathrm{数}\\ & & {\mathbf{q}}_{\mathrm{回}\mathrm{転}}\mathrm{と}{\mathbf{q}}_{\mathrm{回}\mathrm{転}}^{\mathbf{-}\mathbf{1}}\mathrm{で}\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathbf{p}\mathrm{へ}\mathrm{左}\mathrm{右}\mathrm{か}\mathrm{ら}\mathrm{作}\mathrm{用}\mathrm{さ}\mathrm{せ}{\mathbf{q}}_{\mathrm{回}\mathrm{転}}\mathbf{p}{\mathbf{q}}_{\mathrm{回}\mathrm{転}}^{\mathbf{-}\mathbf{1}}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{用}\mathrm{い}\mathrm{る}.\\ {\mathbf{q}}_{\mathbf{1}}& =& (\cos \left(\frac{{\theta }_1}{2}\right),\sin \left(\frac{{\theta }_1}{2}\right){\mathbf{V}}_{\mathbf{1}})\\ & & \cdots \left|q_1\right|=1,\left|{\mathbf{V}}_{\mathbf{1}}\right|=1,\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}p\mathrm{を}{p}_1\mathrm{に}\mathrm{向}\mathrm{け}\mathrm{て}\mathrm{回}\mathrm{転}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{際}\mathrm{の}\mathrm{軸}\\ & =& (C_1,S_1{\mathbf{V}}_{\mathbf{1}})\\ {\mathbf{q}}_{\mathbf{2}}& =& (\cos \left(\frac{{\theta }_2}{2}\right),\sin \left(\frac{{\theta }_2}{2}\right){\mathbf{V}}_{\mathbf{2}})\\ & & \cdots \left|q_2\right|=1,\left|{\mathbf{V}}_{\mathbf{2}}\right|=1\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}p\mathrm{を}{p}_2\mathrm{に}\mathrm{向}\mathrm{け}\mathrm{て}\mathrm{回}\mathrm{転}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{際}\mathrm{の}\mathrm{軸}\\ & =& (C_2,S_2{\mathbf{V}}_{\mathbf{2}})\\ {\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}& =& (\cos \left(\frac{{\theta }_{1\to 2}}{2}\right),\sin \left(\frac{{\theta }_{1\to 2}}{2}\right){\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})\\ & & \cdots \left|q_{1\to 2}\right|=1,\left|{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\right|=1,\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}p\mathrm{を}{p}_1\mathrm{か}\mathrm{ら}{p}_2\mathrm{に}\mathrm{向}\mathrm{け}\mathrm{て}\mathrm{回}\mathrm{転}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{際}\mathrm{の}\mathrm{軸}\\ & =& (C_{1\to 2},S_{1\to 2}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})\\ {\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}& =& (\cos \left(t\frac{{\theta }_{1\to 2}}{2}\right),\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right){\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})\\ & & \cdots 0\le t\le 1,\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}p\mathrm{を}{p}_1\mathrm{か}\mathrm{ら}{p}_2\mathrm{に}\mathrm{向}\mathrm{け}\mathrm{て}\mathrm{回}\mathrm{転}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{途}\mathrm{中}\mathrm{の}{p}_t\mathrm{ま}\mathrm{で}\mathrm{の}\mathrm{際}\mathrm{の}\mathrm{軸}\mathrm{な}\mathrm{の}\mathrm{で}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\mathrm{で}\mathrm{変}\mathrm{わ}\mathrm{ら}\mathrm{な}\mathrm{い}.\mathrm{回}\mathrm{転}\mathrm{量}\mathrm{だ}\mathrm{け}t\mathrm{で}\mathrm{調}\mathrm{整}\mathrm{さ}\mathrm{れ}\mathrm{る}.\\ & =& (C_{1\to t},S_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})\\ {\mathbf{q}}_{\mathbf{t}}& =& (\cos \left(\frac{{\theta }_t}{2}\right),\sin \left(\frac{{\theta }_t}{2}\right){\mathbf{V}}_{\mathbf{t}})\\ & & \cdots \left|q_t\right|=1,\left|{\mathbf{V}}_{\mathbf{t}}\right|=1,\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}p\mathrm{を}{p}_t\mathrm{に}\mathrm{向}\mathrm{け}\mathrm{て}\mathrm{回}\mathrm{転}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{際}\mathrm{の}\mathrm{軸}\\ & =& (C_t,S_t{\mathbf{V}}_{\mathbf{t}})\end{array}$$


$\mathbf{p}$，${\mathbf{p}}_{\mathbf{1}}$，${\mathbf{p}}_{\mathbf{2}}$，${\mathbf{q}}_{\mathbf{1}}$，${\mathbf{q}}_{\mathbf{2}}$の関係性から${\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}=(C_{1\to 2},S_{1\to 2}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})$を求める． $$\begin{array}{rcl}{\mathbf{p}}_{\mathbf{1}}& =& {\mathbf{q}}_{\mathbf{1}}\mathbf{p}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\\ \\ {\mathbf{p}}_{\mathbf{2}}& =& {\mathbf{q}}_{\mathbf{2}}\mathbf{p}{\mathbf{q}}_{\mathbf{2}}^{\mathbf{-}\mathbf{1}}\\ \\ {\mathbf{p}}_{\mathbf{2}}& =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}{\mathbf{p}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}^{\mathbf{-}\mathbf{1}}\\ & =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}\left({\mathbf{q}}_{\mathbf{1}}\mathbf{p}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\right){\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}^{\mathbf{-}\mathbf{1}}\cdots {\mathbf{p}}_{\mathbf{1}}={\mathbf{q}}_{\mathbf{1}}\mathbf{p}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\\ & =& \left({\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}{\mathbf{q}}_{\mathbf{1}}\right)\mathbf{p}\left({\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}{\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}^{\mathbf{-}\mathbf{1}}\right)\\ \\ {\mathbf{q}}_{\mathbf{2}}& =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}{\mathbf{q}}_{\mathbf{1}}\\ \\ {\mathbf{q}}_{\mathbf{1}\to \mathbf{2}}& =& {\mathbf{q}}_{\mathbf{2}}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\\ & =& {\mathbf{q}}_{\mathbf{2}}\frac{\overline{{\mathbf{q}}_{\mathbf{1}}}}{|{\mathbf{q}}_{\mathbf{1}}{|}^2}\cdots {\mathbf{q}}^{\mathbf{-}\mathbf{1}}=\frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}\\ & =& {\mathbf{q}}_{\mathbf{2}}\overline{{\mathbf{q}}_{\mathbf{1}}}\cdots |{\mathbf{q}}_{\mathbf{1}}{|}^2=1^2=1\\ & =& (C_2,S_2{\mathbf{V}}_{\mathbf{2}})(C_1,-S_1{\mathbf{V}}_{\mathbf{1}})\\ & =& (C_1{C}_2-\{-S_1\}{S}_2{\mathbf{V}}_{\mathbf{2}}\cdot {\mathbf{V}}_{\mathbf{1}},-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+\{-S_1\}{S}_2{\mathbf{V}}_{\mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}})\\ & & \cdots {\mathbf{q}}_1{\mathbf{q}}_2=(w_1,{\mathbf{V}}_{\mathbf{1}})(w_2,{\mathbf{V}}_{\mathbf{2}})=(w_1{w}_2-{\mathbf{V}}_1\cdot {\mathbf{V}}_2,w_1{\mathbf{V}}_2+w_2{\mathbf{V}}_1+{\mathbf{V}}_1\times {\mathbf{V}}_2)\\ & =& (C_1{C}_2+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}},-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}})\cdots \mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A},\mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}\\ \\ C_{1\to 2}& =& C_1{C}_2+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}\\ S_{1\to 2}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}& =& -C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\end{array}$$


$\mathbf{p}$，${\mathbf{p}}_{\mathbf{1}}$，${\mathbf{p}}_{\mathbf{t}}$，${\mathbf{q}}_{\mathbf{1}}$，${\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}$の関係性から${\mathbf{q}}_{\mathbf{t}}=(C_t,S_t{\mathbf{V}}_{\mathbf{t}})$を求める． $$\begin{array}{rcl}{\mathbf{p}}_{\mathbf{t}}& =& {\mathbf{q}}_{\mathbf{t}}\mathbf{p}{\mathbf{q}}_{\mathbf{t}}^{\mathbf{-}\mathbf{1}}\\ \\ {\mathbf{p}}_{\mathbf{t}}& =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}{\mathbf{p}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}^{\mathbf{-}\mathbf{1}}\\ & =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}\left({\mathbf{q}}_{\mathbf{1}}\mathbf{p}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\right){\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}^{\mathbf{-}\mathbf{1}}\cdots {\mathbf{p}}_{\mathbf{1}}={\mathbf{q}}_{\mathbf{1}}\mathbf{p}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}\\ & =& \left({\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}{\mathbf{q}}_{\mathbf{1}}\right)\mathbf{p}\left({\mathbf{q}}_{\mathbf{1}}^{\mathbf{-}\mathbf{1}}{\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}^{\mathbf{-}\mathbf{1}}\right)\\ \\ {\mathbf{q}}_{\mathbf{t}}& =& {\mathbf{q}}_{\mathbf{1}\to \mathbf{t}}{\mathbf{q}}_{\mathbf{1}}\\ & =& (C_{1\to t},S_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}})(C_1,S_1{\mathbf{V}}_{\mathbf{1}})\\ & =& (C_1{C}_{1\to t}-S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\cdot {\mathbf{V}}_{\mathbf{1}},C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}+S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}})\\ & =& (C_1{C}_{1\to t}-S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}},C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}+S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}})\cdots \mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A}\\ \\ C_t& =& C_1{C}_{1\to t}-S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\\ S_t{\mathbf{V}}_{\mathbf{t}}& =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}+S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}}\end{array}$$


上記で求めた$S_{1\to 2}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}$を用いて，${\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}$を求めておく． $$\begin{array}{rcl}{S}_{1\to 2}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}& =& -C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\\ \\ {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}& =& \frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\}\\ \\ {\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}& =& {\mathbf{V}}_{\mathbf{1}}\cdot \left[\frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2({\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}})\}\right]\\ & =& \frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot ({\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}})\}\\ & =& \frac{1}{S_{1\to 2}}\{-C_2{S}_1+C_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{2}}\cdot ({\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{1}})\}\cdots {\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}}={\left|V_1\right|}^2=1^2=1,\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C})=\mathbf{C}\cdot (\mathbf{A}\times \mathbf{B})(\mathrm{ス}\mathrm{カ}\mathrm{ラ}\mathrm{ー}\mathrm{三}\mathrm{重}\mathrm{積})\\ & =& \frac{1}{S_{1\to 2}}\{-C_2{S}_1+C_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}\}\cdots \mathbf{A}\times \mathbf{A}=0\end{array}$$


上記で求めた${\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}$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}{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\\ & =& C_1{C}_{1\to t}-S_1{S}_{1\to t}\left[\frac{1}{S_{1\to 2}}\{-C_2{S}_1+C_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}\}\right]\\ & & \cdots {\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}=\frac{1}{S_{1\to 2}}\{-C_2{S}_1+C_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{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{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{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{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}}\}\cdots {\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right)\\ & =& 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{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{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{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{2}})\}\\ & =& C_1{C}_{1\to t}-\frac{S_{1\to t}}{S_{1\to 2}}\{-C_2+C_1{C}_{1\to 2}\}\cdots C_{1\to 2}=C_1{C}_2+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{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 \left(\frac{{\theta }_{1\to 2}}{2}\right)\cos \left(t\frac{{\theta }_{1\to 2}}{2}\right)-\cos \left(\frac{{\theta }_{1\to 2}}{2}\right)\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{S_{1\to 2}}{C}_1+\frac{S_{1\to t}}{S_{1\to 2}}{C}_2\\ & =& \frac{\sin (\frac{{\theta }_{1\to 2}}{2}-t\frac{{\theta }_{1\to 2}}{2})}{S_{1\to 2}}{C}_1+\frac{S_{1\to t}}{S_{1\to 2}}{C}_2\cdots \sin \left(\alpha \right)\cos \left(\beta \right)-\cos \left(\alpha \right)\sin \left(\beta \right)=\sin (\alpha -\beta )\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{S_{1\to 2}}{C}_1+\frac{S_{1\to t}}{S_{1\to 2}}{C}_2\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_1+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_2\end{array}$$


上記で求めた${\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}$を用いて，$S_t{\mathbf{V}}_{\mathbf{t}}$を変形する． $$\begin{array}{rcl}{S}_t{\mathbf{V}}_{\mathbf{t}}& =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}+S_1{S}_{1\to t}{\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}}\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_{1\to t}\left[\frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\}\right]+S_1{S}_{1\to t}\left[\frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\}\right]\times {\mathbf{V}}_{\mathbf{1}}\\ & & \cdots {\mathbf{V}}_{\mathbf{1}\to \mathbf{2}}=\frac{1}{S_{1\to 2}}\{-C_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\}\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}\{-C_1{C}_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}+C_1{S}_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\}+\frac{S_{1\to t}}{S_{1\to 2}}\{-C_2{S}_1^2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{1}}+C_1{S}_1{S}_2{\mathbf{V}}_{\mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}}+S_1^2{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}}\}\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}\{-C_1{C}_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}+C_1{S}_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}-C_2{S}_1^2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{1}}+C_1{S}_1{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}+S_1^2{S}_2{\mathbf{V}}_{\mathbf{1}}\times {\mathbf{V}}_{\mathbf{2}}\times {\mathbf{V}}_{\mathbf{1}}\}\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}[-C_1{C}_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1^2{S}_2\{({\mathbf{V}}_{\mathbf{1}}\cdot {\mathbf{V}}_{\mathbf{1}}){\mathbf{V}}_{\mathbf{2}}-({\mathbf{V}}_{\mathbf{2}}\cdot {\mathbf{V}}_{\mathbf{1}}){\mathbf{V}}_{\mathbf{1}}\}]\cdots \mathbf{A}\times \mathbf{A}=0,\mathbf{A}\times \mathbf{B}\times \mathbf{C}=(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{B}\cdot \mathbf{C})\mathbf{A}\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}[-C_1{C}_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1^2{S}_2\{{\left|{\mathbf{V}}_{\mathbf{1}}\right|}^2{\mathbf{V}}_{\mathbf{2}}-\left|{\mathbf{V}}_{\mathbf{2}}\right|\left|{\mathbf{V}}_{\mathbf{1}}\right|\cos \left({\theta }_{1\to 2}\right){\mathbf{V}}_{\mathbf{1}}\}]\cdots \mathbf{A}\cdot \mathbf{A}={\left|\mathbf{A}\right|}^2,\mathbf{A}\cdot \mathbf{B}=\left|\mathbf{A}\right|\left|\mathbf{B}\right|\cos \left(\theta \right)\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}\{-C_1{C}_2{S}_1{\mathbf{V}}_{\mathbf{1}}+C_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}+S_1^2{S}_2{\mathbf{V}}_{\mathbf{2}}-S_1^2{S}_2\cos \left({\theta }_{1\to 2}\right){\mathbf{V}}_{\mathbf{1}}\}\cdots \left|{\mathbf{V}}_{\mathbf{1}}\right|=1,\left|{\mathbf{V}}_{\mathbf{2}}\right|=1\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}[\{-C_1{C}_2{S}_1-S_1^2{S}_2\cos \left({\theta }_{1\to 2}\right)\}{\mathbf{V}}_{\mathbf{1}}+\{C_1^2{S}_2+S_1^2{S}_2\}{\mathbf{V}}_{\mathbf{2}}]\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}[-S_1\{C_1{C}_2+S_1{S}_2\cos \left({\theta }_{1\to 2}\right)\}{\mathbf{V}}_{\mathbf{1}}+S_2\{C_1^2+S_1^2\}{\mathbf{V}}_{\mathbf{2}}]\\ & =& C_{1\to t}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}\{-S_1{C}_{1\to 2}{\mathbf{V}}_{\mathbf{1}}+S_2{\mathbf{V}}_{\mathbf{2}}\}\cdots C_{1\to 2}=C_1{C}_2+S_1{S}_2\cos \left({\theta }_{1\to 2}\right),{\cos }^2\left(\theta \right)+{\sin }^2\left(\theta \right)=1\\ & =& \frac{S_{1\to 2}{C}_{1\to t}-C_{1\to 2}{S}_{1\to t}}{S_{1\to 2}}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}{S}_2{\mathbf{V}}_{\mathbf{2}}\\ & =& \frac{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)\cos \left(t\frac{{\theta }_{1\to 2}}{2}\right)-\cos \left(\frac{{\theta }_{1\to 2}}{2}\right)\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{S_{1\to 2}}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}{S}_2{\mathbf{V}}_{\mathbf{2}}\\ & =& \frac{\sin (\frac{{\theta }_{1\to 2}}{2}-t\frac{{\theta }_{1\to 2}}{2})}{S_{1\to 2}}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}{S}_2{\mathbf{V}}_{\mathbf{2}}\cdots \sin \left(\alpha \right)\cos \left(\beta \right)-\cos \left(\alpha \right)\sin \left(\beta \right)=\sin (\alpha -\beta )\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{S_{1\to 2}}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{S_{1\to t}}{S_{1\to 2}}{S}_2{\mathbf{V}}_{\mathbf{2}}\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_2{\mathbf{V}}_{\mathbf{2}}\end{array}$$


以上により${\mathbf{p}}_{\mathbf{t}}$へ$\mathbf{p}$を回転させる四元数${\mathbf{q}}_{\mathbf{t}}$が${\mathbf{q}}_{\mathbf{1}}$，${\mathbf{q}}_{\mathbf{2}}$(及び${\theta }_{1\to 2}$と$t$)によって表すことができた． $$\begin{array}{rcl}{\mathbf{q}}_{\mathbf{t}}& =& (C_t,S_t{\mathbf{V}}_{\mathbf{t}})\\ & =& (\frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_1+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_2,\frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_2{\mathbf{V}}_{\mathbf{2}})\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_1+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_2+\frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_2{\mathbf{V}}_{\mathbf{2}}\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_1+\frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_1{\mathbf{V}}_{\mathbf{1}}+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{C}_2+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{S}_2{\mathbf{V}}_{\mathbf{2}}\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}(C_1,S_1{\mathbf{V}}_{\mathbf{1}})+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}(C_2,S_2{\mathbf{V}}_{\mathbf{2}})\\ & =& \frac{\sin \left((1-t)\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{\mathbf{q}}_{\mathbf{1}}+\frac{\sin \left(t\frac{{\theta }_{1\to 2}}{2}\right)}{\sin \left(\frac{{\theta }_{1\to 2}}{2}\right)}{\mathbf{q}}_{\mathbf{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．