1 概要
複素数における回転の表現と補間について簡単に整理をする．
2 回転の表現
定義は
\[ \mathbb{C}_1 := \{p \in \mathbb{C} \mid |p|=1 \}=\{e^{\theta\mathbb{i}} \mid \theta \in \mathbb{R} \} \]
\(e^{\theta\mathbb{i}}\) はオイラーの公式1で
\[ e^{\theta\mathbb{i}} =\cos{(\theta)} + \sin{(\theta)} \mathbb{i} \]
3 補間の方法
\(p=a+b\mathbb{i}, \hspace{3pt} q=c+d\mathbb{i} \in \mathbb{C}_1\) 間を \(h \in \mathbb{R}\) で補間する方法．
3.1 線形補間
定義は
\[ \operatorname{LERP}(p,q;h) =(1-h)p + hq \]
特徴は
- 大きさを維持しない
- 速度は不定
3.2 正規化線形補間
定義は
\[ \operatorname{nLERP}(p,q;h) =\frac{\operatorname{LERP}(p,q;h)}{|\operatorname{LERP}(p,q;h)|} \]
特徴は
- 大きさを維持
- 速度は不定
3.3 球面線形補間
定義は
\[ \operatorname{SLERP}(p,q;h) =p \cdot (p^{-1} \cdot q)^h =p \cdot \exp{(h \log{(p^{-1} \cdot q)})} \]
特徴は
- 大きさを維持
- 速度は一定
3.4 スプライン補間
定義は
\[ \operatorname{SQUAD}(p_{i}, p_{i+1}, s_{i}, s{i+1}; h) =\operatorname{SLERP}(\operatorname{SLERP}(p_{i}, p_{i+1}; h), \operatorname{SLERP}(s_{i}, s_{i+1}; h), 2h(1-h)) \]
ここで \(p_{i}, p_{i+1}, s_{i}, s_{i+1} \in \mathbb{C}_1\) で \(s_{i}\) は次式で表される．
\[ s_{i} =p_{i} \exp{\left(- \frac{\log{(p_{i}^{-1} p_{i-1})} + \log{(p_{i}^{-1} p_{i+1})}}{4}\right)} \]
4 \(e^{\theta\mathbb{i}}\) の補間
一般的な回転を表現する複素数での補間は，常に最短円弧を描く補間となる．
なす角が180度を超える場合に最長円弧を描く補間にするには2
- 複素数の偏角の主値の範囲を \((-\pi, \pi]\) とし \(p, q\) の角度を \([0,\pi)\) で求める
\[ \begin{aligned} \phi= \begin{cases} Arg{(p)} + 2\pi & \text{if } Arg{(p)} < 0\\ Arg{(p)} & \text{otherwise} \end{cases}\\ \\ \psi= \begin{cases} Arg{(q)} + 2\pi & \text{if } Arg{(q)} < 0\\ Arg{(q)} & \text{otherwise} \end{cases} \end{aligned} \]
- 向きを求める
\[ \begin{aligned} d= \begin{vmatrix} p.x & q.x\\ p.y & q.y \end{vmatrix} =p.x \cdot q.y - p.y \cdot q.x \end{aligned} \]
- なす角を決定する
\[ \begin{aligned} \alpha= \begin{cases} \begin{cases} \cos^{-1}{(\langle p, q \rangle)} & \text{if } d < 0\\ 2 \pi - \cos^{-1}{(\langle p, q \rangle)} & \text{otherwise} \end{cases} & \text{if } \psi - \phi < 0\\ \begin{cases} 2 \pi - \cos^{-1}{(\langle p, q \rangle)} & \text{if } d < 0\\ \cos^{-1}{(\langle p, q \rangle)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} \end{aligned} \]
5 \(e^{\frac{\theta}{2}\mathbb{i}}\) の補間
半角形式の補間は，なす角が180度を超えると最長円弧を描く補間となる．半角形式は半円で360度を表現し残りは繰り返しとなるので \(\pm e^{\frac{\theta}{2}\mathbb{i}}\) は同じ回転を表現する．
なす角が180度を超える場合に最短円弧を描く補間にするには3
なす角が180度を超える場合に片方の符号を反転し補間を行う
符号を反転しても同じ回転を表現することを利用
\[ \begin{aligned} r= \begin{cases} -q & \text{if } \langle p, q \rangle \lt 0\\ +q & \text{otherwise} \end{cases} \end{aligned} \]
6 球面線形補間の別形式
指数・対数関数を用いない形式について．\(z_0=a+b\mathbb{i}, \hspace{3pt} z_1=c+d\mathbb{i} \in \mathbb{C}_1\) のなす角を \(\alpha\) とし \(h \in \mathbb{R}\) で補間した \(z_h \in \mathbb{C}_1\) を求める．この時 \(z_1\) を元に \(z_0\) に垂直な軸を \(z^{'}_1\) とすると
\[ z_h =\cos{(\alpha h)} z_0 + \sin{(\alpha h)} z^{'}_1 \]
\(z^{'}_1\) は二次元の場合は90度回転として求められる．
\[ z^{'}_1 =- b + a \mathbb{i} \]
より一般的には
\[ z^{'}_1 =\frac{z_1 - \langle z_0, z_1 \rangle z_0}{|z_1 - \langle z_0, z_1 \rangle z_0|} \]
または簡潔に
\[ z^{'}_1 =\frac{z_1 - \cos{(\alpha)} z_0}{\sin{(\alpha)}} \]
\(s=\langle z_0, z_1 \rangle=\cos{(\alpha)}\) とすると分母は
\[ \begin{aligned} |z_1 - \langle z_0, z_1 \rangle z_0| &=|z_1 - s z_0|\\ &=\sqrt{(c-sa)^2 + (d-sb)^2}\\ &=\sqrt{s^2 |z_0|^2 - 2s \langle z_0,z_1 \rangle + |z_1|^2}\\ &=\sqrt{s^2 - 2s^2 + 1}\\ &=\sqrt{1 - s^2}\\ &=\sqrt{1 - \cos^{2}{(\alpha)}}\\ &=\sqrt{\sin^{2}{(\alpha)}} =\sin{(\alpha)} \end{aligned} \]
\(z^{'}_1\) を \(z_h\) に代入すると
\[ \begin{aligned} z_{h} &=\cos(\alpha h) z_0 + \sin(\alpha h) z^{'}_1\\ &=\cos(\alpha h) z_0 + \sin(\alpha h) \frac{z_1 - \cos(\alpha) z_0}{\sin(\alpha)}\\ &=\cos(\alpha h)z_{0} + \frac{\sin(\alpha h)}{\sin(\alpha)}z_{1} - \frac{\sin(\alpha h)\cos(\alpha)}{\sin(\alpha)}z_{0}\\ &=\frac{\sin(\alpha)\cos(\alpha h) - \cos(\alpha)\sin(\alpha h)}{\sin(\alpha)}z_{0} + \frac{\sin(\alpha h)}{\sin(\alpha)}z_{1}\\ &=\frac{\sin(\alpha(1 - h))}{\sin(\alpha)}z_{0} + \frac{\sin(\alpha h)}{\sin(\alpha)}z_{1} \end{aligned} \]
途中，三角関数の加法定理を利用した．
このままでは分母に \(\sin{(\alpha)}\) があるため \(\alpha \to 0\) で発散するので，\(1 - |\langle z_0, z_1 \rangle| \le \epsilon\) ならば線形補間に切り替え回避をする．
7 スプライン補間の補足
未知数 \(s_{i}\) の導出方法．
\(\operatorname{SLERP}\) の微分
\[ \begin{aligned} \frac{d}{dh} \operatorname{SLERP}(p,q;h) &=\frac{d}{dh} p \cdot (\overline{p} \cdot q)^{h}\\ &=p \cdot \frac{d}{dh} \exp{(h \log{(\overline{p} \cdot q)})}\\ &=p \cdot \exp{(h \log{(\overline{p} \cdot q)})} \cdot \log{(\overline{p} \cdot q)}\\ &=\operatorname{SLERP}(p,q;h) \cdot \log{(\overline{p} \cdot q)} \end{aligned} \]
\(\operatorname{Squad}(p_{i-1}, p_{i}, s_{i-1}, s_{i}, 1) = \operatorname{Squad}(p_{i}, p_{i+1}, s_{i}, s_{i+1}, 0)\)
\[ \begin{aligned} \operatorname{Squad}(p_{i-1}, p_{i}, s_{i-1}, s_{i}; 1) &=\operatorname{SLERP}(\operatorname{SLERP}(p_{i-1}, p_{i}; 1), \operatorname{SLERP}(s_{i-1}, s_{i}; 1); 0)\\ &=\operatorname{SLERP}(p_{i}, s_{i}; 0)\\ &=p_{i}\\ \\ \operatorname{Squad}(p_{i}, p_{i+1}, s_{i}, s_{i+1}; 0) &=\operatorname{SLERP}(\operatorname{SLERP}(p_{i}, p_{i+1}; 0), \operatorname{SLERP}(s_{i}, s_{i+1}; 0), 0)\\ &=\operatorname{SLERP}(p_{i}, s_{i}; 0)\\ &=p_{i} \end{aligned} \]
\(\operatorname{SQUAD}\) の微分
\(g_{i}(h) = \overline{\operatorname{SLERP}(p_{i}, p_{i+1}; h)} \cdot \operatorname{SLERP}(s_{i}, s_{i+1}; h)\) とすると
\[ \begin{aligned} \frac{d}{dh} \operatorname{Squad}(p_{i}, p_{i+1}, s_{i}, s_{i+1}; h) &=\frac{d}{dh} \operatorname{SLERP}(\operatorname{SLERP}(p_{i}, p_{i+1}; h), \operatorname{SLERP}(s_{i}, s_{i+1}; h), 2h(1-h))\\ &=\frac{d}{dh} \operatorname{SLERP}(p_{i}, p_{i+1}; h) \cdot g_{i}(h)^{2h(1-h)}\\ &=\left(\frac{d}{dh} \operatorname{SLERP}(p_{i}, p_{i+1}; h)\right) \cdot g_{i}(h)^{2h(1-h)} - \operatorname{SLERP}(p_{i}, p_{i+1}; h) \cdot \left(\frac{d}{dh} g_{i}(h)^{2h(1-h)}\right)\\ &=\operatorname{SLERP}(p_{i}, p_{i+1}; h) \cdot \log{(\overline{p_{i}} \cdot p_{i+1})} \cdot g_{i}(h)^{2h(1-h)} - \operatorname{SLERP}(p_{i}, p_{i+1}; h) \cdot \left(\frac{d}{dh} g_{i}(h)^{2h(1-h)}\right) \end{aligned} \]
\(g_{i}(h)^{2h(1-h)}\) の微分
\(g_{i}(h)=\cos{(\theta_{g_{i}(h)})} + \sin{(\theta_{g_{i}(h)})} \mathbb{i}, \hspace{5pt} f(h)=2h(1-h)\) と置いて
\[ \begin{aligned} \frac{d}{dh} g_{i}(h)^{2h(1-h)} &=\frac{d}{dh} \exp{(f(h) \cdot \log{(g_{i}(h))})}\\ &=\frac{d}{dh} \exp{(f(h) \cdot \theta_{g_{i}(h)} \mathbb{i})}\\ &=\frac{d}{dh} \left(\cos{(f(h) \cdot \theta_{g_{i}(h)})} + \sin{(f(h) \cdot \theta_{g_{i}(h)})} \mathbb{i}\right)\\ &=-\sin{(f(h) \cdot \theta_{g_{i}(h)})} \cdot \left(\frac{d}{dh} f(h) \cdot \theta_{g_{i}(h)}\right) - \cos{(f(h) \cdot \theta_{g_{i}(h)})} \cdot \left(\frac{d}{dh} f(h) \cdot \theta_{g_{i}(h)}\right) \mathbb{i}\\ &=-\sin{(f(h) \cdot \theta_{g_{i}(h)})} \cdot \left(\left(\frac{d}{dh} f(h)\right) \cdot \theta_{g_{i}(h)} + f(h) \cdot \left(\frac{d}{dh} \theta_{g_{i}(h)}\right)\right) \\&\hspace{11pt} - \cos{(f(h) \cdot \theta_{g_{i}(h)})} \cdot \left(\left(\frac{d}{dh} f(h)\right) \cdot \theta_{g_{i}(h)} + f(h) \cdot \left(\frac{d}{dh} \theta_{g_{i}(h)}\right)\right) \mathbb{i} \end{aligned} \]
\(\frac{d}{dh} \operatorname{Squad}(p_{i-1}, p_{i}, s_{i-1}, s_{i}, 1) = \frac{d}{dh} \operatorname{Squad}(p_{i}, p_{i+1}, s_{i}, s_{i+1}, 0)\)
\[ \begin{aligned} \frac{d}{dh} \operatorname{Squad}(p_{i-1}, p_{i}, s_{i-1}, s_{i}, 1) &=\operatorname{SLERP}(p_{i-1}, p_{i}; 1) \cdot \log{(\overline{p_{i-1}} \cdot p_{i})} - \operatorname{SLERP}(p_{i-1}, p_{i}; 1) \cdot \left(\frac{d}{dh} g_{i-1}(h)^{2h(1-h)}\right)(1)\\ &=p_{i} \cdot \log{(\overline{p_{i-1}} \cdot p_{i})} - 2 p_{i} \cdot \theta_{g_{i-1}(1)} \mathbb{i}\\ &=p_{i} \cdot \left(\log{(\overline{p_{i-1}} \cdot p_{i})} - 2 \log(g_{i-1}(1))\right)\\ &=p_{i} \cdot \left(\log{(\overline{p_{i-1}} \cdot p_{i})} - 2 \log(\overline{p_{i}} \cdot s_{i})\right)\\ \\ \frac{d}{dh} \operatorname{Squad}(p_{i}, p_{i+1}, s_{i}, s_{i+1}, 0) &=\operatorname{SLERP}(p_{i}, p_{i+1}; 0) \cdot \log{(\overline{p_{i}} \cdot p_{i+1})} - \operatorname{SLERP}(p_{i}, p_{i+1}; 0) \cdot \left(\frac{d}{dh} g_{i}(h)^{2h(1-h)}\right)(0)\\ &=p_{i} \cdot \log{(\overline{p_{i}} \cdot p_{i+1})} - 2 p_{i} \cdot \theta_{g_{i}(0)} \mathbb{i}\\ &=p_{i} \cdot \left(\log{(\overline{p_{i}} \cdot p_{i+1})} - 2 \log{(g_{i}(0))}\right)\\ &=p_{i} \cdot \left(\log{(\overline{p_{i}} \cdot p_{i+1})} - 2 \log{(\overline{p_{i}} \cdot s_{i})}\right) \end{aligned} \]
上式より
\[ \begin{aligned} p_{i} \cdot \left(\log{(\overline{p_{i-1}} \cdot p_{i})} - 2 \log(\overline{p_{i}} \cdot s_{i})\right) &=p_{i} \cdot \left(\log{(\overline{p_{i}} \cdot p_{i+1})} - 2 \log{(\overline{p_{i}} \cdot s_{i})}\right)\\ \\ \log{(\overline{p_{i-1}} \cdot p_{i})} - 2 \log(\overline{p_{i}} \cdot s_{i}) &=\log{(\overline{p_{i}} \cdot p_{i+1})} - 2 \log{(\overline{p_{i}} \cdot s_{i})}\\ 4 \log{(\overline{p_{i}} \cdot s_{i})} &=\log{(\overline{p_{i-1}} \cdot p_{i})} - \log{(\overline{p_{i}} \cdot p_{i+1})}\\ \\ \overline{p_{i}} \cdot s_{i} &=\exp{\left(\frac{\log{(\overline{p_{i-1}} \cdot p_{i})} - \log{(\overline{p_{i}} \cdot p_{i+1})}}{4}\right)}\\ \\ s_{i} &=p_{i} \cdot \exp{\left(\frac{\log{(\overline{p_{i-1}} \cdot p_{i})} - \log{(\overline{p_{i}} \cdot p_{i+1})}}{4}\right)} \end{aligned} \]
\(\overline{p_1 \cdot p_2}=\overline{p_1} \cdot \overline{p_2}, \hspace{3pt} \log(\overline{p})=-\log(p), \hspace{3pt} \overline{p}=p^{-1}\) を利用し
\[ \begin{aligned} s_{i} &=p_{i} \cdot \exp{\left(\frac{\log{(\overline{p_{i-1}} \cdot p_{i})} - \log{(\overline{p_{i}} \cdot p_{i+1})}}{4}\right)}\\ &=p_{i} \cdot \exp{\left(\frac{\log{(\overline{p_{i-1} \cdot \overline{p_{i}}})} - \log{(\overline{p_{i}} \cdot p_{i+1})}}{4}\right)}\\ &=p_{i} \cdot \exp{\left(- \frac{\log{(\overline{p_{i}} \cdot p_{i-1})} - \log{(\overline{p_{i}} \cdot p_{i+1})}}{4}\right)}\\ &=p_{i} \cdot \exp{\left(- \frac{\log{(p_{i}^{-1} \cdot p_{i-1})} - \log{(p_{i}^{-1} \cdot p_{i+1})}}{4}\right)} \end{aligned} \]
四元数 \(p_{i} \in {H}_1\) なら \(\overline{p_1 \otimes p_2}=\overline{p_2} \otimes \overline{p_1}\)
四元数のスプライン補間4と同様となる．