ポアンカレ計量による距離ではX軸に沿った移動では線分が最短にならない

計量による距離

$$\begin{array}{rcl}{g}_{(x,y)}(\mathit{v},\mathit{w})& =& \frac{⟨\mathit{v},\mathit{w}⟩}{y^2}\cdots \mathrm{ポ}\mathrm{ア}\mathrm{ン}\mathrm{カ}\mathrm{レ}\mathrm{計}\mathrm{量},⟨\mathit{x},\mathit{y}⟩=\sum _{i=1}^n{x}_i{y}_i\mathrm{は}\mathrm{標}\mathrm{準}\mathrm{内}\mathrm{積}\\ \gamma \left(t\right)& =& (x(t),y(t))\cdots \mathrm{曲}\mathrm{線}\gamma ,t\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{媒}\mathrm{介}\mathrm{表}\mathrm{示},\mathrm{始}\mathrm{点}\gamma (a),\mathrm{終}\mathrm{点}\gamma (b)\\ {\gamma }^{\prime }\left(t\right)& =& (x^{\prime }(t),y^{\prime }(t))\cdots (x(t),y(t))\mathrm{で}\mathrm{の}\mathrm{接}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\\ L\left(\gamma \right)& =& {\int }_a^b\sqrt{g({\gamma }^{\prime }\left(t\right),{\gamma }^{\prime }\left(t\right))}\cdots \gamma \mathrm{の}\mathrm{計}\mathrm{量}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{距}\mathrm{離}\\ & =& {\int }_a^b\sqrt{\frac{⟨{\gamma }^{\prime }\left(t\right),{\gamma }^{\prime }\left(t\right)⟩}{y^2}}\mathrm{d}t\\ & =& {\int }_a^b\frac{\sqrt{⟨{\gamma }^{\prime }\left(t\right),{\gamma }^{\prime }\left(t\right)⟩}}{\sqrt{y^2}}\mathrm{d}t\\ & =& {\int }_a^b\frac{||{\gamma }^{\prime }\left(t\right)||}{y}\mathrm{d}t\cdots ||x||=\sqrt{⟨\mathit{x},\mathit{x}⟩}\end{array}$$

$(\cos \left(\theta \right),\sin \left(\theta \right))$から $(-\cos \left(\theta \right),\sin \left(\theta \right))$までの線分で移動(ただし$\theta \in (0,\frac{\pi }{2})$)

$$\begin{array}{rcl}{\gamma }_3(t)& =& ((x_1-x_0)t+x_0,(y_1-y_0)t+y_0)\\ & =& ((-\cos \left(\theta \right)-\cos \left(\theta \right))t+\cos \left(\theta \right),(\sin \left(\theta \right)-\sin \left(\theta \right))t+\sin \left(\theta \right))\\ & =& ((1-2t)\cos \left(\theta \right),\sin \left(\theta \right))\\ {\gamma }_3^{\prime }(t)& =& (-2\cos \left(\theta \right),0)\\ L_3=L\left({\gamma }_3\right)& =& {\int }_0^1\frac{||{\gamma }_3^{\prime }\left(t\right)||}{y(t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\sqrt{{(-2\cos \left(\theta \right))}^2+0^2}}{\sin \left(\theta \right)}\mathrm{d}t\\ & =& {\int }_0^1\frac{2\cos \left(\theta \right)}{\sin \left(\theta \right)}\mathrm{d}t\\ & =& {\int }_0^1\frac{2}{\tan \left(\theta \right)}\mathrm{d}t\\ & =& \frac{2}{\tan \left(\theta \right)}{\int }_0^1\mathrm{d}t\\ & =& \frac{2}{\tan \left(\theta \right)}[t{]}_0^1\\ & =& \frac{2}{\tan \left(\theta \right)}[1-0]\\ & =& \frac{2}{\tan \left(\theta \right)}\cdot 1\\ & =& \frac{2}{\tan \left(\theta \right)}\end{array}$$

$(\cos \left(\theta \right),\sin \left(\theta \right))$から $(-\cos \left(\theta \right),\sin \left(\theta \right))$まで，原点を中心とした半径1の円弧で移動(ただし $\theta \in (0,\frac{\pi }{2})$)

$$\begin{array}{rcl}{\gamma }_4(t)& =& (\cos (\theta +(\pi -2\theta )t),\sin (\theta +(\pi -2\theta )t))\\ {\gamma }_4^{\prime }(t)& =& (-(\pi -2\theta )\sin (\theta +(\pi -2\theta )t),(\pi -2\theta )\cos (\theta +(\pi -2\theta )t))\\ L_4=L\left({\gamma }_4\right)& =& {\int }_0^1\frac{||{\gamma }_4^{\prime }\left(t\right)||}{y(t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\sqrt{{(-(\pi -2\theta )\sin (\theta +(\pi -2\theta )t))}^2+{((\pi -2\theta )\cos (\theta +(\pi -2\theta )t))}^2}}{\sin (\theta +(\pi -2\theta )t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\sqrt{(\pi -2\theta {)}^2({\sin }^2(\theta +(\pi -2\theta )t)+{\cos }^2(\theta +(\pi -2\theta )t))}}{\sin (\theta +(\pi -2\theta )t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\sqrt{(\pi -2\theta {)}^2\cdot 1}}{\sin (\theta +(\pi -2\theta )t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\pi -2\theta }{\sin (\theta +(\pi -2\theta )t)}\mathrm{d}t\\ & =& {\int }_0^1\frac{\alpha }{\sin (\beta +\alpha t)}\mathrm{d}t\\ & =& {\int }_{\beta }^{\alpha +\beta }\frac{\cancel{\alpha }}{\sin \left(u\right)}\frac{1}{\cancel{\alpha }}\mathrm{d}u\\ & =& {[\ln |\tan \left(\frac{u}{2}\right)|]}_{\beta }^{\alpha +\beta }\cdots \int \frac{1}{\sin \left(x\right)}\mathrm{d}x=\ln |\tan \left(\frac{x}{2}\right)|+C\\ & =& {[\ln |\tan \left(\frac{u}{2}\right)|]}_{\theta }^{\pi -2\theta +\theta }\\ & =& {[\ln |\tan \left(\frac{u}{2}\right)|]}_{\theta }^{\pi -\theta }\\ & =& \ln |\tan \left(\frac{\pi -\theta }{2}\right)|-\ln |\tan \left(\frac{\theta }{2}\right)|\\ & =& \ln \left|\frac{\tan \left(\frac{\pi -\theta }{2}\right)}{\tan \left(\frac{\theta }{2}\right)}\right|\\ & =& \ln \left|\frac{\cot \left(\frac{\theta }{2}\right)}{\tan \left(\frac{\theta }{2}\right)}\right|\\ & =& \ln \left|\frac{1}{{\tan }^2\left(\frac{\theta }{2}\right)}\right|\\ & =& \ln |{\tan }^{-2}\left(\frac{\theta }{2}\right)|\\ & =& -2\ln |\tan \left(\frac{\theta }{2}\right)|\end{array}$$

$L_3-L_4$

$$\begin{array}{rcl}{L}_3-L_4& =& \left\{\frac{2}{\tan \left(\theta \right)}\right\}-\{-2\ln (\tan \left(\frac{\theta }{2}\right))\}\\ & =& 2\{\frac{1}{\tan \left(\theta \right)}+\ln (\tan \left(\frac{\theta }{2}\right))\}\\ \frac{\mathrm{d}}{\mathrm{d}\theta }(L_3-L_4)& =& 2\{\frac{-1}{{\sin }^2\left(\theta \right)}+\frac{1}{2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)}\}\\ & =& 2\{\frac{-1}{{\sin }^2\left(\theta \right)}+\frac{1}{\sin \left(\theta \right)}\}\\ & =& 2\{\frac{-1}{{\sin }^2\left(\theta \right)}+\frac{1}{\sin \left(\theta \right)}\frac{\sin \left(\theta \right)}{\sin \left(\theta \right)}\}\\ & =& 2\{\frac{-1}{{\sin }^2\left(\theta \right)}+\frac{\sin \left(\theta \right)}{{\sin }^2\left(\theta \right)}\}\\ & =& 2\left\{\frac{\sin \left(\theta \right)-1}{{\sin }^2\left(\theta \right)}\right\}\le 0\cdots \mathrm{分}\mathrm{母}\mathrm{は}\mathrm{常}\mathrm{に}\mathrm{正}，\mathrm{分}\mathrm{子}\mathrm{は}\theta \in (0,\frac{\pi }{2})\mathrm{な}\mathrm{の}\mathrm{で}(-1,0)\mathrm{で}\mathrm{あ}\mathrm{り}，\mathrm{全}\mathrm{体}\mathrm{で}\mathrm{は}\mathrm{常}\mathrm{に}\mathrm{負}\mathrm{と}\mathrm{な}\mathrm{る}．\\ \underset{\theta \to \frac{\pi }{2}-0}{lim}{L}_3-L_4& =& 0\\ & & \cdots \underset{\theta \to \frac{\pi }{2}-0}{lim}\frac{1}{\tan \left(\theta \right)}=0\\ & & \cdots \underset{\theta \to \frac{\pi }{2}-0}{lim}\ln (\tan \left(\frac{\theta }{2}\right))=0\\ \mathrm{よ}\mathrm{っ}\mathrm{て}\\ L_3-L_4& \ge & 0\cdots ，\theta \in (0,\frac{\pi }{2})\mathrm{で}\mathrm{は}\mathrm{微}\mathrm{分}\mathrm{が}\mathrm{常}\mathrm{に}\mathrm{負}(\mathrm{減}\mathrm{少}\mathrm{関}\mathrm{数})\mathrm{で}\mathrm{あ}\mathrm{り}，\theta =\frac{\pi }{2}\mathrm{で}0\mathrm{な}\mathrm{の}\mathrm{で}，\mathrm{範}\mathrm{囲}\mathrm{内}\mathrm{で}\mathrm{は}\mathrm{常}\mathrm{に}\mathrm{正}．\end{array}$$

まとめ

ポアンカレ計量では水平の線分より，$x$軸から離れる側となる円弧に沿った経路の方が短いことになる．
これは境界$x$軸$y=0$に近いほど計量中にある$\frac{1}{y^2}$の効果で値が大きくなり，距離としては長くなることが影響した結果．