計量による距離
$$\begin{eqnarray}
g_{(x,y)}\left(\boldsymbol{v},\boldsymbol{w}\right)&=&
\frac{\left\langle \boldsymbol{v},\boldsymbol{w}\right\rangle}{y^2}
\;\cdots\;ポアンカレ計量,\;\left\langle \boldsymbol{x},\boldsymbol{y} \right\rangle=\sum_{i=1}^n{x_i y_i}は標準内積
\\\gamma\left(t\right)&=&\left(x(t),y(t)\right)\;\cdots\;曲線\gamma,\;tによる媒介表示,\;始点\gamma(a),\;終点\gamma(b)
\\\gamma^\prime\left(t\right)&=&\left(x^\prime(t),y^\prime(t)\right)\;\cdots\;(x(t),y(t))での接ベクトル
\\L{\left(\gamma\right)}&=&\int_a^b \sqrt{g\left(\gamma^\prime\left(t\right),\gamma^\prime\left(t\right)\right)}
\;\cdots\;\gammaの計量による距離
\\&=&\int_a^b \sqrt{\frac{\left\langle
\gamma^\prime\left(t\right),\gamma^\prime\left(t\right)\right\rangle}{y^2}}\mathrm{d}t
\\&=&\int_a^b \frac{\sqrt{\left\langle
\gamma^\prime\left(t\right),\gamma^\prime\left(t\right)\right\rangle}}{\sqrt{y^2}}\mathrm{d}t
\\&=&\int_a^b \frac{\left||\gamma^\prime\left(t\right)\right||}{y}\mathrm{d}t
\;\cdots\;\left||x\right||=\sqrt{\left\langle \boldsymbol{x},\boldsymbol{x} \right\rangle}
\end{eqnarray}$$
\((0, y_0)\)から\((0, y_1)\)までの線分\(\gamma_1\)
$$\begin{eqnarray}
\gamma_1\left(t\right)&=&\left(0,\;(y_1 - y_0)t+y_0 \right)\;\cdots\;t\in[0, 1]
\\\gamma_1^\prime\left(t\right)&=&\left(0,\;y_1 - y_0 \right)
\\L_1=L\left(\gamma_1\right)&=&\int_0^1 \frac{\left||\gamma_1^\prime\left(t\right)\right||}{y(t)}\mathrm{d}t
\\&=&\int_0^1\frac{\sqrt{0^2+(y_1 - y_0)^2}}{(y_1 - y_0)t+y_0}\mathrm{d}t
\\&=&\int_0^1\frac{y_1 - y_0}{(y_1 - y_0)t+y_0}\mathrm{d}t
\\&=&\int_{y_0}^{y_1}\frac{\cancel{y_1 - y_0}}{u}\frac{1}{\cancel{y_1-y_0}}\mathrm{d}u
\\&=&\int_{y_0}^{y_1}\frac{1}{u}\mathrm{d}u
\\&=&\left[\log{\left(u\right)}\right]_{y_0}^{y_1}
\\&=&\log{\left(y_1\right)}-\log{\left(y_0\right)}
\\&=&\log{\left(\frac{y_1}{y_0}\right)}
\end{eqnarray}$$
\((0, y_0)\)から\((0, y_1)\)までの任意の曲線\(\gamma_2\)
$$\begin{eqnarray}
\gamma_2\left(t\right)&=&\left(x(t),y(t)\right)\;\cdots\;t\in[0, 1]
\\\gamma_2^\prime\left(t\right)&=&\left(x^\prime(t),y^\prime(t)\right)
\\L_2=L\left(\gamma_2\right)&=&\int_0^1 \frac{\left||\gamma_2^\prime\left(t\right)\right||}{y(t)}\mathrm{d}t
\\&=&\int_0^1\frac{\sqrt{x^\prime(t)^2+y^\prime(t)^2}}{y(t)}\mathrm{d}t
\\&\geq&\int_0^1\frac{\sqrt{y^\prime(t)^2}}{y(t)}\mathrm{d}t\;\cdots\;x^\prime(t)^2(\geq0)を取り除くので等しいか小さくなる
\\&=&\int_0^1\frac{|y^\prime(t)|}{y(t)}\mathrm{d}t
\\&=&\left[\log{\left(y(t)\right)}\right]_{0}^{1}
\\&=&\log{\left(y(1)\right)}-\log{\left(y(0)\right)}
\\&=&\log{\left(y_1\right)}-\log{\left(y_0\right)}
\\&=&\log{\left(\frac{y_1}{y_0}\right)}=L_1
\\L_2&\geq&L_1\;\cdots\;y軸方向では線分が一番短い
\end{eqnarray}$$
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