サイクロイド

定義および性質

要約

視点

定直線に沿って円が滑らずに回転するときの円周上の定点の軌跡をサイクロイドという（→生成アニメーション）。サイクロイドはトロコイドの一種と見なすことができる。半アーチ分の伸開線は、自身と合同なサイクロイドとなる。逆に言うと、サイクロイドの縮閉線は、自身と合同なサイクロイドとなる。

{\begin{cases}x=r_{\mathrm {m} }(\theta -\sin \theta ),\\y=r_{\mathrm {m} }(1-\cos \theta ).\end{cases}}

${\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\mathrm {d} y/\mathrm {d} \theta }{\mathrm {d} x/\mathrm {d} \theta }}={\frac {\sin \theta }{1-\cos \theta }}$
${\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\mathrm {d} \theta }{\mathrm {d} x}}{\frac {\mathrm {d} }{\mathrm {d} \theta }}{\frac {\mathrm {d} y}{\mathrm {d} x}}=-{\frac {1}{r_{\mathrm {m} }(1-\cos \theta )^{2}}}$
媒介変数 $θ$ の地点における曲率半径は

4r_{\mathrm {m} }\left|\sin {\frac {\theta }{2}}\right|.

{\begin{aligned}l&=\int _{0}^{2\pi }{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} \theta }}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} \theta }}\right)^{2}}}{\mathrm {d} \theta }\\&=\int _{0}^{2\pi }r_{\mathrm {m} }{\sqrt {2-2\cos \theta }}\,{\mathrm {d} \theta }\\&=2r_{\mathrm {m} }\int _{0}^{2\pi }\sin {\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=8r_{\mathrm {m} }.\end{aligned}}

（= "当該円の半径" の 8倍）

{\begin{aligned}S&=\int _{0}^{2\pi r_{\mathrm {m} }}y\,{\mathrm {d} x}=\int _{0}^{2\pi }r_{\mathrm {m} }^{2}(1-\cos \theta )^{2}{\mathrm {d} \theta }\\&=4r_{\mathrm {m} }^{2}\int _{0}^{2\pi }\sin ^{4}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=3\pi r_{\mathrm {m} }^{2}.\end{aligned}}

（= "当該円の面積" の 3倍）

{\begin{aligned}V_{x}&=\pi \int _{0}^{2\pi r_{\mathrm {m} }}y^{2}\,{\mathrm {d} x}=\pi \int _{0}^{2\pi }r_{\mathrm {m} }^{3}(1-\cos \theta )^{3}{\mathrm {d} \theta }\\&=8\pi r_{\mathrm {m} }^{3}\int _{0}^{2\pi }\sin ^{6}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=5\pi ^{2}r_{\mathrm {m} }^{3}.\end{aligned}}

{\begin{aligned}S_{x}&=2\pi \int _{0}^{2\pi }r_{\mathrm {m} }(1-\cos \theta ){\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} \theta }}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} \theta }}\right)^{2}}}{\mathrm {d} \theta }\\&=8\pi r_{\mathrm {m} }^{2}\int _{0}^{2\pi }\sin ^{3}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&={\frac {64}{3}}\pi r_{\mathrm {m} }^{2}.\end{aligned}}

\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}={\frac {2r_{\mathrm {m} }}{y}}-1,\quad {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}=-{\frac {r_{\mathrm {m} }}{y^{2}}}.

参考文献

Apostol, Tom M.、Mnatsakanian, Mamikon A. 著、川辺治之訳『Aha! ひらめきの幾何学―アルキメデスも驚くマミコンの定理―』共立出版、2016年8月。ISBN 978-4-320-11138-7。

定義および性質

要約

視点

{\begin{cases}x=r_{\mathrm {m} }(\theta -\sin \theta ),\\y=r_{\mathrm {m} }(1-\cos \theta ).\end{cases}}

${\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\mathrm {d} y/\mathrm {d} \theta }{\mathrm {d} x/\mathrm {d} \theta }}={\frac {\sin \theta }{1-\cos \theta }}$
${\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\mathrm {d} \theta }{\mathrm {d} x}}{\frac {\mathrm {d} }{\mathrm {d} \theta }}{\frac {\mathrm {d} y}{\mathrm {d} x}}=-{\frac {1}{r_{\mathrm {m} }(1-\cos \theta )^{2}}}$
媒介変数 $θ$ の地点における曲率半径は

4r_{\mathrm {m} }\left|\sin {\frac {\theta }{2}}\right|.

{\begin{aligned}l&=\int _{0}^{2\pi }{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} \theta }}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} \theta }}\right)^{2}}}{\mathrm {d} \theta }\\&=\int _{0}^{2\pi }r_{\mathrm {m} }{\sqrt {2-2\cos \theta }}\,{\mathrm {d} \theta }\\&=2r_{\mathrm {m} }\int _{0}^{2\pi }\sin {\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=8r_{\mathrm {m} }.\end{aligned}}

（= "当該円の半径" の 8倍）

{\begin{aligned}S&=\int _{0}^{2\pi r_{\mathrm {m} }}y\,{\mathrm {d} x}=\int _{0}^{2\pi }r_{\mathrm {m} }^{2}(1-\cos \theta )^{2}{\mathrm {d} \theta }\\&=4r_{\mathrm {m} }^{2}\int _{0}^{2\pi }\sin ^{4}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=3\pi r_{\mathrm {m} }^{2}.\end{aligned}}

（= "当該円の面積" の 3倍）

{\begin{aligned}V_{x}&=\pi \int _{0}^{2\pi r_{\mathrm {m} }}y^{2}\,{\mathrm {d} x}=\pi \int _{0}^{2\pi }r_{\mathrm {m} }^{3}(1-\cos \theta )^{3}{\mathrm {d} \theta }\\&=8\pi r_{\mathrm {m} }^{3}\int _{0}^{2\pi }\sin ^{6}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&=5\pi ^{2}r_{\mathrm {m} }^{3}.\end{aligned}}

{\begin{aligned}S_{x}&=2\pi \int _{0}^{2\pi }r_{\mathrm {m} }(1-\cos \theta ){\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} \theta }}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} \theta }}\right)^{2}}}{\mathrm {d} \theta }\\&=8\pi r_{\mathrm {m} }^{2}\int _{0}^{2\pi }\sin ^{3}{\frac {\theta }{2}}\,{\mathrm {d} \theta }\\&={\frac {64}{3}}\pi r_{\mathrm {m} }^{2}.\end{aligned}}

\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}={\frac {2r_{\mathrm {m} }}{y}}-1,\quad {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}=-{\frac {r_{\mathrm {m} }}{y^{2}}}.

参考文献

Apostol, Tom M.、Mnatsakanian, Mamikon A. 著、川辺治之訳『Aha! ひらめきの幾何学―アルキメデスも驚くマミコンの定理―』共立出版、2016年8月。ISBN 978-4-320-11138-7。

定義および性質