捩率

この項目では、曲線の捩率について説明しています。ベクトルバンドルの接続の捩率については「捩率テンソル」をご覧ください。

捩率（れいりつ、英: torsion）または捩れ率（ねじれりつ）とは、空間内の曲線の、平面曲線からの離れ具合を表す量である。これは平面曲線の曲率の空間版であり、空間内の曲線は曲率と捩率が与えられれば、向きを保つ合同変換を除いて一意に定まる（曲線論の基本定理）。

定義

R³上の空間曲線を、弧長パラメータsで表示したものをr(s)とする。

捩率 $\tau$ (s)は

\tau (s)=-{\boldsymbol {b}}'(s)\cdot {\boldsymbol {n}}(s)

で与えられる。

ここで、b(s) は従法線ベクトル

{\boldsymbol {b}}(s)={\boldsymbol {r}}'(s)\times {\boldsymbol {n}}(s),

n(s)は主法線ベクトル

{\boldsymbol {n}}(s)={\frac {{\boldsymbol {r}}''(s)}{|{\boldsymbol {r}}''(s)|}}

である。

' はsによる微分、 $\times$ は外積を意味する。

参考資料

梅原雅顕・山田光太郎　『曲線と曲面』　裳華房、2002年、ISBN 4-7853-1531-8

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捩率

定義