包絡

出自維基百科,自由嘅百科全書
跳去: 定向搵嘢
Construction of the envelope of a family of curves.

幾何上,平面上一族曲線嘅包絡係咁樣一條曲線:喺佢上面每一點處,都有呢族曲線中嘅一條同佢相切。Classically, 包絡上嘅一個點可以認為係兩條「鄰近」曲線嘅交點,即係鄰近曲線嘅交點嘅極限。呢個諗法可以推廣到空間中曲面嘅包絡,同埋更高嘅維數

一族曲線嘅包絡[編輯]

設族入面嘅每條曲線Ctft(xy)=0確定,其中t係參數。記 F(txy)=ft(xy)並且假設F可微。

其他定義[編輯]

  1. 包絡E1係鄰近曲線Ct交點嘅極限。
  2. 包絡E2係一條同所有Ct相切嘅曲線。
  3. 包絡E3係由Ct所填充嘅區域嘅邊界。

於是乎,E_1 \subseteq \mathcal{D}, E_2 \subseteq \mathcal{D}同埋E_3 \subseteq \mathcal{D},其中\mathcal{D}係由呢篇文開頭嘅定義所確定嘅曲線組成嘅集合。

[編輯]

例1[編輯]

睇埋[編輯]

參考[編輯]

  1. Bruce, J. W. & Giblin, P. J. (1984), Curves and Singularities, Cambridge University Press, ISBN 0-521-42999-4 
  2. Eisenhart, Luther P. (2008), A Treatise on the Differential Geometry of Curves and Surfaces, Schwarz Press, ISBN 1-4437-3160-9 
  3. Forsyth, Andrew Russell (1959), Theory of differential equations, Six volumes bound as three, New York: Dover Publications , §§100-106.
  4. Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9 .
  5. John, Fritz (1991), Partial differential equations (4th ed.), Springer, ISBN 978-0-387-90609-6 .
  6. Born, Max (October 1999), Principle of Optics, Cambridge University Press, ISBN 978-0-521-64222-4 , Appendix I: The calculus of variations.
  7. Arnold, V. I. (1997), Mathematical Methods of Classical Mechanics, 2nd ed., Berlin, New York: Springer-Verlag, ISBN 978-0-387-96890-2; ISBN 978-0-387-96890-2 , §46.

出面網頁[編輯]