旋轉曲面是一條平面曲線C繞它所在平面的一條直線L旋轉一周所生產的曲面,其中曲線C稱之為該旋轉曲面的母線,直線L稱為該旋轉曲面的旋轉軸。
面積
如果曲線由參數方程、給出,其中,且旋轉軸是軸,則旋轉曲面的面積由以下的積分給出:
條件是非負。這個公式與古爾丁定理是等價的。
來自勾股定理,表示曲線的一小段弧,像弧長的公式那樣。是這一小段的(重心的)路徑。
如果曲線的方程是y = f(x),a ≤ x ≤ b,則積分變為:
- (繞着x軸旋轉),
- (繞着y軸旋轉)。
這可以由以上的公式推出。
例如,單位半徑的球面由曲線x(t) = sin(t),y(t) = cos(t)旋轉而得,其中。所以,它的面積為:
對於半徑為r的圓繞着x軸旋轉所得的曲面,
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參見
參考文獻
- Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 931-937, 1985.
- Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 42, 1980.
- Gray, A. "Surfaces of Revolution." Ch. 20 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 457-480, 1997.
- Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7-11, 1999.
- Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 79-80 and Appendix III, 1992.
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