旋轉曲面

曲线x=2+cos z的一部分绕着z轴旋转。

旋转曲面是一条平面曲线C绕它所在平面的一条直线L旋转一周所生产的曲面,其中曲线C称之为该旋转曲面的母线,直线L称为该旋转曲面的旋转轴

例子包括球面,由绕着其直径旋转而成,以及环面,由圆绕着外面的一条直线旋转而成。

面积

如果曲线由参数方程给出,其中,且旋转轴是轴,则旋转曲面的面积由以下的积分给出:

条件是非负。这个公式与古尔丁定理是等价的。

来自勾股定理,表示曲线的一小段弧,像弧长的公式那样。是这一小段的(重心的)路径。

如果曲线的方程是y = f(x),axb,则积分变为:

(绕着x轴旋转),
(绕着y轴旋转)。

这可以由以上的公式推出。

例如,单位半径的球面由曲线x(t) = sin(t),y(t) = cos(t)旋转而得,其中。所以,它的面积为:

对于半径为r的圆绕着x轴旋转所得的曲面,

参见

参考文献

  • 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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