旋轉曲面

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，由圆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 $x(t)$ 、 $y(t)$ 给出，其中 $a<t<b$ ，且旋转轴是 $y$ 轴，则旋转曲面 $A$ 的面积由以下的积分给出：

A=2\pi \int _{a}^{b}x(t)\ {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,

条件是 $x(t)$ 非负。这个公式与古尔丁定理是等价的。

\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}

来自勾股定理，表示曲线的一小段弧，像弧长的公式那样。 $2\pi x(t)$ 是这一小段的（重心的）路径。

如果曲线的方程是y = f(x)，a ≤ x ≤ b，则积分变为：

A=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx

（绕着x轴旋转），

A=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dx}{dy}}\right)^{2}}}\,dy

（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 $0<t<\pi$ 。所以，它的面积为：

A=2\pi \int _{0}^{\pi }\sin(t){\sqrt {\left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}}\,dt=2\pi \int _{0}^{\pi }\sin(t)\,dt=4\pi .

对于半径为r的圆 $y(x)={\sqrt {r^{2}-x^{2}}}$ 绕着x轴旋转所得的曲面，

A=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx

=2\pi \int _{-r}^{r}r\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx

=2\pi \int _{-r}^{r}r\,dx

=4\pi r^{2}\,

参见

旋转体

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.

查论编几何学术语
点	頂點交點中點角同界角極值點最值點臨界點驻点鞍點
直线和曲线	线段射线直线切线（主）法线副法線曲线圆锥曲线双曲线抛物线正弦曲線蚌线蜗线螺线（阿基米德螺线、等角螺线……）摆线（最速降線問題）悬链线曳物线漸開線渐屈线渐近线测地线邊周界弦弧矢垂直平分線代數曲線椭圆曲线超橢圓星形线三尖瓣线方圓形勒洛三角形
平面圖形	圆（广义圆）椭圆扇形弓形环形多边形三角形四邊形五边形六边形多边形正多边形梯形平行四边形菱形矩形正方形鷂形卵形线梭形星形五角星六角星
立體圖形	多面体正多面體四面體長方體立方體平行六面体棱柱反棱柱棱锥棱台圆柱体圆锥圆台椭球（長球體、扁球體）球體球缺球冠球台準線母線
曲面	二次曲面旋轉曲面抛物面雙曲面马鞍面球面橢球面類球面环面莫比乌斯带流形黎曼曲面
高維空間	超平面超面超曲面胞多胞形超球體超方形超立方體克莱因瓶四維柱體柱
圖形關係	相似全等對稱平行垂直相交相切相離镜像旋转反演截面缩放
三角形關係	相似三角形全等三角形
量	距离长度周长弧长高度面积表面積体积容積角度曲率撓率離心率凹凸性有向曲面可展曲面直紋曲面
作圖	尺直尺三角尺圆规尺规作图二刻尺作圖
分支	平面幾何立体几何三角学解析几何微分几何拓扑学图论摺紙數學欧几里得几何非欧几里得几何（双曲几何、球面幾何……）分形
理論	定理公理定义數學證明
分类主题共享资源专题

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