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椭圆体的表面积公式

2011-06-11 22:12:24| 分类：几何学 | 标签： |举报 |字号大中小订阅

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一、椭圆体的定义：

椭圆体(又称"旋转椭球体")即椭圆绕其长轴或短轴旋转一周所形成的几何体。绕长轴之，为长轴椭圆体(又称"长球体")；绕短轴之，为短轴椭圆体(又称"扁球体")。

二、长轴椭圆体的表面积公式：

S=2πb²+2πab(arcsine)/e

其中，a为长半轴，b为短半轴，arcsin为反正弦，e=c/a，c=√(a²-b²)

三、短轴椭圆体的表面积公式：

S=2πa²+2πab(arcshk)/k

其中，arcsh为反双曲正弦，k=c/b.

四、椭球体的表面积公式：（三个半轴a≥b≥c＞0）

S=2π[c²+(bc²/A)F(φ,k)+(bA)E(φ,k)]

其中

（1）A=√(a²-c²)、 B=√(b²-c²)

（2）φ=arcsin(A/a)、k=(aB)/(bA)

（3）F(φ,k)、E(φ,k)为椭圆积分。

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