“Oblate Spheroid”的概念、定义、习题解题方法及翻译-数学辞典

单词

Oblate Spheroid

释义

Oblate Spheroid

A ``squashed'' Spheroid for which the equatorial radius is greater than the polar radius , so . Tofirst approximation, the shape assumed by a rotating fluid (including the Earth, which is ``fluid'' overastronomical time scales) is an oblate spheroid. The oblate spheroid can be specified parametrically by the usualSpheroid equations (for a Spheroid with z-Axis as the symmetry axis),

			(1)
			(2)
			(3)

with

, and

. Its Cartesian equation is

(4)

The Ellipticity of an oblate spheroid is defined by

(5)

so that

(6)

Then the radial distance from the rotation axis is given by

(7)

as a function of the Latitude

The Surface Area and Volume of an oblate spheroid are

			(8)
			(9)

An oblate spheroid with its origin at a Focus has equation

(10)

Define and expand up to Powers of ,


			(11)
			(12)
			(13)

Expanding

in Powers of Ellipticity to

therefore yields

(14)

In terms of Legendre Polynomials,



			(15)

The Ellipticity may also be expressed in terms of the Oblateness (also called Flattening), denoted

(16)

(17)

(18)

(19)

(20)

and

(21)

(22)

Define

and expand up to Powers of



			(23)
			(24)
			(25)

Expanding

in Powers of the Oblateness to

yields

(26)

In terms of Legendre Polynomials,


			(27)

To find the projection of an oblate spheroid onto a Plane, set up a coordinate system such that the z-Axis istowards the observer, and the -axis is in the Plane of the page. The equation for an oblate spheroid is