## Oblate Spheroid

A squashed'' Spheroid for which the equatorial radius is greater than the polar radius , so . To first approximation, the shape assumed by a rotating fluid (including the Earth, which is fluid'' over astronomical time scales) is an oblate spheroid. The oblate spheroid can be specified parametrically by the usual Spheroid 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 or .
 (16)

 (17)

 (18)

 (19)

so
 (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 is towards the observer, and the -axis is in the Plane of the page. The equation for an oblate spheroid is

 (28)

Define
 (29)

and . Then
 (30)

Now rotate that spheroid about the -axis by an Angle so that the new symmetry axes for the spheroid are , , and . The projected height of a point in the Plane on the -axis is
 (31)

To find the highest projected point,
 (32)

Simplifying,
 (33)

But
 (34)

Plugging (34) into (33),
 (35)

and performing a number of algebraic simplifications

 (36)

 (37)

 (38)

 (39)

 (40)

 (41)

finally gives the expression for in terms of and ,
 (42)

Combine (30) and (31) and plug in for ,
 (43)

Now re-express in terms of and , using ,
 (44)

so
 (45)

Plug (44) and (45) into (43) to obtain the Semiminor Axis of the projected oblate spheroid,
 (46)

We wish to find the equation for a spheroid which has been rotated about the -axis by Angle , then the -axis by Angle

 (47)

Now, in the original coordinates , the spheroid is given by the equation
 (48)

which becomes in the new coordinates,

 (49)
Collecting Coefficients,

 (50)

where
 (51) (52) (53) (54) (55) (56)

If we are interested in computing , the radial distance from the symmetry axis of the spheroid () corresponding to a point

 (57)

where
 (58) (59)

can now be computed using the quadratic equation when is given,
 (60)

If , then we have and , so (51) to (56) and (58) to (59) become
 (61) (62) (63) (64) (65) (66) (67) (68)

See also Darwin-de Sitter Spheroid, Ellipsoid, Oblate Spheroidal Coordinates, Prolate Spheroid, Sphere, Spheroid

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987.