Spheroid

A spheroid is a Quadric Surface in three dimensions obtained by rotating an Ellipse about one of its principal axes. If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball or Cigar). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).

A spheroid can also be characterised as an Ellipsoid having two equal semi-axes, as represented by the equation

x 2 ––– a 2

+

y 2 ––– a 2

+

z 2 ––– b 2

= 1

A prolate spheroid has one semiaxis longer than the other two, (b > a); an oblate spheroid has one semiaxis shorter than the other two (b < a) and can resemble a Disk.

Prolate spheroid.

Oblate spheroid.

The Sphere is a special case of the spheroid in which the generating ellipse is a circle.

Volume

• volume is

  4 –– 3

  π a b 2

• volume is

  4 –– 3

  π a 2 b

where

• a is the semi-major axis length
• b is the semi-minor axis length

Surface area

π

(

2 a 2 +

b 2 ––– e

ln

(

1+e ––––– 1-e

)

).

A prolate spheroid has surface area

2 π b

(

b + a

arcsin{e } –––––––––––––– e

)

.

Here e is the eccentricity of the ellipse, defined as

√

–––––––––––––––––– 1-(b 2 )/(a 2 )

.