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OblateSpheroid.PNG
ProlateSpheroid.png
oblate spheroid prolate spheroid

A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

Because of its rotation, the Earth's shape is more like an oblate spheroid than a sphere. In cartography, in fact, the Earth is often assumed to be a standard oblate spheroid. In the current World Geodetic System model, the radius is approximately 6,378.137 km at the equator and 6,356.752 km at the poles (a difference of over 21 km).

Contents

Equation

A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation

\left(\frac{x}{a}\right)^2+\left(\frac{y}{a}\right)^2+\left(\frac{z}{b}\right)^2 = 1\quad\quad\hbox{ or }\quad\quad\frac{x^2+y^2}{a^2}+\frac{z^2}{b^2}=1

where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.[1]

Surface area

A prolate spheroid has surface area

2\pi\left(a^2+\frac{a b o\!\varepsilon}{\sin(o\!\varepsilon)}\right)

where o\!\varepsilon=\arccos\left(\frac{a}{b}\right) is the angular eccentricity of the prolate spheroid, and e=\sin(o\!\varepsilon) is its (ordinary) eccentricity.

An oblate spheroid has surface area

2\pi\left[a^2+\frac{b^2}{\sin(o\!\varepsilon)} \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right] where o\!\varepsilon=\arccos\left(\frac{b}{a}\right) is the angular eccentricity of the oblate spheroid.

Volume

The volume of a spheroid (of any kind) is \frac{4}{3}\pi a^2b \approx 4.19\, a^2b. If A=2a is the equatorial diameter, and B=2b is the polar diameter, the volume is \frac{1}{6}\pi A^2B \approx 0.523\, A^2B.

Curvature

If a spheroid is parameterized as

 \vec \sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, b \sin \beta);\,\!

where \beta\,\! is the reduced or parametric latitude, \lambda\,\! is the longitude, and -\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\! and -\pi<\lambda<+\pi\,\!, then its Gaussian curvature is

 K(\beta,\lambda) = {b^2 \over (a^2 + (b^2 - a^2) \cos^2 \beta)^2};\,\!

and its mean curvature is

 H(\beta,\lambda) = {b (2 a^2 + (b^2 - a^2) \cos^2 \beta) \over 2 a (a^2 + (b^2 - a^2) \cos^2 \beta)^{3/2}}.\,\!

Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

See also

References

  1. ^ [1]

External links

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1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

SPHEROID (Gr. a4akpa - a51 7 s, like a sphere), a solid resembling, but not identical with, a sphere in shape. In geometry, the word is confined to the figures generated by an ellipse revolving about a diameter. If the axis of revolution be the major axis of the ellipse, the spheroid is "prolate"; if the minor axis, "oblate"; if any other, "universal." If the generating ellipse has for its equation x 2 /a 2 -1-y 2 /b 2 =1, and revolves about the major axis, the axis of x, the volume of the solid generated is s irab 2, and its surface is 271-{ b 2 +(ab/e) sinl e }, where e denotes the eccentricity. If the curve revolve about the minor axis, the volume is ,ira 2 b, and the surface is Tr{2a2+ (b 2 /e) log (1 +e)/(1 - e)}. The figure of the earth is frequently referred to as an oblate spheroid; this, however, is hardly correct, for the geoid has three unequal axes. The Cartesian equation to a spheroid assumes the forms x 2 /a 2 + (y 2 + z 2)/b 2 =1, for the prolate, and (x 2 +z 2)/a 2 +y 2 /b 2 =1, for the oblate, the origin being the centre and the co-ordinate axes the axes of the original ellipse, x 2 /a 2 +y 2 /b 2 =1, and the line perpendicular to the plane containing them.

In physics, the term "spheroidal state" is given to the following phenomenon. If drops of a liquid be placed on a highly heated surface, for example, the top of a stove, the liquid forms a number of tremulous globules which continually circulate internally. There is no visible boiling, although the globule diminishes slowly in size. The theory of the experiment is that the liquid is surrounded by an elastic envelope of its vapour which acts, as it were, as a cushion preventing actual contact of the drop with the plate. On the formation of a similar protective cushion of vapour depends the immunity of such experiments as plunging a hand into a bath of molten metal.


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Simple English

oblate spheroid prolate spheroid

A spheroid is a quadric surface. It is obtained by rotating an ellipse about one of its principal axes. This means it is an ellipsoid with two equal semi-diameters.

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