Spheroid: Definition from Answers.com


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).

1 Equation
2 Surface area
3 Volume
4 Curvature
5 See also
6 References
7 External links

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

When the spheroid in question is oblate, the volume is $\frac{4}{3}\pi a^2b$ , where a represents the major axis of the ellipse which, when rotated about its minor axis, b, produces the oblate spheroid. When the spheroid in question is prolate, the spheroid is produced by the rotation of an ellipse about it's major axis, hence the volume formula becomes $\frac{4}{3}\pi b^2a$ , where b represents the minor axis of the ellipse which, when rotated about its major axis, produces the prolate spheroid. Hence, the volume of the oblate spheroid which results from the rotation of an ellipse about it's minor axis is always greater than the volume of the prolate spheroid which results from that same ellipse rotated about it's major axis; this is the case whenever a represents the major axis and does not equal b, which represents the minor axis. When the major axis of an ellipse, a, and the minor axis of an ellipse,b, are taken to be equal, the spheroid in question which results from the rotation of such an ellipse is a sphere, whether it is rotated about it's major axis or rotated about it's minor axis. Hence, the resulting volume equation reduces to $\frac{4}{3}\pi r^2r$ , or $\frac{4}{3}\pi r^3$ , which is the equation for the volume of a sphere.

In experimental biology, tumor growth is approximated to take the shape of a spheroid. Often, cancer studies involve the implantation of tumors subcutaneously in mice. Such studies require a simple mechanism by which to evaluate tumor burden. One such method is for two blinded researchers to measure tumor dimensions length and width with calipers. The depth is not measured. Tumor volume in cubic millimeters can be approximated with the following formula: $V o l u m e = 0.52(W i d t h 2) L e n g t h$ ^[2]

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.

References

^ [1]
^ W. Su & Q. Wang: Inhibition of Human Prostate Cancer Growth and Prevention of Metastasis Development by Antiangiogenic Activities of Pigment Epithelium-Derived Factor. The Internet Journal of Oncology, 2007 Volume 4 Number 1

External links

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		Dictionary. Webster 1913 Dictionary edited by Patrick J. Cassidy Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Spheroid". Read more

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