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|-
||<center>[[Image:OblateSpheroid.PNG|240px]]</center>
||<center>[[Image:ProlateSpheroid.png|160px]]</center>
|-
|style="text-align: center"|''oblate spheroid''
|style="text-align: center"|''prolate spheroid''
|}
A '''spheroid''', or '''ellipsoid of revolution''' is a [[quadric]] [[surface]] obtained by rotating an [[ellipse]] about one of its principal axes; in other words, an [[ellipsoid]] with two equal [[semi-diameter]]s. 
 
If the ellipse is rotated about its major axis, the result is a '''[[prolate spheroid|prolate]]''' (elongated) spheroid, like an [[American football]] or [[rugby football|rugby]] ball. If the ellipse is rotated about its minor axis, the result is an '''[[oblate spheroid|oblate]]''' (flattened) spheroid, like a [[lentil]]. If the generating ellipse is a circle, the result is a '''[[sphere]]'''.
 
Because of the combined effects of [[gravitation]] and [[rotation of the Earth|rotation]], the [[Earth]]'s shape is roughly that of a sphere slightly flattened in the direction of its axis.  For that reason, in [[cartography]] the Earth is often approximated by an oblate spheroid instead of a sphere. The current [[World Geodetic System]] model uses a spheroid whose radius is 6,378.137&nbsp;km at the [[equator]] and 6,356.752&nbsp;km at the [[geographical pole|pole]]s.
 
The word ''spheroid'' originally meant an ''approximately spherical body''
and that is how it is used in some older papers on geodesy.
In order to avoid confusion, ''spheroid'' should be defined as an
''ellipsoid of revolution'', if that is the intended meaning.
 
==Equation==
[[File:Spheroid.svg|thumb|250px| The assignment of semi-axes on a spheroid. Ii is oblate if ''c<a'' and prolate if ''c>a''.]]
The equation of a tri-axial ellipsoid centred at the origin with semi-axes ''a'',''b'', ''c'' aligned along the coordinate axes is
:::<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1</math>
The equation of a spheroid  with ''Oz'' as the [[symmetry axis]]  is given by setting ''a=b'':
:::<math>\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1.</math>
The semi-axis ''a'' is the equatorial radius of the spheroid, and ''c'' is the distance from centre to pole along the symmetry axis. There are two possible cases:
:::* &nbsp;  ''c < a'' &nbsp;:&nbsp; '''oblate''' spheroid
:::* &nbsp;  ''c > a'' &nbsp;:&nbsp; '''prolate''' spheroid
The case of ''a=c'' reduces to a sphere.
 
==Surface area==
An '''oblate''' spheroid with ''c < a'' has [[surface area]]
:::<math>S_{\rm oblate} =  2\pi a^2\left(1+\frac{1-e^2}{e}\tanh^{-1}e\right)
\quad\mbox{where}\quad e^2=1-\frac{c^2}{a^2}. </math>
The oblate spheroid  is generated  by rotation about the ''Oz'' axis of an ellipse with semi-major axis ''a''  and semi-minor axis ''c'', therefore ''e'' may be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]]). A derivation of this result may be found at.<ref>http://mathworld.wolfram.com/OblateSpheroid.html</ref>
 
A '''prolate''' spheroid with ''c > a'' has surface area
:::<math>S_{\rm prolate} =  2\pi a^2\left(1+\frac{c}{ae}\sin^{-1}e\right)
\qquad\mbox{where}\qquad e^2=1-\frac{a^2}{c^2}. </math>
The prolate spheroid  is generated  by rotation about the ''Oz'' axis of an ellipse with semi-major axis ''c''  and  semi-minor axis ''a'', therefore ''e'' may again be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]]). A derivation of this result may be found at.<ref>http://mathworld.wolfram.com/ProlateSpheroid.html</ref>
 
These formulas are identical in the sense that the formula for <math>S_{\rm oblate}</math> can be used to calculate the surface area of a prolate spheroid and vice versa. However, ''e'' then becomes [[Imaginary number|imaginary]] and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.
 
==Volume==
The volume of a spheroid (of any kind) is <math>(4\pi/3) a^2c \approx 4.19\, a^2c</math>.  If ''A''=2''a'' is the equatorial diameter, and ''C''=2''c'' is the polar diameter, the volume is <math>(\pi/6) A^2C \approx 0.523\, A^2C</math>.
 
==Curvature==
{{see also|Radius of the Earth#Radii of curvature}}
If a spheroid is parameterized as
:<math> \vec \sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, c \sin \beta);\,\!</math>
where <math>\beta\,\!</math> is the '''reduced''' or '''[[Latitude#Reduced (or parametric) latitude|parametric latitude]]''', <math>\lambda\,\!</math> is the '''[[longitude]]''', and <math>-\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\!</math>
and <math>-\pi<\lambda<+\pi\,\!</math>, then its [[Gaussian curvature]] is
:<math> K(\beta,\lambda) = {c^2 \over (a^2 + (c^2 - a^2) \cos^2 \beta)^2};\,\!</math>
and its [[mean curvature]] is
:<math> H(\beta,\lambda) = {c (2 a^2 + (c^2 - a^2) \cos^2 \beta) \over 2 a (a^2 + (c^2-a^2) \cos^2 \beta)^{3/2}}.\,\!</math>
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.
 
==See also==
*[[Ellipsoid]]
*[[Prolate spheroid]]
*[[Oblate spheroid]]
*[[Oval|Ovoid]]
 
==References==
<references/>
 
[[Category:Surfaces]]
[[Category:Quadrics]]

Latest revision as of 19:11, 30 December 2014

It is time to address the slow computer issues even if you do not recognize how. Just considering a computer is working so slow or keeps freezing up; refuses to mean to not address the issue plus fix it. You may or may not be aware which any computer owner should learn that there are certain things which your computer requires to maintain the number one performance. The sad fact is that numerous individuals that own a program have no idea which it needs routine maintenance really like their cars.

So one day my computer suddenly began being weird. I was thus frustrated, considering my files were missing, plus I cannot open the files which I required, and then, suddenly, everything stopped working!

The error is basically a result of issue with Windows Installer package. The Windows Installer is a tool employed to install, uninstall plus repair the most programs on your computer. Let you discuss a few factors that helped a lot of people that facing the similar problem.

Registry cleaners have been tailored for 1 purpose - to clean out the 'registry'. This really is the central database which Windows relies on to function. Without this database, Windows wouldn't even exist. It's thus important, which a computer is regularly adding and updating the files inside it, even when you're browsing the Internet (like now). This is fantastic, but the difficulties happen when several of those files become corrupt or lost. This occurs a lot, plus it takes a advantageous tool to fix it.

Another popular cause of PC slow down is a corrupt registry. The registry is a important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not start at all. Fixing the registry is easy with the utilize of the system and system mechanic.

Software mistakes or hardware mistakes that happen when running Windows and intermittent mistakes are the general reasons for a blue screen bodily memory dump. New software or drivers which have been installed or changes inside the registry settings are the typical s/w causes. Intermittent mistakes refer to failed program memory/ difficult disk or over heated processor and these too may result the blue screen bodily memory dump error.

To speed up your computer, you just should be capable to do away with all these junk files, permitting your computer to find just what it wants, whenever it wants. Luckily, there's a tool that allows you to do this easily plus immediately. It's a tool called a 'registry cleaner'.

Frequently the greatest method is to read ratings on them plus when several users remark regarding its efficiency, it happens to be probably to be work. The right piece is the fact that there are many top registry cleaners which work; you merely have to take your choose.

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