# مجسم كروي

 كروي مفلطح كروي متطاول

الجسم شبه الكروي (البيضاوي) Spheroid، هو جسم ناتج من تدوير قطع ناقص ( أهليلج ) حول أحد محاوره.

فمثلا البيضة هي من الأجسام شبه الكروية، غير المنتظمة.

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## المعادلة

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

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}$

The equation of a spheroid with Oz as the symmetry axis is given by setting a=b:

${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}$

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:

•   c < a  :  oblate spheroid
•   c > a  :  prolate spheroid

The case of a=c reduces to a sphere.

## المساحة

An oblate spheroid with c < a has surface area

${\displaystyle S_{\rm {oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\right)\quad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}$

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. (See ellipse). A derivation of this result may be found at.[1]

A prolate spheroid with c > a has surface area

${\displaystyle S_{\rm {prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\sin ^{-1}e\right)\qquad {\mbox{where}}\qquad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}$

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. (See ellipse). A derivation of this result may be found at.[2]

## الحجم

The volume inside a spheroid (of any kind) is ${\displaystyle (4\pi /3)a^{2}c\approx 4.19\,a^{2}c}$ . If A=2a is the equatorial diameter, and C=2c is the polar diameter, the volume is ${\displaystyle (\pi /6)A^{2}C\approx 0.523\,A^{2}C}$ .

## الانحناء

If a spheroid is parameterized as

${\displaystyle {\vec {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta );\,\!}$

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

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

and its mean curvature is

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

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

## الهامش

1. ^ "Oblate Spheroid - from Wolfram MathWorld". Mathworld.wolfram.com. Retrieved 2014-06-24.
2. ^ "Prolate Spheroid - from Wolfram MathWorld". Mathworld.wolfram.com. 2003-10-07. Retrieved 2014-06-24.