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.

 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

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

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

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

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

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

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.

A prolate spheroid with c > a has surface area

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

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.

 الحجم

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

 الانحناء

انظر أيضاً: Radius of the Earth § Radii of curvature

If a spheroid is parameterized as

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

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

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

and its mean curvature is

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

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

 انظر أيضاً

• Ellipsoid
• Prolate spheroid
• Oblate spheroid
• Ovoid