Editing Spheroid (section)

==Properties==
===Circumference===
{{anchor|Equatorial circumference}}The equatorial circumference of a spheroid is measured around its [[equator]] and is given as:
:<math>C_\text{e} = 2\pi a</math>
{{anchor|Meridional circumference}}{{anchor|Polar circumference}}The meridional or polar circumference of a spheroid is measured through its [[Semi-major and semi-minor axes|poles]] and is given as:
<math display="block">C_\text{p} \,=\, 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</math>
{{anchor|Volumetric circumference}}The volumetric circumference of a spheroid is the circumference of a [[sphere]] of equal volume as the spheroid and is given as:
:<math>C_\text{v} = 2\sqrt[3]{a^2c}</math>

===Area===
An oblate spheroid with {{math|''c'' < ''a''}} has [[surface area]]
:<math>S_\text{oblate} = 2\pi a^2\left(1+\frac{1-e^2}{e}\operatorname{arctanh}e\right)=2\pi a^2+\pi \frac{c^2}{e}\ln \left( \frac{1+e}{1-e}\right) \qquad \mbox{where} \quad e^2=1-\frac{c^2}{a^2}. </math>
The oblate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|a}} and semi-minor axis {{mvar|c}}, therefore {{mvar|e}} may be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].)<ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/OblateSpheroid.html |title=Oblate Spheroid |publisher=Wolfram MathWorld |access-date=24 June 2014}}</ref>

A prolate spheroid with {{math|''c'' > ''a''}} has surface area
:<math>S_\text{prolate} = 2\pi a^2\left(1+\frac{c}{ae}\arcsin \, e\right) \qquad \mbox{where} \quad e^2=1-\frac{a^2}{c^2}. </math>
The prolate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|c}} and semi-minor axis {{mvar|a}}; therefore, {{mvar|e}} may again be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].) <ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/ProlateSpheroid.html |title=Prolate Spheroid |publisher=Wolfram MathWorld |date=7 October 2003 |access-date=24 June 2014}}</ref>

These formulas are identical in the sense that the formula for {{math|''S''{{sub|oblate}}}} can be used to calculate the surface area of a prolate spheroid and vice versa. However, {{mvar|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 inside a spheroid (of any kind) is
:<math>\tfrac{4}{3}\pi a^2c\approx4.19a^2c.</math>
If {{math|''A'' {{=}} 2''a''}} is the equatorial diameter, and {{math|''C'' {{=}} 2''c''}} is the polar diameter, the volume is
:<math>\tfrac{\pi}{6}A^2C\approx0.523A^2C.</math>

===Curvature===
{{see also|Radius of the Earth#Radii of curvature}}
Let a spheroid be parameterized as
:<math> \boldsymbol\sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, c \sin \beta),</math>
where {{mvar|β}} is the ''reduced latitude'' or ''[[parametric latitude]]'', {{mvar|λ}} is the [[longitude]], and {{math|−{{sfrac|π|2}} < ''β'' < +{{sfrac|π|2}}}} and {{math|−π < ''λ'' < +π}}. Then, the spheroid's [[Gaussian curvature]] is
:<math> K(\beta,\lambda) = \frac{c^2}{\left(a^2 + \left(c^2 - a^2\right) \cos^2 \beta\right)^2},</math>
and its [[mean curvature]] is
:<math> H(\beta,\lambda) = \frac{c \left(2a^2 + \left(c^2 - a^2\right) \cos^2 \beta\right)}{2a \left(a^2 + \left(c^2 - a^2\right) \cos^2\beta\right)^\frac32}.</math>

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

===Aspect ratio===
The ''[[aspect ratio]]'' of an oblate spheroid/ellipse, {{math|''c'' : ''a''}}, is the ratio of the polar to equatorial lengths, while the ''[[flattening]]'' (also called ''oblateness'') {{mvar|f}}, is the ratio of the equatorial-polar length difference to the equatorial length:
:<math>f = \frac{a - c}{a} = 1 - \frac{c}{a} .</math>

The first [[eccentricity (mathematics)#Ellipses|''eccentricity'']] (usually simply eccentricity, as above) is often used instead of flattening.<ref>Brial P., Shaalan C.(2009), [http://80calcs.pagesperso-orange.fr/Downloads/IntroGeodesie.pdf Introduction à la Géodésie et au geopositionnement par satellites], p.8</ref> It is defined by:

: <math>e = \sqrt{1 - \frac{c^2}{a^2}}</math>

The relations between eccentricity and flattening are:
: <math>\begin{align} e &= \sqrt{2f - f^2} \\ f &= 1 - \sqrt{1 - e^2} \end{align}</math>

All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.