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Flattening
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Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is and its definition in terms of the semi-axes and of the resulting ellipse or ellipsoid is
The compression factor is in each case; for the ellipse, this is also its aspect ratio.
Definitions
[edit]There are three variants: the flattening [1] sometimes called the first flattening,[2] as well as two other "flattenings" and each sometimes called the second flattening,[3] sometimes only given a symbol,[4] or sometimes called the second flattening and third flattening, respectively.[5]
In the following, is the larger dimension (e.g. semimajor axis), whereas is the smaller (semiminor axis). All flattenings are zero for a circle (a = b).
(First) flattening Fundamental. Geodetic reference ellipsoids are specified by giving Second flattening Rarely used. Third flattening Used in geodetic calculations as a small expansion parameter.[6]
Identities
[edit]The flattenings can be related to each-other:
![{\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\[5mu]n={\frac {f}{2-f}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6d18928cdbeb14c7315af8fd7b55ac6434cab8)
The flattenings are related to other parameters of the ellipse. For example,
![{\displaystyle {\begin{aligned}{\frac {b}{a}}&=1-f={\frac {1-n}{1+n}},\\[5mu]e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}},\\[5mu]f&=1-{\sqrt {1-e^{2}}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/764e85cde72285c84f0379da9b263b0807a6ea78)
where is the eccentricity.
See also
[edit]References
[edit]- ^ Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office. doi:10.3133/pp1395.
- ^ Tenzer, Róbert (2002). "Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid". Studia Geophysica et Geodaetica. 46 (1): 27–32. doi:10.1023/A:1019881431482. S2CID 117114346. ProQuest 750849329.
- ^ For example, is called the second flattening in: Taff, Laurence G. (1980). An Astronomical Glossary (Technical report). MIT Lincoln Lab. p. 84. However, is called the second flattening in: Hooijberg, Maarten (1997). Practical Geodesy: Using Computers. Springer. p. 41. doi:10.1007/978-3-642-60584-0_3.
- ^ Maling, Derek Hylton (1992). Coordinate Systems and Map Projections (2nd ed.). Oxford; New York: Pergamon Press. p. 65. ISBN 0-08-037233-3. Rapp, Richard H. (1991). Geometric Geodesy, Part I (Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying. Osborne, P. (2008). "The Mercator Projections" (PDF). §5.2. Archived from the original (PDF) on 2012-01-18.
- ^ Lapaine, Miljenko (2017). "Basics of Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.). Choosing a Map Projection. Lecture Notes in Geoinformation and Cartography. pp. 327–343. doi:10.1007/978-3-319-51835-0_13. ISBN 978-3-319-51834-3.Karney, Charles F.F. (2023). "On auxiliary latitudes". Survey Review: 1–16. arXiv:2212.05818. doi:10.1080/00396265.2023.2217604. S2CID 254564050.
- ^ F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241–254, doi:10.1002/asna.201011352, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852–861 (2010), E-print arXiv:0908.1824, Bibcode:1825AN......4..241B
Flattening
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Definition and Basic Concepts
Flattening Parameter
The flattening parameter, denoted as $ f $, is defined for an oblate spheroid as $ f = \frac{a - c}{a} $, where $ a $ is the semi-major axis representing the equatorial radius and $ c $ is the semi-minor axis representing the polar radius.[1] This parameter quantifies the compression along the polar axis relative to the equatorial dimension, with $ f = 0 $ indicating a perfect sphere and positive values of $ f $ describing the oblate shape typical of rotating celestial bodies like Earth.[1] The flattening parameter derives from the standard equation of an ellipsoid of revolution, which exhibits rotational symmetry around the polar (z) axis:
where the substitution $ c = a(1 - f) $ parameterizes the axial deviation from sphericity while preserving the symmetry. This form allows $ f $ to succinctly capture the geometric distortion in models of non-spherical surfaces.
For Earth, the flattening is approximately $ f \approx 1/298.257 $, signifying a minor but measurable oblateness arising from its rotation.[5] The reciprocal flattening $ 1/f $ provides an alternative representation valued for its numerical stability in geodetic calculations.[5]
Geometric Interpretation
Flattening describes the deviation of a rotating body's shape from a perfect sphere, resulting in an oblate spheroid that appears as a squashed sphere with an equatorial bulge. This bulge arises primarily from the centrifugal force generated by the body's rotation, which counteracts gravitational forces more effectively at the equator than at the poles, causing material to redistribute outward along the equatorial plane.[4][6] The degree of this flattening increases qualitatively with the square of the angular velocity of rotation, meaning faster-spinning bodies exhibit a more pronounced equatorial bulge. For Earth, which completes one sidereal rotation every approximately 23 hours, 56 minutes, and 4 seconds relative to distant stars, this rotational effect produces a modest but measurable oblateness, balancing gravitational attraction and centrifugal repulsion to maintain hydrostatic equilibrium.[4][7][8] While oblate spheroids (with positive flattening) are the typical configuration for most rotating planetary bodies due to uniform centrifugal effects, prolate spheroids (with negative flattening, appearing elongated along the rotation axis) can occur in specialized cases, such as non-uniform mass distributions or extreme spin rates in smaller celestial objects. However, planetary science emphasizes oblate forms as the norm for bodies like Earth and Jupiter, where rotation dominates shape evolution.[9][8] In a meridional cross-section of an oblate spheroid, the equatorial radius a exceeds the polar radius c, with the flattening parameter f = (a - c)/a quantifying how the elliptical profile deviates from circular, becoming more oval as f increases and accentuating the polar compression relative to the equatorial expansion.[4]Mathematical Properties
Key Identities
The reciprocal flattening, $ 1/f = a / (a - b) $, is a parameter derived from the standard flattening $ f = (a - b)/a $, where $ a $ is the semi-major axis and $ b $ is the semi-minor axis of the reference ellipsoid.[10] This form enhances computational stability in geodetic algorithms by avoiding division by small values of $ f $ (typically on the order of 1/300 for Earth models), as $ 1/f $ is a large integer around 298 for standard ellipsoids.[11] This form is particularly useful in iterative methods for coordinate transformations and gravity field computations, where high precision is required without numerical instability.[12] A fundamental identity links the flattening to the first (linear) eccentricity $ e $ of the ellipsoid via $ e^2 = 2f - f^2 $.[10] To derive this, start with the definition $ e^2 = 1 - (b/a)^2 $. Since $ b = a(1 - f) $, substitute to obtain $ b/a = 1 - f $, so $ e^2 = 1 - (1 - f)^2 = 1 - (1 - 2f + f^2) = 2f - f^2 $.[11] This relation connects the geometric flattening to the eccentricity used in ellipsoidal coordinates, such as in the expressions for radii of curvature $ N = a / \sqrt{1 - e^2 \sin^2 \phi} $ and $ M = a (1 - e^2) / (1 - e^2 \sin^2 \phi)^{3/2} $, where $ \phi $ is the geodetic latitude.[10] For Earth's ellipsoids like GRS80 ($ f \approx 1/298.257 $), the $ f^2 $ term is approximately $ 10^{-5} $, making $ e^2 \approx 2f $ a useful first-order approximation in many derivations.[11] The authalic flattening $ q $, also known as the third flattening, is defined as $ q = (1 - b/a) / (1 + b/a) $ and plays a key role in equal-area (authalic) projections by facilitating mappings that preserve surface areas between the ellipsoid and auxiliary sphere.[11] In terms of flattening, this simplifies to $ q = f / (2 - f) $, since $ 1 - b/a = f $ and $ 1 + b/a = 2 - f $.[12] This parameter arises in the context of authalic latitudes, which are derived to ensure equal areas by equating zonal surface areas on the ellipsoid to those on a sphere of equivalent total area. The full derivation begins with the differential surface element on the ellipsoid, $ dA = 2\pi N(\phi) \cos \phi , M(\phi) , d\phi $, where $ N(\phi) $ and $ M(\phi) $ are the prime vertical and meridional radii of curvature. Integrating from the equator ($ \phi = 0 $) to latitude $ \phi $ gives the partial surface area $ A(\phi) = 2\pi \int_0^\phi N(\phi') \cos \phi' M(\phi') d\phi' $. Substituting the expressions for $ N $ and $ M $ yields $ A(\phi) = 2\pi a^2 (1 - e^2) \int_0^\phi \cos \phi' / (1 - e^2 \sin^2 \phi')^2 d\phi' $, which can be transformed via integration by parts or substitution to the standard form involving $ q(\phi) = (1 - e^2) \left[ \sin \phi / (1 - e^2 \sin^2 \phi) - (1/(2e)) \ln \left( (1 - e \sin \phi)/(1 + e \sin \phi) \right) \right] $, such that $ A(\phi) = 2\pi a^2 q(\phi) \phi = \pi/2 $), $ q_p = (1 - e^2) \left[ 1 - (1/(2e)) \ln \left( (1 - e)/(1 + e) \right) \right] $, and the total surface area is $ 4\pi a^2 q_p $. The authalic latitude $ \beta $ satisfies $ \sin \beta = q(\phi) / q_p $, ensuring $ A(\phi) = 2\pi r_a^2 (1 - \cos \beta) $, where $ r_a = a \sqrt{q_p} $ is the authalic radius. The parameter $ q $ (the constant third flattening) improves convergence in series expansions of these integrals, as $ q \approx f/2 $ is smaller than $ e \approx \sqrt{2f} $, reducing higher-order terms in Taylor expansions for small $ f $.[11][13] The relation $ b = a(1 - f) $ defines the semi-minor axis exactly from the flattening, but for small $ f $, it serves as a linear approximation in deriving other parameters, with error analysis revealing negligible higher-order effects at Earth's scale.[10] For instance, in the eccentricity identity, approximating $ e^2 \approx 2f $ neglects the $ -f^2 $ term, introducing a relative error of approximately $ f/2 \approx 0.0017 $ for $ f = 1/298.257 $ (GRS80), or an absolute error in $ e^2 $ of about $ 1.1 \times 10^{-5} $, which is below 0.2% and often insignificant in first-order geodetic models but requires inclusion for high-precision applications like satellite orbit determination.[11] Similarly, in rotational flattening models, the linear term dominates, with quadratic corrections from Clairaut's equation contributing less than 10% to the observed $ f $ for Earth.[10]Related Parameters
In geodesy, auxiliary parameters derived from the flattening f provide alternative ways to characterize the shape of an oblate spheroid, facilitating computations in specific contexts such as series expansions, latitude transformations, and spherical approximations. These parameters are interrelated through simple algebraic relations and are chosen based on the numerical stability or convergence properties required for particular calculations. The second flattening, denoted $ f' $, is defined as $ f' = (a - b)/b $, where a is the semi-major (equatorial) axis and b is the semi-minor (polar) axis. This contrasts with the standard flattening f = (a - b)/a by normalizing the axial difference to the polar radius rather than the equatorial one, which can be advantageous in formulations emphasizing polar geometry or when deriving expressions symmetric in certain polar-centric models. The relation between $ f' $ and f is $ f' = f / (1 - f) $.[14] The parametric latitude β, also known as the reduced latitude, is an auxiliary angle used to parameterize points on the ellipsoid via spherical-like coordinates for curve computations, such as in geodesic algorithms or map projections. It relates to the geodetic latitude φ by the equation tan β = (1 - f) tan φ, where the factor (1 - f) = b/a accounts for the ellipsoidal compression. This transformation maps the ellipsoid onto an auxiliary sphere of radius a, simplifying integrals and series for meridian arcs or azimuth calculations.[13] A first-order approximation for the mean radius r of the ellipsoid, useful for spherical equivalents in preliminary modeling or volume-related estimates, is given by r ≈ a (1 - f/3). This derives from the arithmetic mean of the axes, r = (2a + b)/3, which expands to the stated form for small f and establishes the scale for spheroidal approximations in global geodesy.[15] The following table compares key related parameters, highlighting their definitions and advantages/disadvantages for common geodetic uses:| Parameter | Definition | Pros/Cons for Uses |
|---|---|---|
| f (flattening) | f = (a - b)/a | Standard parameter for defining ellipsoid shape; intuitive as fractional compression from equator; but f is small (~1/300 for Earth), leading to ill-conditioned expressions in some high-precision series without reciprocal 1/f. Widely adopted in reference systems like WGS84. |
| f' (second flattening) | f' = (a - b)/b = f / (1 - f) | Normalizes to polar axis, useful for polar-focused computations or symmetry in certain elliptic integrals; larger than f (~1/298 for Earth), avoiding near-unity denominators but less common than f, requiring conversion for standard tools.[14] |
| n (third flattening) | n = (a - b)/(a + b) = f / (2 - f) | Smaller value (~f/2, ~1/600 for Earth) enables faster-converging power series expansions in geodesic distance formulas and latitude functions, avoiding divisions by near-zero quantities; preferred in numerical algorithms over e for stability, though requires transformation from f.[16][17] |
| e (linear eccentricity) | e = √(a² - b²)/a, with e² = 2f - f² (see Key Identities) | Essential for eccentricity-based formulas in latitude reductions and orbital mechanics; e² provides a dimensionless measure of deviation from sphericity, but higher powers of e converge slower than n in expansions, making it less ideal for iterative computations.[17] |