Reference Ellipsoids
SkillDev toolsAbstract math for reference ellipsoids used in geodesy. USE FOR: computing semi-minor axis from flattening; eccentricity and eccentricity squared; first and second eccentricity; arithmetic mean radius; radii of curvature in the meridian and prime vertical; ellipsoid surface area; ellipsoid volume; understanding WGS84/GRS80 parameters; implementing Ellipsoid<T> types. DO NOT USE FOR: geodetic coordinate bounds (use geodetic-coordinate-bounds); datum transformations (use helmert-datum-transformation); map projections.
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Definition
An ellipsoid of revolution is generated by rotating an ellipse around its polar axis. Defined by two independent parameters (choose any two of a, b, f, e):
| Parameter | Symbol | Definition |
|---|---|---|
| Semi-major (equatorial) radius | a | Radius at equator |
| Semi-minor (polar) radius | b | Radius at poles |
| Flattening | f | f = (a−b)/a |
| Eccentricity | e | e = √(f(2−f)) = √(1−b²/a²) |
| Eccentricity squared | e² | e² = f(2−f) = 1−b²/a² |
Derived Parameters
b = a(1−f)
e² = f(2−f)
e = √(e²)
e'² = e²/(1−e²) (second eccentricity squared)
n = (a−b)/(a+b) = f/(2−f) (third flattening, used in Helmert series)
Radii of Curvature
| Name | Symbol | Formula | At |
|---|---|---|---|
| Meridian radius of curvature | M | a(1−e²) / (1−e²sin²φ)^(3/2) | Latitude φ |
| Prime vertical radius | N | a / √(1−e²sin²φ) | Latitude φ |
| At equator (M) | — | a(1−e²) | φ = 0 |
| At pole (M) | — | a/√(1−e²) | φ = 90° |
| At equator (N) | — | a | φ = 0 |
| At pole (N) | — | a/√(1−e²) | φ = 90° |
Global Metrics
Arithmetic mean radius R₁ = (2a + b) / 3
Surface area S = 2πa² (1 + (1−e²)/e · atanh(e))
Volume V = (4/3)π a² b
Standard Ellipsoids
| Name | a (m) | 1/f |
|---|---|---|
| WGS 84 | 6 378 137.0 | 298.257 223 563 |
| GRS 80 | 6 378 137.0 | 298.257 222 101 |
| Airy 1830 | 6 377 563.396 | 299.324 964 6 |
| Clarke 1866 | 6 378 206.4 | 294.978 698 2 |
| International 1924 | 6 378 388.0 | 297.0 |
| Bessel 1841 | 6 377 397.155 | 299.152 812 8 |
Reference Files
| File | Load When |
|---|---|
| references/formulas.md | Surface area derivation, curvature proof, series expansions |
| references/numerical-stability.md | Eccentricity near zero, curvature at poles, overflow for large a |
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- Last commit
- Aug 2026
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reference-ellipsoids- Source
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