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reference-ellipsoids

Abstract 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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NetFabric/NetFabric.Numerics36 stars(GitHub 36 + cccskills 0)

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  • SKILL.md2.5 KB
  • references/formulas.md1.9 KB
  • references/numerical-stability.md1.6 KB

SKILL.md (original)

Review the instructions the agent will receive before you install.

Reference Ellipsoids

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):

ParameterSymbolDefinition
Semi-major (equatorial) radiusaRadius at equator
Semi-minor (polar) radiusbRadius at poles
Flatteningff = (a−b)/a
Eccentricityee = √(f(2−f)) = √(1−b²/a²)
Eccentricity squarede²e² = 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

NameSymbolFormulaAt
Meridian radius of curvatureMa(1−e²) / (1−e²sin²φ)^(3/2)Latitude φ
Prime vertical radiusNa / √(1−e²sin²φ)Latitude φ
At equator (M)—a(1−e²)φ = 0
At pole (M)—a/√(1−e²)φ = 90°
At equator (N)—aφ = 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

Namea (m)1/f
WGS 846 378 137.0298.257 223 563
GRS 806 378 137.0298.257 222 101
Airy 18306 377 563.396299.324 964 6
Clarke 18666 378 206.4294.978 698 2
International 19246 378 388.0297.0
Bessel 18416 377 397.155299.152 812 8

Reference Files

FileLoad When
references/formulas.mdSurface area derivation, curvature proof, series expansions
references/numerical-stability.mdEccentricity near zero, curvature at poles, overflow for large a

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