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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.

日本語の概要は準備中です。原文の説明を表示しています。

NetFabric/NetFabric.Numericsスター 36(GitHub 36 + cccskills 0)

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

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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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