4.1 The Shape of the Earth — Geoid and Ellipsoid

Key takeaways

• Earth is a lumpy ellipsoid — neither a perfect sphere nor a perfectly smooth ellipsoid.
• The geoid is the true surface of mean sea level; the ellipsoid is the mathematical approximation we compute on.
• GNSS receivers report ellipsoidal heights; most maps use orthometric (geoid-referenced) heights. The difference matters.

Introduction

Before you can talk about coordinates, you need to know what surface those coordinates live on. "Latitude and longitude" are easy words, but they refer to positions on a mathematical surface that only approximates the real Earth. Understanding which surface — geoid, ellipsoid, or sphere — is what lets you choose the right datum, interpret GPS readings, and trust your elevation data. This lesson covers the geometry of the Earth in practical GIS terms.

Why the Earth is not a sphere

Rotation flattens it. The Earth spins about its polar axis roughly once every 24 hours, and centrifugal force bulges the equator outward by about 21 km compared to the poles. The equatorial radius is ~6378 km; the polar radius is ~6357 km. This makes the Earth an oblate spheroid — an ellipse of revolution.

Even this ellipsoidal approximation smoothes over the real Earth's shape. Continents, ocean trenches, and variations in the Earth's density cause gravity — and therefore sea level — to be uneven. The true shape of mean sea level is called the geoid.

The geoid

The geoid is the equipotential surface of Earth's gravity field that best approximates global mean sea level. Key properties:

• It is irregular — no simple equation describes it.
• It undulates by about ±100 m relative to the best-fit ellipsoid.
• It is the surface orthometric ("above sea level") heights are measured against.
• Water at rest on the geoid does not flow.

The geoid is defined by a gravity model, typically expressed as spherical harmonic coefficients (e.g., EGM2008, EGM2020). For most GIS work you treat it as a correction surface — add or subtract geoid heights from ellipsoidal heights to get orthometric heights.

The ellipsoid

Because the geoid has no closed-form equation, geodesists approximate it with a best-fit reference ellipsoid — an oblate spheroid defined by two parameters: semi-major axis a (equatorial radius) and inverse flattening 1/f.

Common reference ellipsoids:

Ellipsoid	a (m)	1/f	Used by
WGS 84	6 378 137.0	298.257223563	GPS, web maps, most global data
GRS 80	6 378 137.0	298.257222101	NAD83, ETRS89
Clarke 1866	6 378 206.4	294.9786982	Legacy NAD27
Bessel 1841	6 377 397.155	299.1528128	Many European national grids

WGS 84 and GRS 80 differ by only ~0.0001 in flattening — essentially identical for most purposes. Clarke 1866 differs by ~70 m at the equator, which produces visible offsets when mixing legacy and modern datasets.

Sphere vs ellipsoid — does it matter?

Sometimes a sphere is good enough. Aviation, meteorology, and introductory demonstrations often use a sphere of radius 6 371 km. The Haversine formula assumes a sphere.

For high-accuracy geodesy (survey, GNSS, precise mapping), the ellipsoidal calculation (Vincenty's formulae, Karney's algorithms) is required. The maximum error of assuming a sphere is about 0.5% — tolerable for rough distances but disastrous for property boundaries.

Ellipsoidal height vs orthometric height

Two definitions of "elevation" you must distinguish:

• Ellipsoidal height (h) — height above the reference ellipsoid. GNSS receivers natively report this.
• Orthometric height (H) — height above the geoid (mean sea level). What most topographic maps show.
• Geoid height (N) — height of the geoid above the ellipsoid. Comes from the gravity model.

They relate as:

$$H = h - N$$

Example: a GPS reading shows ellipsoidal height 245 m. The geoid height at that location (from EGM2008) is −32 m. The orthometric height is:

$$H = 245 - (-32) = 277 \text{ m}$$

Mismatching ellipsoidal and orthometric heights can introduce errors of tens of metres — enough to flood a model or misplace a survey benchmark.

Datums

A datum ties a reference ellipsoid to the Earth by specifying the ellipsoid's orientation and position. A datum has:

• A reference ellipsoid.
• A fixed origin (historically a ground station; today, Earth's centre of mass).
• A set of anchor observations that constrain it.

WGS 84 is a geocentric datum aligned with Earth's centre of mass, maintained by the US DoD for GPS. NAD83 (North American) and ETRS89 (European) are tied to their continental plates so coordinates don't drift with plate tectonics.

Module 4.2 digs deeper into datums.

Practical takeaways

• Every coordinate is relative to a datum that uses an ellipsoid that approximates a geoid that approximates Earth.
• GPS gives you ellipsoidal heights on WGS 84. If you need "metres above sea level," subtract the geoid height.
• Tools: GeographicLib, pyproj.transformer (with geoid grid files), cs2cs, gdalwarp with the right flags.
• Legacy datums (NAD27, OSGB36) differ from modern ones by tens to hundreds of metres. Always reproject, don't just rename.

Self-check exercises

1. What's the difference between the geoid and an ellipsoid?

The geoid is the real, irregular equipotential surface of Earth's gravity field — effectively mean sea level extended under continents. The ellipsoid is a smooth mathematical surface (oblate spheroid) used as a best-fit approximation. The geoid has no simple equation; the ellipsoid does.

2. Your GPS reports 312 m elevation; the local geoid height is +28 m. What orthometric height should you report on a topographic map?

Orthometric height H = h − N = 312 − 28 = 284 m. This is the value that compares to "height above sea level" on traditional maps.

3. Is it safe to treat the Earth as a sphere for a shipping-route calculation?

For rough routes and visualisation, yes — errors are under 0.5%. For precise navigation, fuel calculations, or survey work, no — use ellipsoidal formulae (Vincenty, Karney). The choice is a trade-off between complexity and the accuracy your application needs.

Summary

• Earth is best modelled as an oblate ellipsoid — specifically WGS 84 for GPS / web, regional ellipsoids for legacy data.
• The geoid is the true mean-sea-level surface; it undulates around the ellipsoid.
• Ellipsoidal heights (GPS) differ from orthometric heights (maps) by the local geoid height.
• Datums tie ellipsoids to Earth; different datums produce different coordinates for the same physical point.

Further reading

• NGA — Implementation of the World Geodetic System 1984 (WGS 84).
• Vaníček & Krakiwsky — Geodesy: The Concepts.
• NGS — Geoid models and tools.
• PROJ documentation — proj=pipeline operations for datum transformations.