Almost every persistent misunderstanding about heights comes from collapsing three different surfaces into one word. The ellipsoid is a mathematical shape. The geoid is a physical, gravity-defined surface. The terrain is the actual dirt and rock you stand on. They are not the same surface, they are not parallel, and a height measured from one is not convertible to a height from another without a model.
Once you can picture the three surfaces stacked in the right order for your location, the arithmetic of heights becomes obvious rather than memorized.
Surface one: the reference ellipsoid
- Reference ellipsoid
- An ellipsoid of revolution — a sphere flattened at the poles — chosen as a smooth mathematical approximation of the Earth. It is defined by two numbers, a semi-major axis and a flattening, and everything else about it is computed.
The ellipsoid exists because geodetic computation needs a surface with closed-form geometry. GRS 80, the ellipsoid underlying NAD 83, has a semi-major axis of 6 378 137 m and a reciprocal flattening of 298.257222101. Its polar semi-minor axis works out to about 6 356 752.314 m, so the Earth is roughly 21 km fatter through the equator than through the poles.
The ellipsoid has no physical existence. Nothing about it tells you which way water flows. It is a computational convenience, chosen to fit sea level well on average across the whole planet, and it does that job to within roughly 100 m worldwide.
Surface two: the geoid
- Geoid
- The particular equipotential surface of the Earth gravity field that most nearly coincides with global mean sea level. It is everywhere perpendicular to the direction of gravity, which is what makes it the correct zero for heights that describe flow.
The geoid is bumpy because the Earth is not uniform. A dense mass anomaly in the crust or mantle pulls the equipotential surface toward it; a low-density region lets it relax away. The result is a surface with undulations of roughly plus or minus 100 m relative to the ellipsoid, and with local gradients steep enough to matter over a single project.
A useful mental model is that the geoid is where the sea would settle if you cut canals through every continent and let the water come to rest, ignoring tides, currents and wind. It is the surface a spirit level agrees with, and a bubble is the instrument that reveals it.
Surface three: the terrain
The topographic surface is the physical ground: the only one of the three you can actually occupy with an instrument. It is what you survey, and it is irrelevant as a reference because it has no defining property beyond being where the rock stopped.
Every measurement you take sits on the terrain. Every coordinate you compute is referenced to one of the other two surfaces. Reconciling those two facts is a large part of what geodesy is for.
The three heights
| Surface | Nature | Height measured from it | Symbol |
|---|---|---|---|
| Reference ellipsoid | Purely mathematical | Ellipsoid height — what GNSS produces | h |
| Geoid | Physical, defined by gravity | Orthometric height — what a level and a benchmark give you | H |
| Terrain | Physical, arbitrary | None — it is the thing being measured | — |
The vertical separation between the ellipsoid and the geoid at a given point is the geoid height, N. In the conterminous United States N is negative essentially everywhere, meaning the geoid lies below the ellipsoid, and it ranges from roughly −8 m in the east to below −50 m in the mountain west. In Alaska and around the Gulf it takes other values again. Because N is negative in CONUS, an ellipsoid height is typically a smaller number than the corresponding orthometric height at the same point.
h = H + NWhy the surfaces are not parallel
If the geoid were a constant distance below the ellipsoid, converting between h and H would be a single subtraction and nobody would need a geoid model. It is not constant. Over a 10 km project the geoid height can change by a decimetre or more, and in the mountains it changes faster. That gradient is exactly the reason a modelled surface — GEOID18 and its successors — is required rather than a single number for the county.
There is a second, subtler non-parallelism. The ellipsoid normal and the plumb line at a point are not the same direction. The angle between them is the deflection of the vertical, usually a few arcseconds but tens of arcseconds in extreme terrain. It is why an astronomic azimuth and a geodetic azimuth differ, and why a total station levelled to the bubble is oriented to gravity while your coordinates are oriented to an ellipsoid.
Getting the picture right
- The ellipsoid is smooth and computed; the geoid is bumpy and physical; the terrain is where you stand.
- GNSS sees the ellipsoid. A level and a benchmark see the geoid. Neither sees the other.
- The separation between them, N, varies across a project and must come from a model.
- Water flows downhill relative to the geoid, not relative to the ellipsoid — which is why drainage design is never done in ellipsoid heights.
- The terrain matters to coordinate work in one specific way: it is the height that drives the elevation factor when reducing ground distances to grid.