Coordinates and Frames: Describing a Place Precisely

A navigation computer needs a precise way to say where something is. “Liverpool” is useful to a human, but it is not enough for a filter, a map renderer or a GNSS receiver. Modern navigation commonly uses latitude, longitude and height tied to a geodetic reference frame such as WGS 84. NGA describes WGS 84 as a 3D coordinate reference frame for latitude, longitude and heights used in navigation, positioning and targeting.¹

This post introduces three linked ideas:

1. Earth is not a perfect sphere.
2. Coordinates only mean something relative to a reference frame.
3. A navigation algorithm often converts between global and local frames.

NOAA’s National Geodetic Survey defines geodesy as the science of accurately measuring Earth’s geometric shape, gravity field and orientation in space.²

The problem

A web map often displays latitude and longitude. An inertial navigation algorithm may prefer a local North-East-Down frame. A satellite-positioning equation may use Earth-Centred, Earth-Fixed coordinates. These are not competing truths; they are different coordinate descriptions of the same physical point.

The model

Use geodetic coordinates:

$$(\phi, \lambda, h)$$

where $\phi$ is latitude, $\lambda$ is longitude and $h$ is ellipsoidal height.

A common conversion from geodetic WGS 84-style coordinates to ECEF is:

$$N(\phi)=\frac{a}{\sqrt{1-e^2\sin^2\phi}}$$ $$x=(N+h)\cos\phi\cos\lambda$$ $$y=(N+h)\cos\phi\sin\lambda$$ $$z=(N(1-e^2)+h)\sin\phi$$

where $a$ is the semi-major axis and $e$ is ellipsoid eccentricity.

For a local navigation frame, define a tangent frame at a chosen origin. A small local displacement can be written:

$$\Delta \mathbf{r}_{ned} = \begin{bmatrix} \Delta n \\ \Delta e \\ \Delta d \end{bmatrix}$$

where north and east are horizontal components and down is positive toward Earth.

Interactive demo: coordinate frame explorer

The demo below shows the same point in geodetic, ECEF and local coordinates at the same time. It uses a Liverpool-centred local map patch and live WGS 84 conversions to make the relationships visible.

What to watch

• Dragging the point updates latitude, longitude, ECEF, and local NED together.
• Setting a local origin makes the selected point read near [0, 0, 0] in the local frame.
• Small east-west moves produce intuitive metre-scale changes in the local east component.
• The flat-Earth approximation stays close nearby, but the readout makes it clear that it is still only an approximation.

Try this

Set the origin at Liverpool and drag the point a short distance east. Longitude changes, but the local-frame display keeps the motion readable as a simple metre-scale displacement.

Where this breaks

This post does not cover vertical datums, geoids, plate motion, epoch, map projections or survey-grade transformations. For beginner navigation, the key is to stop thinking of latitude and longitude as “just numbers” and start thinking in terms of coordinate frames.

Footnotes

1. National Geospatial-Intelligence Agency, “World Geodetic System 1984 (WGS 84)”. https://earth-info.nga.mil/GandG/wgs84/ - Official reference for WGS 84 as a 3D coordinate reference frame for latitude, longitude and height in navigation, positioning and targeting. Accessed 2026-06-11. ↩

2. NOAA National Geodetic Survey, “Geodesy”. https://www.ngs.noaa.gov/INFO/geodesy.shtml - Defines geodesy as measurement and understanding of Earth shape, gravity field and orientation in space. Accessed 2026-06-11. ↩