## Shape of the Earth

Although the Earth is very round, it is an oblate *spheroid* rather
than a perfect sphere. This difference is so small (only one part in 300) that modeling the
Earth as spherical is sufficient for making small-scale (world or continental) maps. However,
making accurate maps at larger scale demands that a spheroidal model be used. Such models are
essential, for example, when you are mapping high-resolution satellite or aerial imagery, or
when you are working with coordinates from the Global Positioning System (GPS). This section
addresses how Mapping Toolbox™ software accurately models the shape, or figure, of the Earth.

### Ellipsoid Shape

You can define ellipsoids in several ways. They are usually specified by a
*semimajor* and a *semiminor axis*, but are
often expressed in terms of a semimajor axis and either *inverse
flattening* (which for the Earth, as mentioned above, is one part in 300) or
*eccentricity*. Whichever parameters are used, as long as an axis
length is included, the ellipsoid is fully constrained and the other parameters are
derivable. The components of an ellipsoid are shown in the following diagram.

The toolbox includes ellipsoid models that represent the figures of the Sun, Moon, and planets, as well as a set of the most common ellipsoid models of the Earth. For more information, see Comparison of Reference Spheroids.

### Geoid Shape

Literally, *geoid* means *Earth-shaped*. The geoid
is an empirical approximation of the figure of the Earth (minus topographic relief), its
"lumpiness." Specifically, it is an equipotential surface with respect to gravity, more or
less corresponding to mean sea level. It is approximately an ellipsoid, but not exactly so
because local variations in gravity create minor hills and dales (which range from -100 m to
+60 m across the Earth). This variation in height is on the order of 1 percent of the
differences between the semimajor and semiminor ellipsoid axes used to approximate the
Earth's shape.

The shape of the geoid is important for some purposes, such as calculating satellite orbits, but need not be taken into account for every mapping application. However, knowledge of the geoid is sometimes necessary, for example, when you compare elevations given as height above mean sea level to elevations derived from GPS measurements. Geoid representations are also inherent in datum definitions.

#### Map the Geoid

Get geoid heights and a geographic postings reference object from the EGM96 geoid model. Load coastline latitude and longitude data.

```
[N,R] = egm96geoid;
load coastlines
```

Display the geoid heights as a surface using a Robinson projection. Ensure the coastline data appears over the surface by setting the `'CData'`

name-value pair to the geoid heights data and the `'ZData'`

name-value pair to a matrix of zeros. Then, display the coastline data.

axesm robinson Z = zeros(R.RasterSize); geoshow(N,R,'DisplayType','surface','CData',N,'ZData',Z) geoshow(coastlat,coastlon,'color','k')

Display a colorbar below the map.

`colorbar('southoutside')`