#### 6.1 Earth’s shape approximations

There are three way to model the Earth’s shape:

1. Plane (“Topographical plane”)
2. Sphere (“Local Sphere”)
3. Ellipsoid
4. Geoid

### Plane Earth’s shape approximation: Topographical plane For very small earth’s shape modeling we can use a simple plan to represent it. The figure above illustrate the condition of two point A and B; the distance between A and B is measured onto the plane, and the elevations are the vertical distances between the plane and the points.

The topographical plane modeling is used for very small area: max 25 km for the distances and 1-2 km for elevations! In fact the errors derived from Earth’s curvature are more incisive on the elevations than the distances.

### Sphere Earth’s shape approximation – The “Local Sphere” In a range of max 100 km we consider the Earth’s shape as a sphere. For determine distances and elevations we consider a radius of the Sphere passing trough the point, and we measure distances and elevations on the interceptions of these radius and the surface of the sphere. The fundamental parameter for this modeling system is the radius of Local Sphere R.

### Ellipsoid Earth’s shape approximation – Oblate ellipsoid

The sphere has a single radius of curvature, if we want to consider a more approximated Earth’s shape we have to introduce the flattening.

A simple way to consider polar flattening of earth’s shape is consider a two-radii of curvature surface: the Oblate Ellipsoid.

The oblate ellipsoid, also called: “squashed” spheroid or oblate spheroid, is a surface of revolution obtained by rotating an ellipse about its minor axis. ### Geoid, the “real” Earth’s shape The geoid is the shape that the surface of the oceans would take under the influence of Earth’s gravitation and rotation alone. This shape is hypothetical extended through the continents.

Another definition of geoid is given by gravity potential energy. All points on the geoid have the same gravity potential energy; conventionally we assume a particular value of gravity potential energy as the surface of altitude “0”.

With this definition we can consider an important property: “the force of gravity acts everywhere perpendicular to the geoid, meaning that plumb lines point perpendicular and water levels par allel to the geoid“.

From classical physics we know that gravity is proportional to the mass. The gravity potential is also proportional to the mass distribution over the Earth; the gravitation filed is neither perfect nor uniform. This is due to non-uniform distribution of the masses: magma distributions, mountain ranges, deep sea trenches, and so on.

The water is a liquid, and its level is influenced by gravitational field too, the water on the Earth not be the same height everywere. The water level would be higher or lower depending on the particular strength of gravity in that location.

We have seen that a precise definition of the Earth’shape is just a litte bit complicated! Geodesy is the science that study modeling of geoid.