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We can pose the problems as follows:

A coordinate system is a theoretical definition,

it is composed of an origin, and axes with direction in space.

If we look at Earth we can naturally say

that the axis of rotation of Earth already provides a referential direction

and equator also provides a reference plane

for a coordinate system.

The question is: how do we achieve a system

knowing that Earth is so large and knowing that the surface

thereof, the different tectonic plates move.

The first part of the solution is to find a geometrical shape,

which best fits Earth.

In this case, we will choose here an ellipse

that we rotate on the main axis,

thus we have an ellipsoid of revolution, which will give the mathematical form

that represents Earth.

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This ellipsoid have a large axis <i>a</i> and a small axis <i>b</i>,

and for Earth, <i>a</i> is equal to 6377 kilometers.

and <i>b</i> is equal to 6355 kilometers,

a difference of about 22 kilometers,

so, it's therefore relatively small compared to the total size of Earth.

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For the longitude, like for the sphere,

it is the angle between the prime meridian, here, and the meridian,

which passes through the point of interest.

I have here my longitude.

Finally, I have the height on the ellipsoid...

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that I find here on my figure, along the normal of the surface.

A coordinate system is a theoretical definition.

To use it, we have to complete the system

through a framework of coordinates and I have here an example

of the measurement framework in Switzerland with one materialization

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The second issue with geodetic references

concern the vertical dimension.

In fact, the gravity field affect many

human activities, for example in hydraulic constructions.

What references do we use for altimetry.

I take in this example here, three cases of figures:

a simple object, a building, and I draw, here, the vector g

representing the gravitational force of the building.

So we have a reference here, simple, unique, for a specific object.

I will now move to a larger context

in a portion of a territory, where I may be next to a mountain

a vector here,<i>g1</i>, and at the edge of a lake, here,

I have a vector here, gravity <i>g2</i>.

We can already here pose the question: is <i>g1</i> parallel to <i>g2</i>?

Do we have the same vertical reference in these two parts of the territory?

If we look at a global level, it is evident that if I am here

on the American continent or even in this region here in Europe,

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the <i>g</i> here, <i>A</i>, and the <i>g</i> here, <i>E</i>, is evidently not parallel.

The solution to this issue, the vertical dimension pass through

a reference surface in physics is called the geoïd.

We can imagine the geoid like the ocean surface mean,

which is extended under the continents.

We can draw the geoïd here...

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which is our surface, here, the reference.

It is an equipotential gravity field

and it is our reference here, zero for our altitudes.

Above the geoïd, I have my surface, here, topographic.

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And the altitude here, at a point <i>A</i> will be on the vertical line here, of my geoid

with this height relative to the physical surface

that we call here the altitude.

If I take a point <i>B</i> here, that I descend down here

to the reference surface, I have here, and altitude <i>HB</i>.

Knowing that <i>A</i> and <i>B</i>, the direction of the gravity field

is not necessarily parallel.

As the Earth is not a uniform solid,

there are masses with different densities,

the reference surface, the geoïd, will vary in space.

In this image, we see Earth, with its correct form,

namely this reference surface, with on the other hand, bumps,

with here, for example, a little over 80 meters,

and then we have pits here, approximately 100 meters deep.

We talk here about the geoid undulations, that should not be neglected,

in our altimetric reference model.

The geoid is influenced by the surrounding masses.

We can see in the left image, a typical landscape

with a lake, mountains, and it is clear that

the masses will influence the position of the geoïd.

We see in this example here, at the first location, 1, with the lake

we have here a density of masses that is lower

and, in this case, the geoïd would go down slightly.

In the second case, we are in the presence of a mountain

and here, we have one density, which is relatively strong,

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What is the relationship between the geoid, the physical surface,

and the surface of the mathematical reference, the ellipsoid?

We mainly define two geometric quantities.

The first, that we call the geoïd separation,

which means the separation between the two surfaces.

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In the example here, I have a dotted line ellipsoid and a full line geoïd,

so here I find my spot height.

The second geometric element is the angle that creates the vertical

on the surface, do the geoïd, with the normal of the reference surface,

the ellipsoïd.

We have here what we call the deviation of the vertical,

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The relationship between geoid and ellipsoid is something that is documented

in the different offices of topography.

We have this spot height which separate geoïd and ellipsoïd

and finally, what interests us for our topographical work

is the usual altitude, which is equal in this case

the height in the ellipsoid minus the spot height

<i>h</i> is the height of the reference surface.

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In this way we may set up the geoid map,

whether it is on a global or local level.

We see here in this example taken from the GOCE sattelite,

so a mission by the European Space Agency,

which ended in 2013,

we see here this example of a geoïd world map.

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If we look on this map, we see for example

that the region here of Geneva have a geoid separation

of about two meters.

And in the east of Switzerland, in the region called "les Grisons",

I see that I have about four meters of geoid separation.

So we see here a little bit of the geoid amplitude, its variation

throughout Switzerland.

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It exists in multiple geodetic references.

In general, we consider the distance between the reference surface,

the ellipsoid, and the surface level, the geoïd.

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We search to minimize this gap and whether we want a model

for all of Earth, we apply here a global ellipsoid,

or as well a local ellipsoid if we are interested in

one portion of the territory.

Thus we have these two categories, the worldwide or gobal systems,

and the national or local systems.

Here we have the example of the international system, ITRS,

with an ellipsoid, GRS80, and then for the Swiss system,

we have CH1903+ with its ellipsoid of Bessel.

Attention, in this figure the geometry is greatly exaggerated

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to illustrate this principle.

We have for example here, between the center of the global ellipsoïd

and the center of the local ellipsoïd, only a few hundred meters.

This is not at all the scale shown here in this figure.

To summarize this part on geodetic references,

we recall that planimetry and altimetry are two different concepts.

We define a mathematical reference, the ellipsoid,

and we define a physical reference for altimetry, called the geoid.

Each country has its own geodetic reference associated with a framework,

namely a series of materialized points, and known coordinates.

Then, when receiving a set of coordinated,

we will always have to pose the question: what is the geodetic reference

hiding behind?