Back in the kindergarten, many of us have heard that the planet Earth is ball-shaped. And that’s what would suffice for a typical homie. Unfortunately, not everyone is a typical homie and people had a problem.

### What problem?

If we’re making a plan of the smallest town in your country, then there’s no point in caring about the curvature of the Earth. But how do you set an airplane’s route over an ocean? It’s different when flying over a perfect sphere and a flattened sphere! The longer the flight, the bigger the difference. Or with surface calculating. What would a Russian tsar say if he realized that, after long years of surveys, his motherland has been measured wrong?

## Alright, if not a ball, then what?

You could point out immediately that it’s not a ball! There are mountains and valleys. OK, be more specific, how big are they?

Let’s compare the height of Mount Everest with the diameter of the “ball”. It’s **1440 times smaller**. So… far too small for a typical homie to notice. On a half-meter globe, it’s slightly more than the thickness of a sheet of paper.

### So why drill down into this?

‘Cause it’s not everything. Come to think of it, the Earth is spinning. And since there is the centrifugal force, pointed **away from the center**, then it should “pull” the mass of the Earth **away from the center**. Meaning: widening it in the middle, on the equator. Well, right, but this isn’t a sphere anymore. It’s an ellipsoid of revolution. An ellipse in 3D, simply put, like a squeezed football. How much squeezed?

**42 km.**

This is the difference between two diameters: one, measured from the pole to the pole, and the second, measured at the equator. More than the height of Mount Everest, but on a half-meter globe that’s not much more than thickness of a fingernail.

### So why keep drilling down into this?!

‘Cause it’s still not everything! You probably didn’t realize that we quietly assumed that the Earth is uniform inside. And it’s not! The iron deposits have their own gravity, a bit different from mountain ranges and oceans. It has its effect on the gravitational force. It means that a weight hanging on a string always does point the vertical direction, but not necessarily the Earth’s center point! On the coast of Chile, with the Pacific on the one side and Andes on the other, it’ll be slightly leaning towards the mountains. That’s why the **geoid **was invented, although… no one will ever see it.

### Well, what’s a geoid?

It’s a concept, not something tangible. It’s a surface, almost identical to the surface of the planet, on which the gravitational force is everywhere equal^{1}. The thing is: in some places it’s under the mountains, in others it’s above the seas! It’s distanced from the ellipsoid^{2} by just several meters. It’s quite handy for geodesists, because it’s everywhere perpendicular to the vertical direction. And geodesists’ tools quite often are based on a plummet, hanging on a string, called a plumb bob. For example, steepness of a given terrain can be determined thanks to that.

The shape of geoid depends on gravity. You know what’s the conclusion? Stacking a heap of dirt changes its shape, because the heap has different gravity than a lawn below it!

### But what is that for?

A geodesist needs the geoid, because his appliance determines the vertical and horizontal directions. “The horizontal” is exactly the surface of the geoid! If you’re more or less on the sea level. And geodesists’ work is useful when tracing new roads, calculating the surfaces of cities, regions, forests, countries, designing buildings (building on a steep slope? better ask a geodesist, if it won’t slide down after heavy rains!), drilling tunnels, and so on, and so forth…

The GPS uses ellipsoid, because it’s accurate enough. If you wish to update your car’s position every second, you don’t have time to complicate things. The ellipsoid is precisely described by three parameters. Period. The geoid cannot be mathematically described, because… it’ll be changed by a flock of cows crossing the street. Of course, it can be described with great approximation (disregarding the gravity of cows, but not the gravity of the Alps). But this is far more complex to calculate.

And the oceanography! Look at this course of thinking: the water in oceans can float freely ⇒ the surface in the oceans is identical to ellipsoid ⇒ a planet covered completely in water is perfectly ellipsoid-shaped. Right?

**WRONG.**

Tides, sea currents, waves, the influence of atmospheric pressure, salinity, temperature changes – it all has some effect on the sea level. The surface of the ocean is not identical to the surface of ellipsoid… But if you could measure the difference between them, you could conclude a lot, for example about the subtle changes of sea currents, or you could forecast the weather. And indeed, there are satellites which measure the topography of the oceans with cosmic (pun intended) accuracy.

## Oh come on, it won’t be useful for me…

Which one? Knowledge about the shape of our planet? Maybe.

Measuring the surface of the East China Sea? Maybe.

Determining a terrain steepness? Well, if you build a house.

Length of a plane flight? In airlines, sure.

Drilling a tunnel under the Alps? During a train ride…

GPS? Hmmm… Alright, it’ll be useful.

So don’t be surprised if people mentioned at the beginning devoted their lives to measurements and calculations in order to measure the Earth. Let there be respect for geodesists!

Alongside with other measurers… But that will be another story about them.

^{1}To be precise: it’s not the gravity itself, but gravity combined with the centrifugal force.

^{2}What ellipsoid, you ask? That’s a tough one, because it needed to be calculated somehow. There were lots and lots of such measurements and calculations. Nowadays, the one agreed upon for the GPS in the 80s is in very common use.

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