TL;DR: This post will add nothing to your life.. just ranting 😛

Back in the old days, people thought that the earth was flat. Then, they started to realize that in lunar eclipses, the shape of the of the Earth’s shadow, projected on the moon, is always circular, which means that the Earth must be a sphere. Now that was was a huge achievement but was still not very true.

After a while, Newton explained in his infamous book, the Principia, that the Earth’s shape would be flattened at the poles (i.e. Ellipsoid) by centrifugal force! Now that was very smart of him! However, since he wrongly assumed that Earth’s mass is evenly distributed, the flatting was a bit too much (f=1/230, while the true value is 1/298.28).

Now we have more than one Ellipsoids, so you better be careful when you see the three musketeers ( φ , λ , h ) without specifying in which Ellipsoid!

The ellipsoid surface is not level – therefore a distance above the ellipsoid is not really a height at all. The reference surface that is everywhere
level is the Geoid.

Later, we realized that the Earth’s mass is not evenly distributed and it is very irregular and complex in shape. therefore, more complicated mathematical model was needed. the Geoid.

Geodesy VS Cartography:

One can define Geodesy as the science of measuring the 3D shape of the Earth as well as its orientation in space. To refer to a measured location on Earth’s surface, the geodetic co-ordinates ( φ , λ , h ) are used.

These geodetic co-ordinates are:

• φ: the angle between the horizontal plan and the point’s ellipsoidal normal (we will talk about the ellipsoidal normal later)
• λ: the angle between the vertical plan passing through the Greenwich meridian and the ellipsoidal normal
• h : the distance between the point and the ellipsoid

Cartography, on the other hand, is the science of re-presenting this 3D shape into a 2D plot (i.e. a map). To refer to that measured location, a cartographer need to convert the geodetic co-ordinates ( φ , λ , h ) to the horizontal rectangular landscape co-ordinates ( x , y ).

A hacky way to do this conversion is to use the co-ordinate pair ( φ , λ ) directly after multiplying them by a scale . The distance between degrees of latitude is 110.567, 110.948, 111.699 km, at the equator, Tropic of Cancer/Capricorn and the poles, respectively. which means that roughly speaking each minute is less than 2 km (cool, hah?).

 x = S*φ 
// where S is the scale and equals to 111 km.

The distance between degrees of longitude varies greatly depending upon your location on the globe as the length of 1 degree of longitude is 111.321 km at the equator while it goes to zero at the poles. So, one can interpolate between these two values based on the location as follows:

y = S*cos(φ) 
// where S is the scale and equals to 111 km.
// cos(90)=0 and cos(0)=1

Yet, a better way to so this conversion (from 3D to 2D) is by using map projections which is not ideal either but way better than this hacky method.