Projecting a sphere on a plane

Did you ever play with an orange peel? I did it a lot when I was a child, often pressing them in the hope to flatten it almost perfectly. It's a hopeless challenge, but kids are stubborn and ambitious. Many years later, I found a similar analogy in a geography book. It was about cartographic projection and used an orange as a model of the Earth. If you think of the orange's peel as the Earth's surface, it is suddenly clear why you can't have a planar representation of Earth's surface without a great amount of distortion.

All the maps you will ever find are on a plain paper sheet. Curved digital screens are quite uncommon in GeoGeek's nests. So, how do cartographers represent a curved surface on a plane? This is done by means of a mathematical operation called projection. Consider the following image:

Indeed, there are several different projections developed in the last few centuries by cartographers and mathematicians. There is no mathematical method to transfer a sphere or an ellipsoid to a two-dimensional space without distortion. Hence, projections modify the data and include some deformations about lengths, areas, or shapes you can observe and measure on maps.

We can classify projections according to the geographical features and properties they preserve, as shown here:

• Conformal projections preserve angles locally. Meridian and parallels intersect at 90-degree angles.
• Equal-area projections preserve proportions between areas. In a map with equal-area projections, each part has the same proportional area as the corresponding part of the Earth.
• Equidistant projections maintain a scale along one or more lines, or from one or two points to all other points on the map. Lines along which the scale (distance) is correct are of the same proportional length as the lines they refer to on the globe.

It is important that you understand there is no best projection; choosing one for your map is a trade-off. According to the portion of the earth's surface, the map that you are designing will contain and/or use the projections that suit best. Let's explore some widely-used projections.