A sphere is a super common type of surface. Can we make a sphere out of a list of two coordinates, like before?
Here's a coordinate system people use on this weird planet called "earth", made of a line-circle pair. The first coordinate, "latitude", controls how north/south the point is, and the second coordinate, "longitude", controls how east/west the point is. There we go - we've got a coordinate system, right?
Unfortunately, there's a problem with this coordinate system. Can you see it?
To see the problem, it might help to look at the the circle of all possible points with the same latitude (north/south) coordinate.
The problem is: at the north pole, that circle of all possible east/west values shrinks into a point!
At first glance, this might not seem that important. Why does it matter if our line-and-circle coordinate system shrinks the circle into a point? But this is a sign of something worse.
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Imagine we're using this coordinate system to fly an airplane around the earth's surface. If you're the plane's pilot, you probably want to know the plane's speed (so you don't crash) and direction (so you fly the right way).
But when we fly over the north pole, our east/west coordinate suddenly teleports!
Teleporting is pretty bad because it means we can't measure speeds and directions. Teleporting means changing position in zero time - or in other words, moving infinitely fast. But the plane clearly isn't moving infinitely fast; it's moving slowly and smoothly. Somehow our coordinate system can't measure this plane's speed without ugly infinities appearing. Something's gone horribly wrong!
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One reason we ran into infinities is that in this coordinate system, more than one list of coordinates represent the same point on the sphere. For example, every single list of coordinates whose first coordinate is fully up, no matter the east/west coordinate, corresponds to the north pole.
When multiple different coordinates secretly represent the same point, that's usually not good. For example, if we used this coordinate system to draw a map, it would look super distorted around the north and south poles. Personally, I don't like being lied to - if we draw two different points on a coordinate system, they should always represent two different points on the surface. I guess we can't make a sphere out of pairs of points on simpler shapes without running into ugly infinities.
...but what if we use more than one coordinate system?
North Pole Coordinate System
Equator (Polar) Coordinate System
South Pole Coordinate System
We saw one coordinate system can't cover the entire sphere unless some points are represented by more than one pair of coordinates. But a coordinate system doesn't have to cover the entire sphere. If one coordinate system can't cover a certain area without representing a point twice, we could create a second coordinate system, which only represents that point once, to cover that area!
Let's add two more overlapping coordinate systems centered at the north pole and south pole to our our original coordinate system (we'll move it away from the poles, too).
Now we can describe everywhere on the sphere using at least one coordinate system. And if we want to reach somewhere that isn't covered by one coordinate system, we can use the overlapping areas to switch to another coordinate system and keep going.
This is great: we can still reach every point on the sphere using at least one coordinate system. And we fixed the problem - on each coordinate system, different sets of coordinates always give different points on the sphere. Perfect!
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