Absolutely! I can explain why the shortest path on a map of the Earth appears curved, and how this depends on map embedding.  I will also explain it as simply and intuitively as possible, but know that you can always ask me for clarifications anytime. 

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##### Introduction - Geodesics
The nearest thing to a straight path in a curved space, such as in Riemannian Geometry, is called a geodesic. Technically speaking, a geodesic is defined as the shortest (or longest) path between two nearby points.

A geometric plane is an example of a two-dimensional flat space, on which the geodesics are lines. The surface of the earth is a two-dimensional curved space. A geodesic on the earth is called a great circle. The equator is a great circle. So is any other circle on the globe whose center coincides with the center of the earth. (The term "great circle" comes from the fact that these are the largest circles you can draw on the globe.) As the geodesic is the shortest path between two airports, this is the route an airline navigator will tell the pilot to fly along. For instance, you could fly from New York to Madrid by following your compass for 3,707 miles almost straight east, along their common line of latitude. But you can get there in 3,605 miles if you fly along a great circle, heading first northeast, then gradually turning east, and then southeast. The appearance of these two paths on a map, in which the surface of the globe has been distorted (flattened out), is deceiving. When you move "straight" east, you are not really moving straight, at least not straight in the sense of the most direct path, the geodesic.

Now let's talk about map embeddings, also known as map projections.

##### Flattening the Earth
It’s impossible to flatten the Earth without distorting it in some fashion. Consider an orange peel: if you want to try and lay it flat, you have to stretch it, squash it, and tear it. Likewise with the Earth—if we want to make a map, we need to distort the Earth’s surface to flatten it. The good news is that map projections allow us to distort systematically; we know exactly how things are being stretched or squashed at any given point. We have many different map projections because each has different patterns of distortion—there is more than one way to flatten an orange peel. Some projections can even preserve certain features of the Earth without distorting them, though they can’t preserve everything.

##### Projection Properties
We often talk about map projections in terms of the ways in which they distort or preserve certain things about the Earth, which we call projection properties. There are four main properties:

- Area — Some projections distort areas (e.g., Mercator projection maps)
- Form — Some projections distort the “form” of features (e.g., Azimuthal Equidistant projection)
- Distance — Most projections distort distances (e.g., Equirectangular projection)
- Directions — Sometimes a straight line isn’t the shortest path!

Let's review those one by one, building up to the last one.

##### Area — Some projections distort areas
###### *Example: Mercator projection maps*
Notice in [this image](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cdefe189b06a7097d3_projections1.jpeg), how Greenland is about as big as South America on a Mercator projection. In reality, South America is eight times larger than Greenland. The Mercator projection doesn’t preserve area correctly, especially as you get closer to the poles. On the other hand, one kind of projection that doesn’t distort area is the Cylindrical Equal Area.


Now, notice [here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cd5dad296baf720c2d_projections2.jpeg) how Greenland looks the right size as compared to South America. Projections which preserve areas are called *equivalent* or *equal-area* projections. A map projection either preserves areas everywhere, or distorts it everywhere. This is an all-or-nothing property.

##### Form — Some projections distort the “form” of features
###### *Example: Azimuthal Equidistant maps*
On the projection [here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cd48fff9427e6833c2_projections3.png), look at how Australia, on the right, is unrecognizable, and New Zealand is stretched out into a ring around the left edge of the map. This projection does not preserve the “look” or the “form” of places. It stretches or twists or squashes them, instead. Contrast that with a Lambert Conformal Conic ([here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cd734558b18471adaf_projections4.png)), on the other hand, which preserves the general form of the landmasses.


Projections like this are called *conformal* projections. Under the hood, this property is actually a little more complex: conformal projections actually preserve local angles. But what that boils down to for cartographers is that places look more like themselves. In the example [here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cef7b8e0331a6544a2_greenland.png), Greenland is shown as it appears on three conformal projections (top row) and three non-conformal projections (bottom row).  Notice how the conformal projections keep Greenland looking Greenlandy. 

Like equal-area, this property is all-or-nothing; your projection either preserves forms everywhere on the map, or it doesn’t preserve them anywhere.

##### Distance — Most projections distort distances
###### *Example: Equirectangular projection*
A trip from Madison to Buenos Aires is much farther than a trip from Madison to Madrid. But on an Equirectangular projection ([shown here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448ce033e7478297a831d_equirectangular.png)), both of those trips looks like they’re the same length, because this is a projection that does not preserve distance. On the other hand, the Azimuthal Equidistant projection shows distances in the correct proportion, as you can see [here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cebd3d6b3e8e44b6b6_ae.png).


There’s a catch, though. While we have map projections that can preserve areas or form everywhere on the map, there isn’t one that can preserve distances everywhere. 

> The properties of area, distance, and form are mutually exclusive. If you have a map projection that preserves one, it will distort the other two.

##### Directions — Sometimes a straight line isn’t the shortest path!
Now we can get back to talking about geodesics and why a straight line on a map isn't always a straight line on a globe.  This time, let's use Istanbul instead of Madrid, as it is a little father out and easier to see the curvature starkly.

New York City and Istanbul are on nearly the same line of latitude, about 41ºN. That means that if you head due east on a straight line from New York, you’ll reach Istanbul. But that doesn’t mean that this is the shortest distance between the two cities.


In [this image](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448cebd3d6b2aec44b6b5_projections6.jpeg), there’s a line which shows the straightest, simplest path between New York and Istanbul, which is simply to point yourself east and start flying. But the curved line above it shows the way you should go if you’d like to travel the least distance while getting there. Because the Earth’s surface is curved, the shortest paths around it are curved, too. This can be a bit confusing, but makes more sense if you try it yourself: find a globe and place a piece of string on it. Pin one end to New York and one to Istanbul, and pull the string taut. You’ll notice that the string covers the exact same path as the curved route in the map above. We call these curved shortest-distance paths *great circle* routes. On the other hand, a path like the straight line, where you keep yourself pointed in the exact same compass direction the whole time, is called a rhumb line or a *loxodrome*.

Some projections, like the Mercator above, show loxodromes as straight lines. They make air and ship navigation easy, because you just have to draw a straight line, point your ship in that direction, and start sailing. Other projections show great circle routes as straight lines, making it easy to figure out the shortest distance between two places. The Stereographic projection is one of these, seen [here](https://uploads-ssl.webflow.com/5f6a4786fa53db61032919f9/5fa448ce48fff9271d6833c3_projections7.jpeg).


Now the straight line is the great circle, and the curved one is the loxodrome. These lines are the same as in the Mercator above, but the projection changes their appearance.

When a projection preserves great circle routes as straight lines, we call it an *azimuthal* projection. Unfortunately, much like the equidistant projections, it only works for one point at a time. In the Stereographic above, the projection is centered on New York. Only straight lines coming into or going out of New York will be great circles. A straight line between Madrid and Casablanca won’t be.


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I hope this was a gentle introduction to map projections in explaining the shortest path on a Riemannian Geometry, such as the surface of the Earth.

Do you have any questions, or is there something else I might be able to help you with?
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