<|im_start|>system
You are a helpful AI assistant.<|im_end|>
<|im_start|>user
Explain the concept of Riemannian Geometry in a simple intuitive way<|im_end|>
<|im_start|>assistant
Riemannian Geometry is a branch of mathematics that deals with the study of curved spaces. It was developed by the mathematician Bernhard Riemann in the 19th century.

Imagine you're walking on a flat surface, like a soccer field. You can move in any direction without any constraints. The distances between two points on the field are straightforward to measure using a ruler or by simply walking the path. This is the geometry we are most familiar with, called Euclidean geometry.

Now, let's imagine a different scenario. Picture yourself walking on the surface of a sphere, like the Earth. As you walk, you'll notice that the shortest path between two points is not a straight line, as it would be on a flat surface. Instead, it's a curve that follows along the surface of the sphere. This curved space is an example of what Riemannian Geometry studies.

In Riemannian Geometry, mathematicians investigate spaces that have intrinsic curvature, just like the surface of a sphere. These spaces can be more complicated than simple spheres, but the essential idea remains the same: the shape of the space affects how distances are measured and how paths between points are determined.

One key concept in Riemannian Geometry is the notion of a metric. A metric is a way of measuring distances on a curved space. It assigns a length to each infinitesimally small piece of the space, allowing us to calculate the total distance between two points. The metric takes into account the curvature of the space and provides a framework for understanding the geometry of the curved space.

Riemannian Geometry has applications in various fields, including physics, computer science, and even in understanding the universe itself. It helps us understand how objects move and interact in curved spaces, and it plays a crucial role in Einstein's theory of general relativity, which describes the curvature of space-time due to the presence of mass and energy.

In summary, Riemannian Geometry is about studying curved spaces and understanding how their geometry differs from the familiar flat spaces of Euclidean geometry. It explores the concept of distance, paths, and shapes in these curved spaces and has broad applications in different areas of science and mathematics.<|im_end|>
<|im_start|>user
Explain in a simple intuitive way why shortest path between two points on a sphere is not a straight line<|im_end|>
<|im_start|>assistant
On a sphere, such as the Earth, the shortest path between two points is not a straight line because of the curved nature of the surface. Let's imagine you are trying to find the shortest path between two cities on Earth.

If you were to draw a straight line on a flat map, it would seem like the shortest route. However, when you consider that the Earth is a three-dimensional object, that straight line would actually pass through the Earth's interior, which is not physically possible. We need to stay on the surface of the sphere.

To understand the concept of the shortest path on a sphere, think about it like walking on the surface of a globe. If you were to walk from one point to another, you would follow a curved path along the globe's surface. This curved path is the shortest distance between those two points on the sphere.

To visualize this, imagine taking a piece of string and placing it on the surface of a globe, connecting the two cities. You will notice that the string curves along the surface, mimicking the shortest path.

This curved path is known as a great circle. A great circle is a circle formed by the intersection of the sphere and a plane passing through its center. It is the largest circle that can be drawn on a sphere.

So, to find the shortest path between two points on a sphere, we need to follow the curved path along the great circle, rather than a straight line. This is because the curvature of the sphere necessitates it, and it allows us to stay on the surface of the sphere while finding the most efficient route.<|im_end|>
<|im_start|>user
Thanks.  Now explain the part about the shortest path on a map of the Earth being curved, how this depends on map embedding.  But like before, explain it in a simple and intuitive way.<|im_end|>
<|im_start|>assistant
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. 

---

##### 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.


---

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?<|im_end|>