If you don’t know what a Riemannian manifold is, well…you should probably go back and learn some math before diving into this tutorial.

So, what are geodesics? Well, they’re the shortest paths between two points on a curved surface like a sphere or a hyperbolic plane. But here’s the thing: in order to find these shortest paths, we need to use calculus and some fancy math stuff that makes our heads spin.

But don’t worry! We’re going to break it down for you step by step (or should I say “curve by curve”?) so that even the most mathematically challenged among us can understand.

First, let’s start with a basic example: finding the shortest path between two points on a sphere. Imagine we have a globe in front of us and we want to find the shortest distance (in terms of length) between New York City and Sydney, Australia. To do this, we need to use calculus to find the geodesic or “great circle” that connects these two points on the surface of the sphere.

Now, if you’re not familiar with calculus (or if your high school math teacher was particularly cruel), let me explain what a great circle is: it’s essentially an imaginary line drawn around the Earth that passes through both poles and connects any two points on its surface. This line forms a perfect circle when viewed from above, hence the name “great” (or “large”) circle.

So how do we find this great circle? Well, first we need to define our starting point let’s say New York City is located at latitude 40 degrees north and longitude 74 degrees west. To get from here to Sydney, Australia (which is located at latitude 33 degrees south and longitude 151 degrees east), we can use the following formula:

θ = arctan2(sin(Δlat) * cos(lat1), cos(lat1) * sin(lon1 lon2))
φ = asin(sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(Δlon))

where Δlat and Δlon are the differences in latitude and longitude between our starting point (New York City) and Sydney, Australia. The arctan2 function is used to calculate the angle between the two points on a sphere this is essentially what we need to find the shortest path or geodesic that connects them.

Now, let’s take a closer look at how this formula works:

– First, we calculate Δlat and Δlon using the following formulas:
Δlat = lat2 lat1
Δlon = lon2 lon1

– Next, we use these values to calculate two angles (θ and φ) that represent the direction of our geodesic. The angle θ is calculated using the arctan2 function, which takes into account both the x and y coordinates on a sphere. This ensures that our calculation is accurate even if we’re dealing with negative latitudes or longitudes.

– Finally, we use the asin function to calculate the second angle (φ) this represents the distance between our starting point and Sydney, Australia along the great circle. By combining these two angles, we can find the shortest path that connects New York City and Sydney on a sphere.

Geodesics on Riemannian manifolds may sound complicated, but they’re actually pretty simple once you break them down into smaller steps. And who knows maybe one day you’ll be able to use this knowledge to find the shortest path between two points in space-time (or at least your local grocery store).