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On 6/13/2021 at 4:25 AM, Kaito said:

By the way, I found my Math IA about celestial navigation so if anyone wants to read it just send me a message with an e-mail or some other preferred method of file transfer and I'll send a copy.

In case anyone was wondering, the elegantly worded IB-style title is:

Mathematics Exploration – How can celestial navigation be mathematically modeled to find one’s location on Earth when time and the position of celestial bodies are known?
 

hi, I'm in N22 and I'm also doing my IA on celestial navigation! I would love to look at your IA for some inspiration. If you still have it please email me at [email protected]

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On 5/26/2021 at 10:13 PM, Kaito said:

I'd love to tell you more about the topic and my IA and I could even send a copy of it if the IBO and your school both allow it. If you want specific help with the topic feel free to message me and I can even provide some quick online tutoring on the topic if you would like. I'm going to look for my Math IA, I know I have it saved somewhere but it's been a couple of years so I don't remember everything. Here's what I remember.

Basically the way navigation works is you have 3 celestial objects, usually stars or in the case of GPS, satellites, and you are able to measure angles and find your position based on your distance to the projection of 3 different points on the Earth, assumed to be a perfect sphere.

I don't think this vector projection was covered in Math HL in IB and the syllabus has probably changed anyway since I graduated so I'm going to explain vector projection briefly. You can think of it as a shadow. An example of vector projection used in both Math SL and Math HL would be projecting a vector onto the x axis using cos(θ) and y axis using sin(θ). You can take a 3D object and its shadow on the ground is a projection of that 3D object onto a 2D surface. Similarly, you need to project the position of a star in longitude and latitude onto the Earth, a sphere. This point is defined such that if you were at that point and looked straight up you would be directly underneath the star. You can draw a line from the center of the earth to the star and this point is the intersection of that imaginary line and the surface of the Earth.

You start by measuring the altitude of a star. If I recall correctly that is the angle between the horizon and the star. This is not the same as height, it is an angle. You can look up the positions of stars in longitude and latitude at any time and date in specific books containing such data. When you measure the altitude and know the star's position on Earth, you can calculate your distance to that point on the sphere, using longitude and latitude. (Technically it's the projection of the star onto the Earth's surface but saying "the star's position on Earth" is easier and more intuitive). 

Once again, we are dealing with angles and not straight lines since the Earth is a sphere. You need to do this with three stars and your end result will (hopefully) consist of 3 circles defined by the stars' positions on Earth (measured using 2 angles, longitude and latitude) and your distance to the stars (an angle derived from the altitude). Your position is the point where all 3 of these circles intersect. It's worth mentioning that these aren't regular circles, they are circles projected onto the surface of a sphere, measure in angles from the center. You can represent these circular projections in several ways including polar coordinates and planes. 

Things can get difficult here depending on your approach and I don't remember exactly how I did it but you can take a plane and have it intersect a sphere. The points of intersection form a circle, much like the circles you would get from these calculations. This means that each circular projection that you use to measure distance from a star's position on Earth can be represented as a plane intersecting a sphere at an angle. You can model your circles as planes and solve them simultaneously using linear algebra. Alternatively you can look into using polar coordinates, you can probably skip the radius though so you would just need to represent the three circles using a few angles.

This last part can be a bit tricky and the exact method you use isn't important as long as you can represent these projected circles as equations and solve them simultaneously.

I'll try to provide a better answer once I've reread my IA. Feel free to contact me with any questions you may have.

Hi!

Did you calculate the angle between the horizon and the stars or were you able to avoid that? If you calculated, what was your procedure?

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