Also, should we let out the orthogovores now? They're getting feisty and the humans look smug.
A few days ago I asked if there might be a mathematical solution for why a planet I'm making in an ongoing world building exercise is seemingly normal planet shaped to it's inhabitants, has a known radius and is therefore expected to and in actuality looks such that one large group of continents and it's surrounding islands encompasses almost the entire globe. In the story the inhabitants of this large grop of continents never sales east to reach western shores or west to reach eastern shores because ships that try never make it. It's assumed the reason is some kind of magicl or maybe just natural barrier having to do with rough seas or storms.
In actuality there is a whole OTHER half of the planet with civilizations of it's own and a very large ocean inbetween the two different gruopings of civilizations. Sort of like Afroeurasia and Turtle Island but with an ocean as large as the pacific on both sides. In the story inhabitants from one side eventually realize that their world is a sphere in hyperbolic 3-space with a gaussian curvature of about -0.6103 if you take the unit scale to be the radius of the planet. In hyperbolic geometry many types of geometry work exactly the same. Since people are on a much smaller scale than the unit where you can easily detect curvature the math and geometry of this world develops on a similar path to our own for millenia.
One of the main differenecs is that planet or near planet scale (or greater sized) curves in space work very different in their universe. The development of high speed powered flight happens but never becomes common because an airplane travling hundreds or thousands of miles is a bad idea if your civilization didn't yet realize they had the geometry of their planet all wrong! Similarly around the time powered flight is invented the technological building blocks for basic rocket based space travel exist. But the early satellite and space exploration missions go all wrong! Eventually everyone in the world realize their planet is a sphere in hyperbolic space as opposed to flat space or elliptic space. However, the group of continents and civilizations with a more cooperative and friendly mathmatical tradition figure it out centuries earlier and go out of their way to avoid contact with the other due to some early negative experiences bumping into sailors and war parties from the other side.
PRE-CONTEXT: If you dont' care about the context skip to the "QUESTOIN:" bit at the end.
CONTEXT: I have been working on world building project for some time. It's a planet such that the inhabits of a particular group of continents have found that when you sail from the eastern most edge of their known world in the westward direction you navigate almost fully around a globe. Similarly when you sail from the western most edge of their known world in the eastward direction you navigate almost fully around a globe. For that reason the inhabitants of this group of continents assumes they live on a spherical planet and that their world map is the entire planet. This world has a north pole and a south pole so for some time these people assumed, they live on a planet much like we do. However, there was the ongoing curiosity of ships that started at the eastern edge of the map and sailed eastward to reach the western edge of the map or vice versa. Those ships always dissapeared never to return or came back because they were "lost at sea" and could not find land despite the scientifically verified curvature of their worlds surface (measured by shadows and how sails on ships or mountains dissapear over the horizon).
The kicker is, there is another grouping of continents on this world with the exact same problem. And in the stories I'm writing around with this world building project. The people on these 2 different gorupings of continents come to realize that due to the higher dimensional geometry of their universe and/or world it can take as many as TWO great circles to circumnavigate their planet instead of just one =)
The topology of this world is otherwise exactly what you might expect from a planet in 3 spatial dimensions so their science and such has developed with many of the same findings as in our world. Some differences I've been thikning about to help facilitate this strange geometry/topology are that folks on one set of continents have mirror image cardinality to folks on the other set of continents and as you go across the surface of the planet from one globe to the other you spint through the 4th dimension probably unknowingly. But my topology and geometry skills are weak so I'm mostly just looking for a solution that fits at least maybe at leats one of the conditions in the Question below.
QUESTION:What sub-categories of math, topics, or concepts I can study to find a mathematically valid solution to this prob... keep reading on reddit ➡
If you dont know what I mean, search hyperbolica on youtube
a circle just above the equator and one just below
Are the lines highlighted in green the same length?
If so, wouldn't that make the 2 red small circles (Tropic of Cancer and Tropic of Capricon in the diagram) parallel to each other? If not, what is their relationship to each other?
Suppose you have 2 points on a sphere and each point observes the movement of the sphere in 2 dimensions but not the third, i.e. each point is blind to rotation about itself (blind to rotation about the axis that goes through it and the center of the sphere).
For example if one of the points were at the north pole then it would observe tilt in either cardinal direction but not rotation exactly about the north pole. Likewise, if the second point is somewhere at the equator then the second point would measure the normal rotation of the earth as it spins (1 dimension) which is what the first point couldn't measure, and the second point could also measure the "tilt" of the earth (2nd dimension). However, the second point would be blind in the third direction, namely if someone spun the whole globe around the axis that goes through that point to the center of the earth. (The Earth itself doesn't normally spin this way.)
To repeat myslef, the 2 axes of rotation that the north pole point would see is 1) the axis that goes from the center of the earth out the equator in one direction, 2) the axis that goes from the center of the earth out the equator at a different 90 degree angle, but it would be blind to 3) the axis that goes from the center of itself through itself.
So this should tell you what points of rotation each point can and can't observe.
Given this, then, can you reconstruct what the rotation of the sphere is in 3 dimensions given the 2 observed two dimensional quantities?
Thanks. Please let me know if I was unclear.
I'm a geology student and currently working on an independent research project wherein I need to use some trig and spherical geometry; maybe a tad bit of calculus too.
To try and better understand what Im trying to do I'll need to explain a tad bit of geology background:
The question I'm trying to ultimately work out is this: How much has a tectonic plate subducted at a specific location (a point) in the past 130 million years?
Subduction is where one tectonic plate collides with another and results in one of the plates being pushed down underneath the overlying colliding plate. I'll try to hopefully make this more relatable/understandable with an analogy: this process would be like being in a pool with your friend and you and he push two boogie boards (the tectonic plates) together. One of these boogie boards will end up dipping below the other and going underwater, aka being subducted. If you pushed your Boogie board at a constant velocity while your friend held his still, you could predict how much of your boogie board is underwater at a given time (relative to the position of his fixed board) with some basic math.
The math can be spiced up a bit by changing the angle you are pushing against your friend's boogie board. Instead of pushing directly against your friend's board, you push at an angle to his fixed board. Your board is still being pushed underwater, but now with a decreased velocity as you now have to resolve the magnitude of the velocity into separate components. Still not too bad.
Now, here's where I get confused. In my analogy above, this could be calculated using cartesian coordinates, however, the Earth is a Sphere and I dont believe the math I'd use to model "how much my boogie board has subducted in the past 130 million years" would be quite the same. Could someone help me out with the trig (& possibly multi-variable calculus) needed to calculate this?
I have this data to work with:
Velocity Magnitude (y-axis) vs Time (x-axis) (All these are relative to referenced point (lat, long) on the boundary of the subducting Plate)
Velocity Azimuth vs Time
Angular Velocity vs Time
Lattitude vs Time
Longitude vs Time
Velocity Colattitude vs Time
Velocity Longitude vs Time
Thank you very very much for your time and help!
Is it possible to find out where a image was taken only knowing the time it was taken, object height and shadow height? I wondered because there was a scene in the movie "G.I. Joe" where the good guys found the villain's secret base only having a picture of him standing in sun light.
I just started encountering stuff about spherical geometry (and other non-Euclidean geometry) in my current projects. My question is: how is spherical geometry and spherical coordinates related?
I used spherical coordinates (and other non-Cartesian coordinate systems) extensively in my engineering undergrad days for tackling heat flow. Although I haven't run into them in years, I think it will be fairly easy and quick to brush up on it. I think if I can know how spherical geometry (which I have no clue about) is related to spherical coordinates (which I know a lot about), it may save me some time in my learning journey.
I am a GIS analyst with 3 years in the field who is averagely competent in python and SQL(postgis). As my skills have grown I have found myself delving into complex distance calcualtions/trig/geometry (something like this:https://gis.stackexchange.com/questions/88484/excel-distance-calculation)
I have also been learning more and more about geodosy and custom coordinate systems for work, and unfortunately my math skills are quite poor. I haven't taken a math class since high school and I am a self taught programmer.
So I am wondering if there are any good resources(a class, book, lecture etc..) on where to learn trig/math/geometry that relates to GIS. I really want to understand angles, bearings, distances etc...
any input would be greatly appreciated