5 ms·

You can always use Euler's angles. Well, maybe not quite _always_...

dheera 3y ago
Euler angles irk me too because latitude rings are parallel to each other but longitude rings aren't, one runs from 0 to 2pi, anther runs from -pi to pi, another runs from 0 to pi, it's a mess. I wish we could find a system with better symmetry and consistency without being overspecified.
- dahart 3y ago
  There is no such system. https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
  - dheera 3y ago
    I mean, one could create an "east pole" and a "west pole" on the Earth and have latitudes behave like longitudes ...
    - seanhunter 3y ago
      You couuld also have an East pole and a West pole and have great circles for both latitude and longitude. It's not clear eitherr of those would be preferable to the system we have now.
    - dahart 3y ago
      It took me a minute to understand what you mean, but think for a minute about the disadvantages and inconsistencies of your idea, because this seems like it would be worse in every possible way for Earth coordinates and navigation, aside from the imagined symmetry. Smart people have already considered this idea, and it hasn’t caught on. Why? - First of all, this is minor, but you cannot call them east and west poles, because the east one is only to the east from one side. One the other side, the “east pole” would be to the west of the “west” pole. - Bigger problem but related: traveling along latitudes is no longer traveling north or south, and no longer traveling in great arcs. When you stay in one latitude in your system, you have to know where you are in order to know what compass heading you’re traveling (or vice versa - if you start with a heading, you don’t know whether your longitude or latitude is either increasing or decreasing), it cycles through all direction on the compass without passing the north or south pole. - That means that latitudinal navigation by the stars is no longer an option, and latitudinal travel by the compass (!) is no longer an option either. Your east-west-pole idea would require GPS for sailing. (Might be technically fine today, but should be pretty obvious by now why earth coordinates could never have evolved this way and would be extra bad for pre-computer navigation.) - You lose the property that longitude lines and latitude lines are perpendicular to each other on the surface - except at the two non-poles - the two spots on the axis in between the 4 “poles”. Longitude and latitude in this system are no longer orthogonal, and in some places you can specify ways to travel from one longitude to another via either longitude or latitude, in other places you can’t. - You add two more singularities to your coordinate system. And now you have a 3-dimensional coordinate system, but only 2 axes. Why have 4 poles and not 6? That feels asymmetric somehow. How do you pick the location of these east & west poles (and what would you call them instead of east & west?) - Finally, think about how you’d draw a map. If you get rid of the north-south axis, then which way is up on the map? How do you flatten your coordinate system and put it on a 2d rectangle? Do you want the lat/long lines to look straight and perpendicular, or faithfully represent the local surface while letting lat/long lines curve all over the map? Could you actually do either of those things with this system? The existing solutions aren’t great, but it seems like this would be even worse. There are very good reasons, for example, that nobody uses a 4-pole system for texture-mapping spheres.
- hgomersall 3y ago
  Check out the exponential representation of rotations in geometric algebra.

Euler angles irk me too because latitude rings are parallel to each other but longitude rings aren't, one runs from 0 to 2pi, anther runs from -pi to pi, another runs from 0 to pi, it's a mess. I wish we could find a system with better symmetry and consistency without being overspecified.

dahart 3y ago
There is no such system. https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
- dheera 3y ago
  I mean, one could create an "east pole" and a "west pole" on the Earth and have latitudes behave like longitudes ...
  - seanhunter 3y ago
    You couuld also have an East pole and a West pole and have great circles for both latitude and longitude. It's not clear eitherr of those would be preferable to the system we have now.
  - dahart 3y ago
    It took me a minute to understand what you mean, but think for a minute about the disadvantages and inconsistencies of your idea, because this seems like it would be worse in every possible way for Earth coordinates and navigation, aside from the imagined symmetry. Smart people have already considered this idea, and it hasn’t caught on. Why? - First of all, this is minor, but you cannot call them east and west poles, because the east one is only to the east from one side. One the other side, the “east pole” would be to the west of the “west” pole. - Bigger problem but related: traveling along latitudes is no longer traveling north or south, and no longer traveling in great arcs. When you stay in one latitude in your system, you have to know where you are in order to know what compass heading you’re traveling (or vice versa - if you start with a heading, you don’t know whether your longitude or latitude is either increasing or decreasing), it cycles through all direction on the compass without passing the north or south pole. - That means that latitudinal navigation by the stars is no longer an option, and latitudinal travel by the compass (!) is no longer an option either. Your east-west-pole idea would require GPS for sailing. (Might be technically fine today, but should be pretty obvious by now why earth coordinates could never have evolved this way and would be extra bad for pre-computer navigation.) - You lose the property that longitude lines and latitude lines are perpendicular to each other on the surface - except at the two non-poles - the two spots on the axis in between the 4 “poles”. Longitude and latitude in this system are no longer orthogonal, and in some places you can specify ways to travel from one longitude to another via either longitude or latitude, in other places you can’t. - You add two more singularities to your coordinate system. And now you have a 3-dimensional coordinate system, but only 2 axes. Why have 4 poles and not 6? That feels asymmetric somehow. How do you pick the location of these east & west poles (and what would you call them instead of east & west?) - Finally, think about how you’d draw a map. If you get rid of the north-south axis, then which way is up on the map? How do you flatten your coordinate system and put it on a 2d rectangle? Do you want the lat/long lines to look straight and perpendicular, or faithfully represent the local surface while letting lat/long lines curve all over the map? Could you actually do either of those things with this system? The existing solutions aren’t great, but it seems like this would be even worse. There are very good reasons, for example, that nobody uses a 4-pole system for texture-mapping spheres.
hgomersall 3y ago
Check out the exponential representation of rotations in geometric algebra.