Any Math Majors in the House? Help Wanted.

[ender_wiggin]

Longitude is a fairly arbitrary concept which would probably not exist in the world you've created, IMO. Latitude is easy, it's distance from the equator.

That's tough when the equator is a curve for which I don't have the equation.

 

[Flatus Maximus]

That's tough when the equator is a curve for which I don't have the equation.

Now I'm confused again: The equator isn't the usual equator? If it helps, I'm imagining something like Earth where there's only the North + South America supercontinent, except it twists around the globe in roughly the shape of a loxodrome. The North and South Poles and the equator are where they should be.

 

[ender_wiggin]

Now I'm confused again: The equator isn't the usual equator? If it helps, I'm imagining something like Earth where there's only the North + South America supercontinent, except it twists around the globe in roughly the shape of a loxodrome. The North and South Poles and the equator are where they should be.

Your understanding is correct.

The equator is still the normal equator, but on the map the equator will appear as a curve. This is because we're taking a spiral and wrenching it into a straight line (making actual straight lines like the equator look curved). Sort of like: the shortest path between Boston and Beijing is a straight line, but on a Mercator projection (a normal world map) it looks like a curve.

Also, I have since realized that the shape I'm looking for is not a loxodrome (at least not exactly). I'm looking for a curve that crosses each longitude and latitude exactly once. A loxodrome never reaches either pole but instead continues to wind round and round. The shape I'm most likely looking for is a spherical helix, but the math unfortunately is more complicated for such curves.

EDIT: Nope, not true. I don't think such a curve I'm looking for exists, because the north and south pole by definition are all longitudes. Back to the drawing board.

 

[malkav666]

OP you should check out PROJ.4 It can be found at:

PROJ.4

Its open source and can do all sorts of projections.

JH labs makes a java version that can do all sorts of projections (from azimuthal to Wagner projections, probably about 50 in all). It can be found at:

Java Map Projection Library

There may even be better free implementations of the library as it is open source. I know there is an Android version close to finished and an applet that ties in with Microsoft Virtual Earth.

It can really give you a great idea of how to place your Meridians on your map regardless of how you would like to project the map. It will also give you a great benchmark for how projecting your map in various ways will distort the land masses.

Good luck with your project.

love,

malkav

 

[Flatus Maximus]

EDIT: Nope, not true. I don't think such a curve I'm looking for exists, because the north and south pole by definition are all longitudes. Back to the drawing board.

I wouldn't get too caught up in the details here. (And I say that as a very detailed-oriented mathematician.) Keep in mind you want a map that the people of your world would come up with naturally. I would suggest that you do a Mercator where, instead of using an axis passing through the North and South poles, you pick the axis passing through the two points, one in the Northern Hemisphere and one in the Southern Hemisphere, furthest from any land mass. This way all the spatial distortion is in the ocean, far away from land. You're going to get distortion anyway, so might as well put it in the ocean, far away from land.

 

[ender_wiggin]

You know I was thinking of resorting to that. It would work, I would still get the effects I want, but the poles wouldn't be _exactly_ where I wanted them.

Then I found a map projection (or rather, somebody on cc pointed it out to me) called the Space Oblique Mercator Projection. It was developed for mapping the path carved by a satellite that orbits from pole to pole and back again. Since the Earth rotates under the satellite, the path traced is pretty much exactly the type of curve I'm looking for.

The math to implement this thing, however, is horrendous. So it will take me a while to see whether this method is fruitful or not. In the off chance that anybody here has insight into that, I'm all ears. And easier alternatives are appreciated as well.

 

[Flatus Maximus]

...Space Oblique Mercator Projection...

Sounds cool, and might lead to a fun project for my multivariable calc students this fall. I'll google around for it, or feel free to send me a link, if you have one/some.