ender_wiggin

08-16-2011, 08:25 PM

I want to draw latitude and longitude lines on a map I'm making for my homebrew, but it's a pretty unique map. I suspect I will need the help of a real mathematician.

The sole landmass in my setting is an archipelego that begins at the north pole and runs to the southpole. It does so however, not using the shortest path but one that winds around the globe such that it completes exactly one rotation by the time it reaches the south pole (a loxodrome aka rhumb line). I want to draw a rectangular map such that the horizontal midline of the map corresponds to the axis of the archipelego, which means I'm projecting a curved structure onto a straight 2D plane. In other words on the map the archipelego is rendered as a straight line, with the poles at either end of the map, but the archipelego actually follows a rhumb line and thus is curved. Due to this excellent site, I don't have any trouble getting the art right.

Now, the hard part about this is figuring out where the longitude and latitude lines go. I have an idea in my head but I want to be precise.

I was an engineer in undergrad, so I have the background required to figure this out, but it's been a long time since I've done any of that. I need someone who knows what they're doing to point me in the right direction. It took me about 4 hours just to figure out the length of the archipelego in terms of the planet's radius, so you can see how out of shape I am.

As an aside, I think the solution lies in calculating an equation for x and y of cartesian coordinates as a function of psi & theta of the spherical coordinates of the globe. Then I can use a graphing program to hold either of the angles constant and draw the results. That's pretty much as far as I can surmise.

Advice appreciated.

The sole landmass in my setting is an archipelego that begins at the north pole and runs to the southpole. It does so however, not using the shortest path but one that winds around the globe such that it completes exactly one rotation by the time it reaches the south pole (a loxodrome aka rhumb line). I want to draw a rectangular map such that the horizontal midline of the map corresponds to the axis of the archipelego, which means I'm projecting a curved structure onto a straight 2D plane. In other words on the map the archipelego is rendered as a straight line, with the poles at either end of the map, but the archipelego actually follows a rhumb line and thus is curved. Due to this excellent site, I don't have any trouble getting the art right.

Now, the hard part about this is figuring out where the longitude and latitude lines go. I have an idea in my head but I want to be precise.

I was an engineer in undergrad, so I have the background required to figure this out, but it's been a long time since I've done any of that. I need someone who knows what they're doing to point me in the right direction. It took me about 4 hours just to figure out the length of the archipelego in terms of the planet's radius, so you can see how out of shape I am.

As an aside, I think the solution lies in calculating an equation for x and y of cartesian coordinates as a function of psi & theta of the spherical coordinates of the globe. Then I can use a graphing program to hold either of the angles constant and draw the results. That's pretty much as far as I can surmise.

Advice appreciated.