How much should a globe be cut up to render it into flat squares? (math question)

[imp_fireball]

I want to create a two-dimensional world, so I'm wondering how I can render it accurately like a globe.

It doesn't have to be a single projection - it can be multiple separate maps. Is this mathematically possible?

I'm looking for simple square or rectangle shapes due to the software I'm using to make maps (hexographer).

[rdanhenry]

With no distortion? An infinite number of squares will do the job nicely.

Otherwise, you need to decide what kind of distortion you want to minimize.

With Hexographer, shouldn't you be asking how many hexagons? (The answer is still infinitely many, but if you really want to approximate with hexagons, the answer is going to differ from squares or triangles).

Rather than asking about a method, why don't you try asking about what you are actually trying to achieve with it, so that our experts in projections can identify the methods that can achieve it?

If you are stuck on showing a globe on a flat map without any distortion, that is impossible. Again, what you can do is choose what the distortion will be. You cannot fit a curved surface to a flat surface without changing it.

[waldronate]

As I mentioned in your earlier discussion, the question is mostly what kind of distortion you're willing to live with and how much. For fairly local maps (say, fifty to 100 miles across), the distortions will probably be on the order of the width of your drawn lines, so the won't much matter. It's when you get to continental and world scale that it really matters.

Some questions to consider:

What do you want to do with these maps? Decorative maps are different from navigational maps, which may themselves be quite different from maps that you would use for doing worldbuilding.

How big do you want your individual maps to be? http://projectionwizard.org/ can offer some suggestions for projections if you can answer this question.

How many maps do you want to make and maintain? More maps is more work and I'd recommend keeping the number of maps to a minimum to avoid burnout.

You mention wanting to use Hexographer. How large do you want your hexes to be on individual maps?

The nice thing about hex maps is that they tend to introduce a lot of distortion on their own, so you might get a sufficiently-accurate map for a whole world using something like an interrupted sinusoidal projection of even the classic icosahedron. Hexes don't perfectly tile a sphere, but using the classic hex-tiled icosahedral projection ( http://oldguygaming.com/exporting-my...ld-back-to-cc3 shows an example ) will move the pentagons into the corners and points of the map, where they are less obvious.

The hardest question of all, of course, is: how much are you willing to spend (both monetary units and time units) in order to make your map? Mapping isn't quick and it isn't easy if you want good results.

And I will end with a secret: If you're the only one who knows what it's supposed to look like, it's perfectly acceptable if the first few (dozen) maps are inaccurate in some way. If you're going to develop a world over time, things are going to change anyhow and some areas are going to get more detail than others just on account of how you use the map.

[imp_fireball]

Each hex represents 100 square kilometers, or 10km in any direction. It's fairly rudimentary (maybe a symbol for ridged and sloping coast lines and archipelagos are what's missing), and it can only render so many hexes at once.

So in actuality, I probably won't draw in hexographer, but do what I can by superimposing a hex grid over paint.net. Or abandon hexes and resort to a square grid.

[imp_fireball]

I might switch over to square grid instead, if it turns out to be simpler.

What I want to do is world build, but not for an rpg - instead, a 4x or 'civilization' game.

[waldronate]

At a 10km radius cell size (20km diameter), that would give you about 2000 cells around the equator for an earth-sized world. The problem is that the number of cells in each of the 500ish bands on each side of the equator diminishes as you approach the pole, down to a single cell at the pole (the Sinusoidal projection for constant-size cells is exactly that pattern). The icosahedral projection that I mentioned interrupts that pattern to reduce the distortion in any particular triangle.

If you're looking to do a 4X game or to develop a map in/for such a game, then you're going to be limited by the mechanics of the engine that you choose. You will also be limited by how the in-game board is defined. If you're looking to do worldbuilding that you will later translate into a game engine, then it doesn't much matter what projection you use for your own worldbuilding: you're going to have to translate your map into the game engine that you'll use anyhow. That translation may include reprojection and redrawing.

[imp_fireball]

If you're looking to do a 4X game or to develop a map in/for such a game, then you're going to be limited by the mechanics of the engine that you choose. You will also be limited by how the in-game board is defined. If you're looking to do worldbuilding that you will later translate into a game engine, then it doesn't much matter what projection you use for your own worldbuilding: you're going to have to translate your map into the game engine that you'll use anyhow. That translation may include reprojection and redrawing.

I wouldn't worry about how I'm going to put it in a game engine. If I had the coding knowledge, I'd probably know how to integrate it (I don't; that's a task for another time). Right now, I'm just trying to see what's possible, and what I can do. Gotta look before you leap, as they say.

The icosahedral projection that I mentioned interrupts that pattern to reduce the distortion in any particular triangle.

How? Does it remove squares that would otherwise exist?

At a 10km radius cell size (20km diameter)

I mentioned that I'd be going for 10km diameter. If it's a hex, I would mathematically treat it like a square for travel purposes and simplicity.

Is there a problem with having too many cells?

[waldronate]

An icosahedral projection is an interrupted projection. As such, it renders some areas as separate from each other on the plane, even though those areas would be next to each other on the globe (see http://www.progonos.com/furuti/MapPr...osahedron.html for some information; http://www.progonos.com/furuti/MapPr...C/cartTOC.html is a good starting point if you're unfamiliar with the idea of map projections). One nice property of using an icosahedron is that you can print it out onto a piece of paper, cut it out, and fold it up to get a 3D object to look at.

I apologize for the notion of cell size. I misinterpreted the "10km in any direction" as indicating radius, despite your indication of 100 square km per cell. I was doing some rough calculations to get an idea of the amount of distortion that would be hidden by using discrete cells. Having more cells increases the accuracy of your map, but there isn't a huge difference between a 10km or 20 km cells at a global scale.

[Pixie]

An earth sized planet has around 40,000 km perimeter at the equator.. at 10 km cells, that's 4,000 hexes wide. Now, supposing a very tiny hex drawn with close to 10x10 pixels, and you're talking already about an image that is 40,000 px wide. On most projections, that would turn into something around 40,000 x 20,000 px. It would be a huge file to work with.
A 20km cell would half the dimensions, making the area 4 times smaller... but a 20,000 x 10,000 px file is still huge for a worldbuilder to work on, imho.

Instead...
You can use a typical equiretangular projection to draw a whole world map, and then use a software like g-projector to make local maps with equal-area projections, over which you can lay an hex-grid easily.

[imp_fireball]

I was hoping that I could just divide up a flattened oval projection (wider than it is tall) along the latitude lines, and then read the level of distortion going on there. However, I wouldn't know how to distort it to render the globe into an oval to begin with (if the formula is simple, then fine... no calculus, hoping).

But if it's done successfully, then I was thinking I could (afterward) reshape each latitude zone (representing 15 degrees of latitude) to get rid of the distortion.

I was thinking of using square grids, unless that's (for whatever reason) more impossible. BTW, the 'squares represent 10km in any direction' only applies to maps of 1 million square km (so 100x100 grids). Squares would have represent much more on a globe, I acknowledge.

EDIT: Holy crap, it's been 5 months since I was last here. Sorry for the minor necro. xD