World-building: Scale

This is the second article in a series.
Click here for the series index.

For any kind of graphic, the size of your image or art board in pixels is an important consideration.  If you are ever going to print your image, you must also consider DPI (dots per inch).  If you are working with raster images, a good general rule is that reducing the size of your image is not a problem, but enlarging it is almost never a good option.  The reason for this is that it's easy to maintain sharpness when reducing, but enlarging invariably makes raster images look grainy and pixelated.

This rule applies to maps as well as other kinds of images.  Try to think of how you want to use your image, and choose your image size accordingly.  If you want to print it out, think about what size it will be printed at.  Print resolution standard is usually around 300 DPI.  If it's to be a wall poster, you'll likely need a rather large image.  If it's only going to be a small inset on a page, you can make it much smaller.

If, like me, you want to make an image that could cover all of these possibilities, the simple answer is to start big.  How big depends on the biggest usage you can imagine, plus one other factor: how big can your computer handle?

Before we delve into the topic of scale as it relates to digital maps, there's one more thing to consider: it may be tempting to try to work with a single, massive world map which can be zoomed in and cropped to provide continental maps.  I can't recommend this approach, firstly because it will cause you problems thinking about what size to make labels, and indeed what levels of labels to include.  Bear in mind that a world map is a world map, and unless it's designed to be printed out as a poster, it needn't display low level detail.  To put it another way, if you have to zoom in to see something clearly, it probably belongs on a continental or local map.

The second reason is that choice of map projections is severely limited for the world map, since by its very nature it shows the whole world at once.  Better to make separate, more detailed continental/regional maps, with increased levels of detail, as well as more appropriate map projections.

Now let's start talking numbers.

Scale

It seems to me that the most practical way to measure scale in digital raster maps is in kilometres per pixel, which I notate as km/pixel, or kmpp.  Raster images are made up of pixels as their smallest units, and most users are used to dealing with pixel sizes to at least some degree.

For an earth-like world, we have a great example of this in NASA's Blue Marble: Next Generation.  This wonderful (and beautiful) resource provides composited satellite imagery of the earth in Equirectangular Projection at three scales: 5,400 x 2,700, 21,600 x 10,800, and 86,400 x 43,200.  The last resolution is the full resolution, and NASA writes: "Blue Marble: Next Generation offers a years worth of monthly composites at a spatial resolution of 500 meters [per pixel]."

Let's think about those numbers.  Remember that Equirectangular maps are accurate at the equator, with increased east-west distortions as you move north or south.  This means that east-west measurements should be made at the equator.  500m per pixel means 0.5 km/pixel.  Multiply by 86,400 and we see that this model has the earth's circumference as 43,200 km.  Actual earth equatorial circumference is 40,075.017 km (according to Wikipedia), so the true resolution is likely around 0.46 km/pixel.

You can make the same calculations using north-south distances, because Equirectangular maps do not distort north-south distances.  The result is the same.

Calidar's circumference is in fact precisely 43,200 km, meaning that an 86,400 x 43,200 map is precisely 0.5 km/pixel for Calidar.  But this size is far too unwieldy for today's computers to work with, so I chose to go with a 21,600 x 10,800 base world map.  Do the maths and you'll see that this is equivalent to 2 km/pixel.  I will likely downscale this to 5,400 x 2,700, or 8 km/pixel, for printing at smaller sizes.

For my purposes, 21,600 x 10,800 seems like a good size of world map to be working with.  It's not insanely huge in terms of working in Photoshop and other programs using my current computers, and it provides a level of detail that can be printed extremely large (72" x 36" at 300 DPI), or reduced to be printed smaller.

My first attempt at a height map for the +World of Calidar.  Note how
the mountains are huge – they dominate the continents as if they were
mere islands.  The colouring is climate-based shading done in Fractal
Terrains.  While this was the very first full world height map, this test
remains the only full climate shaded version of the world map to date.
As such, it was used to create the 3D render of Calidar seen in the logo
for this series and throughout Calidar's online presence.  It will
eventually be replaced with the finished map, once all of the continents
have been eroded. This was a very small test, at 3,000 x 1,500 pixels.

Height Maps and Scale

There is another reason for my choice of such a big image size, which is that the relative scale of terrain when you build a height map depends on image resolution.  More specifically, if you work with erosion, the results of the erosion filters approximate a certain real world scale.  This is why I have chosen to develop Calidar at such high resolution.

To recount my thought process here, I first created a height map of Calidar at 3,000 x 1,500 as a trial, but it quickly became obvious that the terrain was off-scale – individual mountains were hundreds of miles across.  In an attempt to solve this problem, I moved up to 21,600 x 10,800, but the results still did not satisfy me.

My solution has been to work in three separate scales: 2 km/pixel for the world map, 0.5 km/pixel for continental maps, and 0.125 km/pixel for local maps.  The size of the world at these scales is 21,600 x 10,800, 86,400 x 43,200, and 345,600 x 172,800 respectively – but note that the last of these will likely never be fully developed, nor is there any need to do so.

The first full 21,600 x 10,800 height map of Calidar.  In my notation, D
refers to the world map version: this is the fourth version of the fourth
(and final) draft of the continental outlines.  I advance the letter whenever
significant tweaks are made to the outlines, or when I start a new draft of
the height map.  This map was changed to version E partway through.
Note how the mountains are much finer than on the D test version, and
yet still far too big for the mountains of an approximately earth-sized
world.  This is the last time I played with coastal shelves and the sea
floor – after this map, I decided to leave them until after the land is all
finalised.

What this three stage system does is provide a nice level of terrain detail without going overboard with realism.

Wilbur's Erosion and Scale

My primary tool of choice for erosion is Joe Slayton's Wilbur, the sister program of Fractal Terrains.  Joe Slayton is a regular member of the Cartographer's Guild, and he is very gracious about answering questions and giving information about his software.  He has stated that Wilbur's erosion model approximates realism at a resolution of between 1 and 50 m/pixel, or 0.001 and 0.05 km/pixel.

I wasn't happy with the DE height map, so I started again with another
pass at the height map in Photoshop.  This is the eroded result.  You can
see the current design beginning to take shape, but the scale is still off.

A world built at 0.05 km/pixel would measure a whopping 1,080,000 x 540,000 pixels.  Clearly it is beyond the realms of possibility to develop the whole world at that level.  However, small local areas could certainly be done at this resolution.

For Calidar, this would mean one last quadrupling of the scale, from my "local" 0.125 km/pixel to a super-local 0.03125 km/pixel, or 31.25 m/pixel.  It's not beyond the realms of possibility that the need for such a map may arise in the future – for example, a detailed map of the area around a settlement.

Town maps may go further than this, of course, but by that point, terrain is hardly an issue, so it's really a moot point as far as height maps go.

Map F marked the final design for the world map's height map, and I
never ran it through the erosion process – instead I split the world up
into continents at quadruple the resolution, with each continent
reprojected to a more suitable projection.  From there, each continent
has undergone its own development and refinement, and only when
they are all finished will the world map be reassembled again.
At that point the world map will be 86,400 x 43,200, but I will
likely shrink it back down to 21,600 x 10,800 to make it more
 manageable, leaving the fine detail for continental and local maps.

To sum up: the ideal resolution for Wilbur erosion is 0.05 km/pixel, but such realism is probably only necessary for local maps of small areas.  In terms of developing the world as a whole, much lower resolutions should give acceptable results, although the lower you go, the more blocky and oversized your terrain will begin to feel.

Traditional Scale

Those of you who are familiar with real world paper maps may have seen scales quoted as ratios: 1:500,000 scale, or 1:10,000 scale, and so on.  Depending on your background, you may or may not be familiar with what these numbers mean.

I find it useful for the sake of comparison to know what the equivalent scale in km/pixel is for each of these scales, so I researched the issue and came up with the following table.

Hex Scales	km/pixel	Scale 1:?	km width/1000 pixels	Scale	Wilbur
	100	200,000,000	100,000	Small Scale
	50	100,000,000	50,000
	25	50,000,000	25,000
	10	20,000,000	10,000
	7.2	14,400,000	7,200
	5	10,000,000	5,000
	2	4,000,000	2,000
30 miles per hex	1.5	3,000,000	1,500
	1	2,000,000	1,000
10 miles per hex	0.5	1,000,000	500
	0.33	660,000	330
	0.25	500,000	250
5 miles per hex	0.2	400,000	200
	0.15	300,000	150
	0.125	250,000	125
2.5 miles per hex	0.1	200,000	100	Medium Scale
	0.05	100,000	50		Ideal scale for Wilbur erosion
1 mile per hex	0.04	80,000	40
	0.025	50,000	25	Large Scale
	0.005	10,000	5
	0.0025	5,000	3
	0.001	2,000	1
	0.0005	1,000	0.5

Note that the meanings of the phrases "small scale" and "large scale" are routinely mixed up in vernacular usage.  It's a natural mistake, because small scale refers to progressively larger areas, while large scale refers to smaller areas.

A trick for remembering the correct meanings is that small scale refers to a view of everything from very high above, or very far away, making everything appear small.  Large scale conversely means looking at things from much closer, so that everything appears large.  Small scale will have generalised features, while large scale can include very precise detail.

Note also that the ranges for small, medium, and large scale presented here are not universal.  There is apparently no standard.  Wikipedia's article on map scale provides a different set of ranges, for example.

In terms of fantasy maps, the split could be considered as follows:

• Small scale: world maps, continental maps, many local maps
• Medium scale: some local maps, some city maps
• Large scale: some city maps, town & village maps, adventure location maps, floor plans

Hex Maps and Scale

This is the local scale version of Meryath, 0.125 km/pixel at full
scale, or sixteen times the resolution of the world map.  The hexes
here are 10 miles per hex (16 km per hex).  The precise
measurements mean that no scaling is necessary to use this as a
guide in Hexographer or Adobe Illustrator.

You may have noticed that I included hex scales in the table above.  We can calculate the km/pixel scale for a hex map quite easily, simply by dividing the height of a hex in pixels by its scale in miles or km.  How meaningful this may be is another matter, but it can be useful to know for the sake of comparison, and especially if you are working from a guide to create a hex map.

Calidar's hexes are 39 pixels tall, with the standard hex size being 10 miles per hex.  We may also be making 30 mile per hex regional maps, but not any smaller scales.  Larger scale 5 or 2.5 mile per hex local maps are also a distinct possibility.


I have done all the measurements in this article so far in km, so here are the scales in km/pixel for each scale of hex:

30 miles per hex – 1.23 km/pixel
10 miles per hex – 0.41 km/pixel
5 miles per hex – 0.21 km/pixel
2.5 miles per hex – 0.10 km/pixel

The nature of hex art of course means that in real terms hex maps don't include anything like this level of detail, but it's interesting to have the numbers for the sake of comparison nonetheless.

Conclusion

This has been a rather technical and probably boring article for many people.  Thank you if you read this far!  I hope it will prove useful for cartographers, at least.

Next in the pipeline is an article on creating base height maps in Photoshop.

Map of the Day 16: Meryath – the First Hex Map

Today we take a slight detour from our Calidar world tour to present something special: the first work-in-progress hex map for Calidar.

This is Bruce's original first draft of Palatine Island, the main island of the Kingdom of Meryath.  It was revealed today on the Kickstarter draft page.  (The project is not live yet, but you can now see the page and check it out.)

In terms of art, it's bare bones stuff - it's pretty much just the default hexes that come with Hexographer.  I am hard at work on a new set of hexes for Calidar, which will eventually replace these.

But in terms of development, everything you have come to expect from Bruce's hex maps is right there.

By the way, the projection this map is based on is called Albers Equal Area.  I chose this projection because it accurately depicts the area of the region.  This will make Bruce's demographic calculations all the more accurate, since they are based on hexes.

Another policy note about hex maps: you won't be able to join up all Calidar's hex maps into one massive trail map, I'm afraid.  The reason for this is simple: hex maps were never intended to cover large areas.  By definition they imply constant shape, direction, area and distance – which is an impossibility for any map of a sphere or ellipsoid.  At larger scales (i.e. showing things up close or zoomed in) the distortion is manageable, if not insignificant.  But at smaller scales (zoomed out), the distortion becomes so extreme as to make the hexes meaningless.

But don't despair!  That's where other cool map styles come in.  If you're a fan of hex maps, you can rest assured that Calidar will have many wonderful hex maps – Bruce is after all pretty much the king of hexes.  The only thing that will change is that there will be more other styles of maps in addition to the hex maps.  It's the best of both worlds, really.

An expanded version of this hex map, including the other islands of Meryath as well as some other things, is one of the stretch goals in the Kickstarter.  If we get that far, you and I will all be able to hold this map in our hands.  Let's make it a reality!

While we're on the subject of hex maps, if there's anything in particular that you would like to see in Calidar's hex maps – a new kind of hex, a certain way to use a symbol, or anything like that – please let me know.  Now is the time to make such changes, while I'm working on setting the style.

Tomorrow we will get back to looking at Calidar's western hemisphere.