Generating fantasy maps
You may also be interested in this companion piece, which describes the placename generation.
I wanted to make maps that look like something you'd find at the back of one of the cheap paperback fantasy novels of my youth. I always had a fascination with these imagined worlds, which were often much more interesting than whatever luke-warm sub-Tolkien tale they were attached to.
At the same time, I wanted to play with terrain generation with a physical basis. There are loads of articles on the internet which describe terrain generation, and they almost all use some variation on a fractal noise approach, either directly (by adding layers of noise functions), or indirectly (e.g. through midpoint displacement). These methods produce lots of fine detail, but the large-scale structure always looks a bit off. Features are attached in random ways, with no thought to the processes which form landscapes. I wanted to try something a little bit different.
There are a few different stages to the generator. First we build up a height-map of the terrain, and do things like routing water flow over the surface. Then we can render the 'physical' portion of the map. Finally we can place cities and 'regions' on the map, and place their labels.
To represent the heightmap, first we need a grid of points. Although it can be simpler to work on a regular square grid, I much prefer to work on an irregular set of points for something like this. With a regular grid, it's very easy to run into weird artifacts, and you often have to do a lot of postprocessing to hide the effects of the grid. If you use an irregular grid, then there are a few things which are more complicated, but the structure of the grid helps to give the map a rough, organic feel, and you never have to worry about nasty linear artifacts in the finished product.
Note: this shows 256 (2^8) points, to make viewing easier, but the real generator uses 16,384 (2^14) points. I have a programmer's superstitions about always using powers of 2, which are more pleasing to the spirit of the machine.
The approach I use is the same as in this article, which is one of the better references out there on how to do non-fractal terrain generation. I won't go into too much detail here because that article explains it very clearly, with lots of diagrams.
I start by selecting points at random within the map. These points tend to be a bit clumpy and uneven, so I use Lloyd relaxation to improve the point set. For speed, I only use one iteration of this process, but you can repeat it as many times as you like. There are rapidly diminishing returns after a few iterations though.
All of the calculations are actually carried out on the 'dual points' of the original point set, which correspond to the corners of the Voronoi polygons. This has the advantage that the number of neighbours per node is fixed at three, which helps in some parts of the code.
One of the difficulties of creating landscapes in a realistic way is that real landscapes aren't created all at once. Instead, they evolve from earlier landscapes, which in turn evolved from even earlier landscapes, and so on back for billions of years. There's no good way to simulate this process in a reasonable amount of time, so we need to cheat slightly.
Rather than an infinite regress of older landscapes, I start with a simple 'proto-landscape', built with geometric primitives. This lets me control the broad outlines of the terrain, while leaving the details for the more physical processes to fill in later.
Some useful primitives which we can add together:
- Constant slope - if you want to pretend this is physically motivated, think of it as tectonic uplift on one side of the map
- Cone shapes - these can be islands or mountains, or if inverted, lakes or seas
- Rounded blobs - these make better hills, and can be scattered all around to make a noisy surface
- 斜坡 - 如果你需要模拟在地图的一边构造隆起地表的物理情景。
- 锥形 - 这些可以生成岛屿、山脉，或反转它的海拔，使之成为湖泊。
- 圆形斑点 - 这将制造更好的丘陵，它可以四散分布，从而产生更为真实的地形。
We also have a few operations which are handy:
- Normalize - rescale the heights to lie in the range 0-1
- Round - normalize, then take the square root of the height value, to round off the tops of hills
- Relax - replace each height value with the average of its neighbours, to smooth the surface
- Set sea level - translate the heightmap up or down so that a particular quantile is at zero
- 标准 - 重置(边界)线的高度，范围是 0 - 1。
- 圆 - 规范(？)，获取高度值的平方根，以此环绕丘陵的突出部分。
- 缓和 - 通过遍历，以相邻高度的平均值替换高度值，用以产生平滑的曲线。
- 设置海平面 - 将特定位转译成 0，来提升或降低高度。
The particular sequence of primitives and operations used can be varied to produce different kinds of landscape, such as coastlines, islands and mountain ranges.
Note: the black line indicates the zero contour, which we treat as 'sea level'. Also, this map uses 4,096 (212) points, for speed.
注释：那条黑线显示了 0 水平单位的等高线，这将被当作海平面。为了效率，张地图采用的是4096(2^12)个点。
The results of this process can be a little bit on the blobby side, which means they rarely look good on their own. We want to scuff them up a bit, so they look more like real landscapes. We do this by applying an erosion operation.
In most of the world, by far the largest influence on the shape of landforms is fluvial (water-based) erosion. Water flows downhill, carrying sediment along with it, carving out valleys and river basins. This is a massively complex phenomenon, and modelling it correctly is a very active research area, but we can get a long way by sketching a simple version of the process.
We need to start by tracing the routes that water would take over the grid. For each grid point, we say that water flows to its lowest neighbour, and so on down until we reach the edge of the map. This gives a map of water flow.
There's an obvious problem when we reach gridpoints which are lower than all of their neighbours. Do we route the water back uphill? This will probably lead to cycles in the water system, which are trouble. Instead, we want to fill in these gaps (often called sinks or depressions), so that the water always runs downhill all the way to the edge.
It's easy to see how to fill in a single gridpoint, but as the depression gets bigger, and possibly links up with other depressions, the number of possible cases multiplies enormously. Luckily, there's an algorithm for filling depressions, called the Planchon-Darboux algorithm.
Aside: the Planchon-Darboux algorithm
The algorithm works by finding the lowest surface with the following two properties:
- The surface is everywhere at least as high as the input surface
- Every non-edge point has a neighbour which is lower than it
To calculate this, we start with an infinitely high surface everywhere except on the edge, where we use the original heights. Then, on each iteration, we find points which have a neighbour which is lower than them, and set their height to their original height, or the height of their lowest neighbour (plus a small amount), whichever is higher. We halt when we can go a full iteration without changing any point.
There are various ways of speeding up this algorithm, mostly by tweaking the order in which points are visited. For more details, and a proof of correctness, you can read the original paper.
With the water routing calculated, we can work out how much water is flowing through each point. I assume that rainfall is constant across the whole map, and iterate through the points in descending order, passing the rainfall, plus the accumulated water flux, from each point to its 'downhill point'. This gives a map of water flux, which usually converges into a nice branching river structure, with lots of small streams feeding a larger central channel.
To calculate erosion, I combine the water flux with the slope at each point, as calculated based on the triangle of its neighbours. The exact formula I use is the product of the slope with the square root of the water flux. This isn't necessarily very physical, but it does give nice-looking results. I also add a small term which is proportional to the slope squared. This prevents deep gorges from forming, which might be physically realistic, but don't look good in the graphical style I've chosen.
I find it's very important to cap the erosion rate, otherwise strange things can happen. A little goes a very long way with this. Also, erosion always lowers the surface, so it usually helps to drop the sea level afterwards to match.
A final tweak to the heightmap is to smooth out the coastlines slightly. The erosion tends to produce quite rough terrain, which becomes tiny islands when cut off by sea level. A few of these can look good, but too many just looks messy. I repeatedly apply a filter where points which are below sea level, but a majority of whose neighbours are above sea level, get pulled up, and vice versa for points which are above sea level and have undersea neighbours. A couple of repeats of this produces a much cleaner coastline.
- 此句英文属于上一段，为了翻译顺畅，将中文挪到本段 ↩