As programmers, we love to build things, and what could be more exciting than building a world? Minecraft, Terragen, Skyrim, and every flight simulator ever all use some sort of fractal terrain generation. Today we'll explore the beautifully simple diamond-square algorithm so you, too can play God. [Demo] [Source]

Programmers tend to be lazy (I speak from experience), and one nice side effect of laziness is really brilliant ways to avoid work. In this case, instead of spending mind-numbing hours manually creating what would likely be pretty lame rocky surfaces, we'll get spiritual and teach the computer what it means to be a rock. We'll do this by generating fractals, or shapes that repeat patterns in smaller and smaller variations.

I don't have any way to prove that terrain is a fractal but this method looks really damn good, so maybe you'll take it on faith.

Height maps

We'll store our terrain as a simple height map: a 2-dimensional array of values that represents the height of terrain at any given x, y coordinate. With this simple data structure, we can render the heights however we like - canvas, webgl, interpretive dance, etc. The biggest limitation is that we can't represent vertical holes in the terrain like caves, tunnels, or bridges.

function Terrain(detail) { this.size = Math.pow(2, detail) + 1; this.max = this.size - 1; this.map = new Float32Array(this.size * this.size); }

You can apply this algorithm to any dimension of grid, but it's easiest with a square that's a power of 2 plus 1. We'll use the same value size for the x, y, and z axes, forming our terrain in a cube. We convert detail into a power of 2 plus 1, so higher detail renders larger cubes.

The algorithm

Here's the idea: take a flat square. Split it into four sub-squares, and move their center points up or down by a random offset. Split each of those into more sub-squares and repeat, each time reducing the range of the random offset so that the first choices matter most while the later choices provide smaller details.

That's the midpoint displacement algorithm. Our diamond-square algorithm is based on similar principles but generates more natural-looking results. Instead of just dividing into sub-squares, it alternates between dividing into sub-squares and dividing into sub-diamonds.

1. Set the corners

First, set the corners to a 'seed' value which will influence the rest of the rendering. This would start all the corners halfway up the cube:

this.set(0, 0, self.max / 2); this.set(this.max, 0, self.max / 2); this.set(this.max, this.max, self.max / 2); this.set(0, this.max, self.max / 2);

2. Divide the map

Now, we'll recursively look at smaller and smaller divisions of the height map. At each division, we'll split the map into squares and update their center points during the square phase. Then, we'll split the map into diamonds and update their center points during the diamond phase.

divide(this.max); function divide(size) { var x, y, half = size / 2; var scale = roughness * size; if (half < 1) return; for (y = half; y < self.max; y += size) { for (x = half; x < self.max; x += size) { square(x, y, half, Math.random() * scale * 2 - scale); } } for (y = 0; y <= self.max; y += half) { for (x = (y + half) % size; x <= self.max; x += size) { diamond(x, y, half, Math.random() * scale * 2 - scale); } } divide(size / 2); }

The scale ensures that the size of our offsets decreases along with the size of our divisons. For each divison, we multiply the current size by roughness, which determines whether the terrain is smooth (values near zero) or mountainous (values near one).

3. The shapes

Both shapes work similarly, but draw data from different points. The square phase averages four corner points before applying a random offset, while the diamond phase averages four edge points before applying a random offset.

function diamond(x, y, size, offset) { var ave = average([ self.get(x, y - size), // top self.get(x + size, y), // right self.get(x, y + size), // bottom self.get(x - size, y) // left ]); self.set(x, y, ave + offset); }

Rendering

This algorithm just gives us data, which we can render in any number of ways. We'll combine a slew of rendering techniques into a rasterized, isometric, 3d-projected terrain map on a canvas element.

Back to front

First, we create nested loops that drew rectangles from the 'back' (y = 0) to the 'front' (y = this.size) of our map. This is the same loop you'd use to render a simple, flat, top-down square.

for (var y = 0; y < this.size; y++) { for (var x = 0; x < this.size; x++) { var val = this.get(x, y); var top = project(x, y, val); var bottom = project(x + 1, y, 0); var water = project(x, y, waterVal); var style = brightness(x, y, this.get(x + 1, y) - val); rect(top, bottom, style); rect(water, bottom, 'rgba(50, 150, 200, 0.15)'); } }

Light and shadow

Our naive approach to bump-mapping provides a nice visual texture. We compare our current height value against the next point's height value to find a slope. We draw brighter rectangles for higher slopes to fill one side with light and the other with shadow.

var b = ~~(slope * 50) + 128; return ['rgba(', b, ',', b, ',', b, ',1)'].join('');

Isometric projection

We could draw everything head-on, but it's more visually interesting to rotate our square into a diamond before projecting it into 3d. Isometric projection aligns the top-left and bottom-right corners into the middle of the view.

function iso(x, y) { return { x: 0.5 * (self.size + x - y), y: 0.5 * (x + y) }; }

Perspective projection

We'll use a similarly simple 3d projection to convert our x, y, z values into a flat image with perspective on a 2D screen.

The basic idea behind any perspective projection is to divide horizonal and vertical position by depth so that higher depths render closer to the origin (ie, further away objects appear smaller).

function project(flatX, flatY, flatZ) { var point = iso(flatX, flatY); var x0 = width * 0.5; var y0 = height * 0.2; var z = self.size * 0.5 - flatZ + point.y * 0.75; var x = (point.x - self.size * 0.5) * 6; var y = (self.size - point.y) * 0.005 + 1; return { x: x0 + x / y, y: y0 + z / y }; } };

Putting it all together

First, we create a new Terrain instance with our desired detail level. Then, we generate its heightmap, providing a roughness value between 0 and 1. Finally, we draw the terrain onto a canvas.

var terrain = new Terrain(9); terrain.generate(0.7); terrain.draw(canvasContext, width, height);

Try it out

Explore the otherworldly terrain.

What's next?

If you're anything like me, the results of this simple algorithm leave you itching to go build an online Terragen, a jetpack-based first person shooter, fishing simulator, MMORPG, etc. This single-cube, canvas-projected demo practically begs for extension.

Fork the code!

Here are a few things I challenge you to try:

WebGL rendering

Variation by height, where lower altitudes are smoother (like sand) and higher altitudes are more rocky

Cast shadows instead of purely slope-based shading

A second pass that generates caves and tunnels

Parallelize terrain building with WebWorkers

As always, get in touch if you'd like to riff on the ideas here.

Related Work

Lots of folks are playing with this algorithm right now and building cool stuff. Also, the Hacker News discussion brought out some really fantastic related examples. Here are some highlights:

Discuss

Join the discussion at Hacker News.