## Polygon map generation, part 3#

In the previous blog posts (part 1 and part 2), I described generating random polygonal maps with elevation, moisture, biomes, and rivers. For some games, those maps are sufficient. However, in other games I want to hide the polygon structure. In this blog post I'll describe how to render the polygons into a game map that doesn't look polygonal, and conclude with the demo and source code.

The full article is here. There's also a demo and source code.

### Noisy Edges

Recall from earlier that there are two graphs: one for Voronoi corners (`1`, `2` in the diagram below) and edges (blue lines), and one for polycon centers (`A`, `B`) and Delaunay edges (red lines) between them:

I wanted to add some “noise” to the the straight lines. I tried making them move randomly, but sometimes lines would cross, and I realized I needed to constrain them so that they would never cross each other. The second thing I wanted was to make sure that the lines had as much space to wander as possible.

I realized that points `A`, `1`, `B`, and `2` form a quadrilateral, and I could constrain the wanderings of the line segment to that quadrilateral:

I further divided the quadrilateral into four quadrilaterals. Two were usable for the red (Delaunay) edge and two for the blue (Voronoi) edge. As long as the lines stayed within their allocated space and met in the center, they'd never cross each other. That takes care of constraining them.

The entire map can be divided up into these quadrilateral regions, with no space left over:

That ensures that the noisy lines aren't constrained any more than necessary. (I wonder if these quadrilaterals would be useful for game mechanics.)

I can use any noisy line algorithm that fits within these constraints. I decided to subdivide the quadrilaterals recursively and stitch line segments together within the small quadrilaterals into a complete edge. The result is here:

The noisiness is tunable, and I have examples at segment size 7, segment size 4, and segment size 1. In the map demo I use segment size 1 for rivers and coastlines, 3 where biomes meet, and 10 elsewhere.

### More noise

I'm generally a fan of noise in game art, and wanted to add a little bit of noise to these maps as well. In a real game map the noise might reflect vegetation or small variations in terrain. In the demo I just filled the screen with a random noise texture, and smoothed the colors between adjacent polygons:

However, with a bit more random noise, we can generate this (described in the full article):

Here's a rendering with 16,000 polygons, noisy edges, a noise texture overlay, and simple lighting:

### Demo

I wrote a Flash demo to explore the generated maps:

The simplest way to explore the maps is to click Random and the various View options.

In a shape number like `85882-3`, 85882 chooses the overall island shape and 3 is the random number seed for the details (random points, noisy edges, rivers, lava). You can type in a shape number and press Return to generate that map. The demo also shows some unfinished features that may be useful for some games: lava, roads, and watersheds.

### Source

I've placed the Actionscript source under the MIT license; it's available on github. There's an overview page describing what's in these blog posts, along with notes about the code. I don't expect that the code will be immediately useful to anyone, but it might be a useful starting point if you'd like to use these techniques for making your own game maps. The diagrams are built with 300 polygons, the demo uses 2000, and the code can go much higher, although I've not tried above 16,000.

If you find the ideas or code useful, I'd love to hear about it!

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## Polygon map generation, part 2#

In the previous blog post I described creating polygon island maps. Given random points, Voronoi diagrams with Lloyd relaxation produce a nice set of polygons. The polygons, their edges, and their corners can be represented as two related graphs. Given a random shape, those polygons can be marked as land, ocean, or lake. In this blog post I'll describe how to add elevation, rivers, moisture, and biomes to make the maps interesting.

The full article is here. There's also a demo and source code.

### Elevation

The most realistic approach would have been to define elevation first, and then define the coastline to be where the elevation reaches sea level. Instead, I'm starting with the goal, which is a good coastline, and working backwards from there. I set elevation to be the distance from the coast. I originally tried elevations at polygon centers but setting elevations at corners worked out better. Corner-to-corner edges can serve as ridges and valleys. After calculating the elevation of corners, the polygon elevation is the average of the elevation at the corners.

Water polygons don't count towards the distance. This is both because I expect lakes to be flat instead of sloped, and because this tends to build valleys around lakes, which helps guide rivers towards lakes.

One problem with the simple definition is that some islands have too many mountains and others have too few. To fix this, I redistribute the elevations to match a desired distribution, which has more low elevation land (coastline) than high elevation land (mountains).

Elevations always increase from the coast to the mountains. That means that for any location, going downhill will eventually lead to the ocean. This diagram shows the steepest downhill direction from every corner:

By following the downhill arrows from any location, we eventually reach the ocean. This will be useful for rivers but may also be useful for calculating watersheds and other features.

I had two main goals for elevation:

1. Biome types: high elevations get snow, rock, tundra; medium elevations get shrubs, deserts, forests, and grassland; low elevations get rain forests, grassland, and beaches.
2. Rivers flow from high elevations down to the coast. Having elevations that always increase away from the coast means that there's no local minima that complicate river generation.

In addition, games may define their own use of elevation data. For example, Realm of the Mad God uses elevation to distribute monsters.

### Rivers

Rivers and lakes are the two fresh water features I wanted. The most realistic approach would be to define moisture with wind, clouds, humidity, and rainfall, and then define the rivers and lakes based on where it rains. Instead, I'm starting with the goal, which is good rivers, and working backwards from there.

The island shape determines which areas are water and which are land. Lakes are water polygons that aren't oceans.

Rivers use the downhill directions shown earlier. I choose random corner locations in the mountains, and then follow a path down to the ocean. The rivers flow from corner to corner:

I tried both polygon centers and corners, but found that the corner graph made for much nicer looking rivers. Also, by keeping lakes flat, elevation tends to be lower near lakes, so rivers naturally flow into and out of lakes. Multiple rivers can share the lower portion of their path, but once they join, they never diverge, so tributary formation comes for free. It's simple and seems to work pretty well.

### Moisture

Since I'm working backwards, I don't need moisture to form rivers. However, moisture would be useful for defining biomes (deserts, swamps, forests, etc.). Since rivers and lakes should form in areas with high moisture, I defined moisture based on distance from fresh water:

As with elevation, I redistribute moisture to match a desired distribution. In this case, I want roughly equal numbers of dry and wet regions. In this map generator, moisture is only used for biomes. However, games may find other uses for the moisture data. For example, Realm of the Mad God uses moisture and elevation to distribute vegetation.

### Biomes

Together, elevation and moisture provide a good amount of variety to define biome types. I use elevation as a proxy for temperature. Biomes first depend on whether it's water or land:

• Ocean is any water polygon connected to the map border
• Lake is any water polygon not connected to the map border
• Ice lake if the lake is at high elevation (low temperature)
• Marsh if it's at low elevation
• Beach is any land polygon next to an ocean

For all land polygons, I started with the Whittaker diagram and adapted it to my needs:

ElevationMoisture
WetDry
HighSnowTundraBare rockScorched
Medium-highTaigaShrublandTemperate desert
Medium-lowTemperate rain forestTemperate deciduous forestGrasslandTemperate desert
LowTropical rain forestTropical seasonal forestGrasslandSubtropical desert

Here's the result:

These biomes look good in the map generation demo, but each game will have its own needs. Realm of the Mad God for example ignores these biomes and uses its own (based on elevation and moisture).

In the last blog post I'll describe how I get from this biome map to maps like this: (or even this!)

Update: [2010-09-22] I replaced the last diagram on this page with what I originally wanted but didn't finish in time for the blog post. At the time of posting, I used this image instead.

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## Polygon map generation, part 1#

I wanted to generate interesting game maps that weren't constrained to be realistic, and I wanted to try some techniques I hadn't tried before. I usually make tile maps but this time I decided to make polygonal maps. Instead of 1,000,000 tiles, what could I do with 1,000 polygons? I think the distinct player-recognizable areas might be useful for gameplay: locations of towns, places to quest, territory to conquer or settle, pathfinding waypoints, difficulty zones, etc.

There were three main things I wanted: good coastlines, mountains and rivers. For the coastline, I wanted to make island/continent maps that are surrounded by ocean, so that I don't have to deal with people walking to the edge of the map. For the mountains, I started with something simple: mountains are whatever's farthest from the coastline. For the rivers, I started with something simple: draw rivers from the coast to the mountains.

The full article is here. There's also a demo and source code.

### Polygons

The first step is to generate some polygons. I picked random points and generated Voronoi polygons, which are used for lots of things, including maps. Here's an example of random dots (red) and the polygons that result:

The first problem is that polygon shapes and sizes are a bit irregular. Random numbers are more “clumpy” than what people expect. What I really want is semi-random “blue noise”, not random points. I approximate that by using Lloyd relaxation, which is a fairly simple tweak to the random point locations to make them more evenly distributed. Here's the result after running Lloyd relaxation twice:

Compare it to running once or fifty times. The more iterations, the more regular the polygons get. Running it twice gives me good results but every game will vary in its needs.

[2010-09-22] The second problem with the polygons is that some edges are very short. For games where boundaries between polygons matters, having short edges is a problem. We can adjust the edge lengths by moving corners, but we lose the Voronoi properties. Since I'm using Voronoi only to generate polygons, and do not need to preserve the Voronoi properties, I move the corners to the average of the polygon centers they touch. Note: I added this code after the initial blog post, and did not update diagrams to show this step.

### Map Representation

I'm representing the map as two related graphs: nodes and edges. The first graph has nodes for each polygon and edges between adjacent polygons. It represents the Delaunay triangulation, which is useful for anything involving adjacency (such as pathfinding). The second graph has nodes for each polygon corner and edges between corners. It contains the shapes of the Voronoi polygons. It's useful for anything involving the shapes (such as rendering borders).

The two graphs are related. Every triangle in the Delaunay triangulation corresponds to a polygon corner in the Voronoi diagram. Every polygon in the Voronoi diagram corresponds to a corner of a Delaunay triangle. Every edge in the Delaunay graph corresponds to an edge in the Voronoi graph. You can see this in the following diagram:

Polygon `A` and `B` are adjacent to each other, so there is a (red) edge between `A` and `B` in the adjacency graph. For them to be adjacent there must be a polygon edge between them. The (blue) polygon edge connects corners `1` and `2` in the Voronoi shape graph. Every edge in the adjacency graph corresponds to exactly one edge in the shape graph.

In the Delaunay triangulation, triangle `A`-`B`-`C` connects the three polygons, and can be represented by corner `2`. Thus, corners in the Delaunay triangulation are polygons in the Voronoi diagram, and vice versa. Here's a larger example showing the relationship, with Voronoi polygon centers in red and corners in blue, and the Voronoi edges in white and the Delaunay triangulation in black:

This duality means that I can represent the two graphs together. Edges are the key. Each edge in a normal graph points to two nodes. Instead of representing two edges in the two graph separately, I made edges point to four nodes: two polygon centers and two corners. It turns out to be quite useful to connect the two graphs together.

With the combined representation, I can now use the Relationships Between Grid Parts sections of my article on grids. They're not grids so I'm not assigning grid coordinates, but many of the algorithms that work on grids also work here, and the algorithms that work on graphs also work here (on either of the two graphs).

### Islands

The second step is to draw the coastline. I used a simple function to divide the world into land and water. There are many different ways to do this. You can even draw your own shapes, e.g., a skull island. The map generator works with any division of points, but it forces the outer layer of polygons to be ocean. Here's an example that divides the world into land and water:

A simple flood fill starting from the border of the map can determine which water areas are oceans (connected to the border) and lakes (surrounded by land):

In the next two blog posts (part 2, part 3) I'll describe how I add elevation data to build mountains and valleys, add moisture data for lakes and rivers, render the map so that it doesn't look polygonal, and conclude with the demo and source code. Together, elevation and moisture produce a good range of terrain and map features. The goal is to produce maps like this:

Update: [2010-09-22] Since the original blog post, I added a corner adjustment step to lengthen short edges. Just as Lloyd relaxation improves the polygon sizes, I needed to adjust the edge lengths. The adjustment does not preserve Voronoi properties, but I'm not using those properties so it worked out. I didn't update diagrams to reflect this change. I also improved map rendering and replaced the last diagram on this page. I originally wanted to have a rendering that didn't show the polygons at all, but couldn't get it to work in time, so at the time of the blog post I used this rendering instead. The rendering technique is described in the full article but not in the blog posts.

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