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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Map rendering: cutting corners#

I was drawing tile maps in Flash and found a cheap trick that improved the appearance of the maps, so I thought I'd share. Here's how I had been drawing the tile map:

Flash, OpenGL, and DirectX are vector-based engines. The bitmap techniques still work, but there are new possibilities available. I'm drawing each square tile by filling a square polygon with a bitmap texture. The engine doesn't care that it's square; it works on any polygon. I'm taking advantage of this with what I call “vertex displacement”:

I look at the four tiles touching each vertex. If three of them are the same and the fourth is different, I move the vertex to expand the area of the three common tiles and shrink the area of the uncommon one. It's easier to see the effect with the polygon borders:

It turns out there are other fun things you can do with this trick. For example, a little bit of random noise on each vertex makes for a map that looks a little more hand drawn:

I've also used it to animate the boundary between the ocean and the beach. I added vertex displacement to my simple map generator; see the demo (Flash). Try changing the corner setting to adjust how much the corners get moved when three tiles are the same, and the random setting to adjust how much vertices get randomly moved. I have a separate demo (also Flash) to show the animated coastline.

The technique has its limitations. Terrain boundaries that don't fall on a 45° angle look wavy (see example). Some terrain types shouldn't meet in this way. And the biggest limitation is that it only applies when the terrain texture has no major features (such as trees, rocks, etc.). But it's a cheap enough trick that it's worth keeping in my toolbox.

Update: [2012-03-30] Also see Marching squares.

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Teleological vs. ontogenetic map generation#

Edit: The feedback from comments is that the names “teleological” and “ontogenetic” aren't good fits. I agree. There's an interpretation in which the terms work but there are also other ways to interpret them where it doesn't make sense. I had gotten the terms from the procedural content wiki, but in the future I'll avoid using these terms. I've changed the blog text to use “bottom up” and “top down”. Thanks for the feedback.

Long ago, I spent years working on the terrain generation for a game I worked on. I used earthquake faults, volcanoes, soil erosion, lava, river erosion, sediment deposition, water flow, weather simulation, vegetation (affected soil, which affected erosion), forest fires, floods, continental uplift, and other physical processes. This is the teleological approach bottom up approach to making maps.

To make good maps, I'd change parameters and run it to see if the maps looked good. I'd go back and change parameters again and again. Ideally I'd write a search/evolutionary algorithm that explored the parameter space. To do that I need to write code to determine whether a map will look to my eye, and I never did figure that out. I knew it when I saw it.

The other approach to map making is ontogenetic top down. Start with the structure you want to see, and make algorithms produce that directly and fill in the details. This is what I did with last year's simple map generation project. I had seen Perlin Noise, and thought, hm, that looks vaguely like a map — let's see how much work it takes to turn it into a map. Not much, it turns out!

For the past month I've been working on a new map generation project. I wanted island/continent maps with rivers, and Perlin Noise wasn't a good fit for that. I tried making random mountains come out of the ocean, and then using water flow algorithms to carve out rivers. I've been tweaking and testing, tweaking and testing, tweaking and testing, and finally realized that I'm back to using bottom up algorithms. It's incredibly addictive. It feels right: erosion, mountains, gravity, and all the other “realistic” things that form terrain.

The trouble is that I really don't need all of that. I'm working on 2D Flash games, and I don't need terrain; I need maps. It doesn't need to be completely realistic. It just has to be interesting. So I'm stepping back a bit and switching back to the top-down approach. I'm drawing some maps that I like, and then I'll find algorithms that generate maps like those, instead of modeling physical processes that produce realistic terrain. I think this will lead to better maps with less work.

Update: [2010-09-04] See my newer blog post to see the beginning of the new map generation project.

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Varying game difficulty#

Contrast is commonly used for visuals and sometimes for music but you can also use it for game difficulty. The usual pattern in nature is slow rises and sharp falls.

• The stock market tends to slowly go up and occasionally drop dramatically.
• When you build a sand castle on the beach, you slowly make it larger, and then suddenly a wave destroys it all.
• Forests grow slowly, and once in a while a fire will wipe out large sections of forest.
• Stress slowly builds up in rock faults, and then an earthquake will suddenly release a large amount of stress.

In games we see this same pattern. Think of how many games you've seen that start off slow, build up, and then have a “boss” fight, followed by the relative calm of the next area. The difficulty or excitement might look a lot like a stock market plot.

One major difference between games and the patterns in nature is that you can't always predict the falls in nature. I've seen some games where a boss is unexpectedly followed by another boss, but usually you roughly know what's going to happen after a boss fight. In games where encounters are dynamically generated, it might be interesting for the encounters to follow a pattern like those seen in nature. More randomness with slow increases and sudden drops could make games more exciting.

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Retro art: breaking the rules#

When you play a game with retro graphics, there are some things you expect. Pixelated graphics. Sprites. Limited color palette. At first glance, Realm of the Mad God has retro graphics like many other games:

However, Alex of Wild Shadow Studios put in several things that don't fit the usual retro graphics approach. The above screenshot shows that the world map and its pixels rotate with the player, while the trees, players, and monsters stay axis-aligned (like billboards in 3d games). Weapon projectiles move in any direction, not only the usual 4 or 8 that you might expect from a game on a tile grid.

Less apparent is the variable scale. The “pixels” aren't simply magnified by a fixed amount. Each object has its own scale, which you can see in this screenshot by comparing the trees:

Something that's easy to miss is that some effects have a shower of “pixels”. In this composite of three screenshots, you can see red “blood splatter” and some green remains of something that exploded (I think):

Also easy to miss is how the water effect breaks expectations of retro game art. The sprites themselves are pixelated, but the alignment doesn't match the map. With this, Alex produces water animation. Look closely at the sand and water boundary, and you'll see that the “pixels” don't match up:

One of the neater effects in this game is the 3d dungeon walls. The ground, tops of walls, and sides of walls are all rendered with pixel art. In this screenshot you can see one of the players is behind the wall:

I drew some examples of things that could be interesting but aren't actually in the game at this time: dissolve effects (teleportation?), shadows (to show height), shearing (to show motion), rotated body parts or weapons (for improved animation), and outlines (to improve contrast or show effects like damage or buffs):

The sprite art we got from Oryx is pixel art, but in a vector graphics engine there are lots of things you can do with the sprite art. For Realm of the Mad God, Alex played with rotation, scale, alignment, and 3d to produce some neat effects that you don't normally see in games with retro graphics.

Update: [2013-07-13] Also see this blog post about the 3d art style.

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Simple map generation#

Update [2016] I have an updated interactive version of this technique here, with sample code.

My friends Alex and Rob at Wild Shadow Studios entered the TIG Assemblee competition. In the first 30 days, artists create art assets, and in the second 30 days, programmers create games with those assets. Alex and Rob decided to write an MMO in those 30 days. They asked me about generating game maps.

In the past I had generated outdoor terrain maps by using randomness, erosion, and water flow (see the Simblob Project). However there was a terrible trap in there. I spent years refining the terrain generation system, and far too little time working on combat and the game itself. I had fun, but I never finished that game.

Wary of my tendencies to work on one aspect of a game too much while neglecting everything else, I decided this time to do something simple. I started with Perlin Noise. I generated both a moisture map and an altitude map:

To ensure that some areas are dry and some areas are wet, I renormalized the moisture map to try to produce equal areas of every moisture level. I also renormalized the altitude map to produce far more flat and low lands than high altitudes, following a quadratic function. After renormalization every random map has a reasonable mix of land types.

From these I defined simple rules that assigned a vegetation. High altitudes are snowy mountaintops and low altitudes are ocean or beach. In between, the moisture determines how green or yellow the land is. Since most altitudes are not reflected in the map coloring, I added some shading so that it would be easier to see the mountains and valleys:

There were two things I wanted to add. The first was noise. I find noise to add to the visual appeal, and in this case it would also decrease the sharp boundaries between different terrain types. Compare this map to the original:

The second thing I wanted to add was wind. In many parts of the world, wind picks up moisture from the ocean and spreads it over the land. At mountains, much of this moisture is extracted from the wind and dropped as rain:

You can see this pattern in the United States, with winds out of the west dropping moisture in Oregon and California, passing over the Sierra Nevada mountains, leaving Utah and Arizona rather dry. In Australia, the winds come from the east, hit the Great Dividing Range, leaving the eastern coast wet and the interior of the continent dry. In South America, the winds come from the east but don't hit mountains until the Andes, so the rain falls over most of the continent, giving us the Amazon rainforest. I wanted the mountains in the game map to affect the moisture levels. I added a wind algorithm that spreads moisture from west to east and generated this map:

You can see that the dry areas in the northwest are now green, and the southeast, where the moisture is blocked by mountains, is dry. It seems more “realistic” but I think the players don't really notice such things.

In the demo app you can click on the minimap to see the zoomed area, change the number of wind algorithm iterations (then press Update), randomize the map seed (the Randomize button), or type in a random number seed and press Update. I find that 83980, 59695, 94400, 92697, 30628, 9146, 23896, 60489, 57078, 89680, 10377, 42612, and 29732 are seeds that produce decent maps. The demo differs a little bit from what we used in the game but the overall map shape and features are the same.

For Realm of the Mad God we exported the map and then made vegetation and monsters based on the terrain characterstics. The tree density was higher in jungles than on hills, rocks mostly were on mountains, and dead trees littered the deserts. Monsters too were terrain-specific at first, but they ended up spreading throughout the land. On top of the map we added random temple ruins and were hoping to make those areas special in some way, but ran out of time.

Looking back on this project, there were lots of things I did that don't really matter to the players. I tried to make the algorithms “scale independent” so that I could generate maps of different sizes and they'd end up with similar features, but in practice we only wanted maps of 2048x2048. The noise, which looks good in the overview map, looks awful when playing the game, so we ended up smoothing it out. I tried to make a two-step process that would generate the large scale features with one algorithm and the details with a different algorithm, and I never got that working well; I ended up just generating everything with the large scale algorithm. I had a river algorithm that carved out canyons, and I didn't get it working well enough in time for the game contest. I tried several ways to make the edges of the map into oceans (so that players would never reach the edge of the map) but none of them worked out, and players didn't seem to mind reaching the edge. The edges of the map never worked out right because bands of weirdness would form (you can see this in the demo). The simplest thing would've been to cut off the edges.

There were also lots of things I'd like to add, but probably shouldn't because I should move on to other projects. It would've been nice to create a map rating algorithm that could automatically pick out maps that look good (plenty of water, beach, mountains, variety, etc.). But for Realm of the Mad God we just looked at a bunch of maps and picked seed 72689. It was a pretty nice looking map, except the peninsula turned out to be a little frustrating for players, and we don't want the bosses to spawn on islands that players can't reach. For the expansion pack we'll probably just pick a few random seeds that look good and make maps from them. It would've been nice to automatically pick spawn points for players and monsters but those too were just placed manually. It would also be nice if there were some landmarks and regions with names so that the players could talk about where they were.

Try the game and see how different it feels to be exploring the map in the game than it does to look at it in the demo app. For example, the god monsters live on the mountains; it's much easier to see where these are on the map than to figure it out while playing the game. I think the 30 day time limit really helped keep me focused on things that would be useful for players. After the contest was over I found I was spending time on much less important things. I'm planning to stop working on the map generator for now, and only revisit it if there are specific features I need to add. The source is available under the MIT license (however the Oryx tileset is not licensed the same way so the version I put on github uses solid colors instead).

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