Previously we considered the Arc Consistency 3 and Arc Consistency 4 algorithms, which are an essential component of many constraint solvers. I’ve been using AC-4 myself for some time, and I naturally got curious as to what other improvements can be made.

Diving it, I discovered there is a huge amount of papers with new innovations are refinements on these two fundamental techniques. I’m going to attempt to summarize them there, and maybe later experiment with them in my own software, but I’m afraid this article is going to be pretty niche.

# Arc Consistency Explained

I’ve been doing a lot of experiments with WaveFunctionCollapse, which as we’ve covered before, is essentially a constraint solver turned into a procedural generator.

The underlying solver WaveFunctionCollapse came with is called Arc Consistency 3, or AC-3 for short. But AC-3 is actually one of a whole family of Arc Consistency algorithms. Today, my solver and most others uses AC-4, a more advanced algorithm. Let’s talk a bit about how those both work.

# Recursive Subdivision Variants

You are probably familiar with Recursive Subdivision – also known as Binary Space Partitioning – as a procedural generation technique. Like all the best proc gen, it’s a simple idea, that produces complex output. I’m here to discuss some variants that others have used to produce interesting results.

# Random Paths via Chiseling

I went over a previous project to randomly generate paths between points and came up with a much more efficient and versitile algorithm.

The algorithm is simple. Start with the entire area covered in path tiles, then and remove tiles one by one until only a thin path remains. When removing tiles, you cannot remove any tile that will cause the ends of the path to become disconnected. These are called articulation points (or cut-vertices). I use a fast algorithm based on DFS to find the articulation points. I had to modify the algorithm slightly so it only cares about articulation points that separate the ends, rather than anything which cuts the area in two. After identifying articulation points it’s just a matter of picking a random tile from the remaining points, and repeating. When there are no more removable tiles, you are done. Or you can stop early, to give a bit of a different feel.

I call it “chiseling” as you are carving the path out of a much larger space, piece by piece.

See on github .

# Dual Contouring Tutorial

How to create a sharp mesh from a function without even trying

In part 1 and part 2 of the series, we looked at the Marching Cubes algorithm, and how it can turn any function into a grid based mesh. In this article we’ll look at some of the shortcomings and how we can do better.

# Marching Cubes 3d Tutorial

In the first article I showed how the Marching Cubes algorithm works in 2d.

In this tutorial, I cover how it can be extended to 3d.

# Marching Cubes Tutorial

In Minecraft, you can dig in any direction – removing a block at a time with well defined edges. But other games manage to destruct terrain smoothly, without all the blockiness of Minecraft.

The following tutorial in Marching Cubes, a technique for achieving destructible terrain, and more generally, creating a smooth boundary mesh to something solid. In this series, we’ll cover 2d in this first article, follwed by 3d in the next , and Dual Contouring in the third. This last is a more advanced technique for achieving the same effect.

# Celtic Knots 1.0 – Now with Twills

I’ve just released a new update for an old blender addon, bumping the version to 1.0.

The addon can now generate a much wider variety.
Check it out on github.

# Blue Noise Particles

I’ve released a plugin generates a random arrangement of particles with a blue noise distribution. This is also known as Poisson Disk Sampling.

This distribution of particles guarantees no two particles are very near each other. It’s often considered a higher quality particle arrangement than Blender’s default uniform sampling. It’s particularly useful for organic arrangements, and randomly arranging meshes without collisions.

# Axaxaxas

I’ve released a Python based string parser on GitHub. This was part of a much more ambitious project that fell through, but I extracted the good part.

Axaxaxas is a Python 3.3 implementation of an Earley Parser. Earley parsers are a robust parser that can recognize any context-free grammar, with good support for amiguous grammars. They have linear performance for a wide class of grammars, and worst case $O(n^3)$.

The main goals of this implementation are ease of use, customization, and requiring no pre-processing step for the grammar. You may find the Marpa project better suits high performance needs.

Documentation can be found at: http://axaxaxas.readthedocs.org