Minecraft mesh download

Supposing that we were given two such algorithms, it is not a-priori clear which method we should prefer.

Fortunately, there is a simple solution to this impasse. If we have a fast, but dumb, mesher; we can just run that on any meshes that have changed recently, and mark the geometry that changed for remeshing later on. Then, when we have some idle cycles or alternatively in another thread we can run the better mesher on them at some later point.

Once the high quality mesh is built, the low quality mesh can then be replaced. In this way, one could conceivably achieve the best of both worlds. The conclusion of this digression is that of the two of these criteria, it is item 1 which is ultimately the most important and unfortunately also the most difficult to address.

Thus our main focus in the analysis of meshing algorithms will be on how many quads they spit out. The absolute dumbest way to generate a Minecraft mesh is to just iterate over each voxel and generate one 6-sided cube per solid voxel. For example, here is the mesh you would get using this approach on a 8x8x8 solid region:. The time complexity of this method is linear in the number of voxels.

Clearly this is pretty terrible, and not something you would ever want to do. The only thing good about this approach is that it is almost impossible to screw up. For example, I was able to code it up correctly in minutes and it worked the first time compared to the culling method which took me a few iterations to get the edge cases right.

Clearly the above method is pretty bad. One obvious improvement is to just cull out the interior faces, leaving only quads on the surface.

For example, here is the mesh you get from culling the same 8x8x8 solid cube:. Practically speaking, this method is pretty much the same as the previous, except we have to do a bit of extra book keeping when we traverse the volume. The reason for this is that now we not only have to read each voxel, but we we also have to scan through their neighbors. This requires a little more thought when coding, but time complexity-wise the cost of generating this mesh is asymptotically the same as before.

The real improvement comes in the reduction of quads. Unlike the stupid method, which generates a number of quads proportional to the boundary, culling produces quads proportional to the surface area.

It is not hard to see that the culled mesh is strictly smaller than the stupid mesh and often by quite a lot! We can try to estimate how much smaller using a dimensional analysis argument: assuming that our geometry is more-or-less spherical, let be its radius. This gives an improvement of a factor of about , which is pretty good! So, if our chunks were say 16x16x16, then we might expect the culled volume to have about 16x fewer faces than the naive method heuristically. Of course, spherical geometry is in some sense the best possible case.

In the worst case namely a checkerboard world the size of the meshes produced would be identical! In practice, one would usually expect something somewhere in between.

The previous two approaches are probably the most frequently cited methods for generating Minecraft-like meshes, but they are quite far from optimal.

The last method that I will talk about today is a greedy algorithm which merges adjacent quads together into larger regions to reduce the total size of the geometry. For example, we could try to mesh the previous cube by fusing all the faces along each side together:. This gives a mesh with only 6 quads which is actually optimal in this case! Clearly this improvement by a factor of 64 is quite significant. To explain how this works, we need to use two ideas.

First, observe that that it suffices to consider only the problem of generating a quad mesh for some 2D cross section. That is we can just sweep the volume across once in each direction and mesh each cross section separately:. This reduces the 3D problem to 2D. The next step and the hard one! Doing this optimally that is with the fewest quads is quite hard. The idea is that we are going to define some type of total order on the set of all possible quadrangulations, and then pick the minimal element of this set as our mesh.

To do this, we will first define an order on each quad, which we will then extend to an order on the set of all meshes. One simple way to order two quads is to sort them top-to-bottom, left-to-right and then compare by their length. Given such a representation of a quad, we denote its underlying set by:. By a 2D quad mesh, here we mean a partition of some set into a collection of quads. The next thing we want to do is extend this ordering on quads to an ordering on meshes, and we do this in a very simple way.

Given two sorted sequences of quads such that and and , we can simply compare them lexicographically. Again, this new ordering is in fact a total order, and this means that it has a unique least element. Here is an example of what one of these lexicographically minimal meshes looks like:.

You may be skeptical that finding the least element in this new order always produces a better quad mesh. For example, there is no guarantee that the least element in this new order is also the mesh with the fewest quads. Consider what you might get when quad meshing a T-shape using this method:. Exercise 2: Find a mesh of this shape which uses fewer quads. However, we can say a few things. For example, it always true that each quad in the greedy mesh completely covers at least one edge on the perimeter of the region.

Therefore, we can prove the following:. Proposition: The number of quads in the greedy mesh is strictly less than the number of edges in the perimter of the region.

This means that in the worst case, we should expect the size of the greedy mesh to be roughly proportional to the number of edges. But we can be more precise about how good the greedy mesh actually is:. Theorem: The greedy mesh has no more than 8x as many quads than the optimal mesh. Here is a picture, with the convex vertices colored in red and the concave vertices colored in green:.

It turns out that if you sum them up, you always get the same thing:. Proposition: For any simply connected region with convex and concave vertices on the perimeter,.

This can be proven using the winding number theorem from calculus. Lemma: Any connected genus region with a perimeter of edges requires at least quads to mesh. Because each quad can cover at most 4 convex vertices, the number of quads, , in the mesh is always at least.

By the winding number theorem, it is true that for simply connected regions , and so. Therefore any mesh of a simply connected region requires at least quads. To extend this to non-simply connected regions, we observe that any quad mesh can be cut into a simply connected one by slicing along some edges in the quad mesh connecting out to the boundary. Making this cut requires introducing at most two edges per each hole in the object, and so we could just as well treat this as a simply connected region having edges, and applying the previous result gives us the bound in the lemma.

To prove our main theorem, we just combine these two lemmas together, using the fact that. In summary, we have the following progression:. The idea as in all greedy algorithms is to grow the mesh by making locally optimal decisions. To do this, we start from an empty mesh and the find the quad which comes lexicographically first. The complexity of this method is again linear in the size of the volume, though in practice it is a bit more expensive since we end up doing multiple passes.

If you are the type who understands code better than prose, you can take a look at this javascript implementation which goes through all the details. To try out some of these different methods, I put together a quick little three. Click here to try out the Javascript demo!

As you can see, greedy meshing strictly outperforms naive meshing despite taking asymptotically the same amount of time. However, the performance gets worse as the curvature and number of components of the data increase. This is because the number of edges in each slice increases.

In the extreme case, for a collection of disconnected singleton cubes, they are identical and optimal. Also I would ask the reader to excuse the fairly rough complexity analysis here. I think the real weakness in my analysis lies in the computation of the upper bound on the size of greedy mesh. Conjecture: The size of the greedy mesh is at most. If this conjecture holds true, then it would reduce the current bounds on the approximation factor to 4x instead of 8x.

It is also worth pointing out that the time complexity of each of these algorithms is optimal ie linear for a voxel world which is encoded as a bitmap. However, it would be interesting to investigate using some other type of data structure. In the previous post, I talked about using run-length encoding as an alternative in-memory storage of voxels. But greedy meshing seems to suggest that we could do much better: why not store the voxels themselves as a 3D cuboid mesh?

Doing this could drastically reduce the amount of memory, and it is plausible that such a representation could also be used to speed up meshing substantially.

Of course the additional complexity associated with implementing something like this is likely to be quite high. Open Problem: Do there exist efficient data structures for dynamically maintaining greedy meshes? Was written x,y,w,h when it should have been y,x,w,h Thanks Matt! The right-most quad second from the top is not lexicographically minimal the bottom part could be extended.

I added your test case to the demo to show why, though here is a picture to show what goes wrong:. If it had just one more quad it would be a valid counter example. As a quick check, try flipping that example over, and rotating it. Ah, I think I see it. I made a mistake when I was writing it up. But at least this explains where I messed up! There are certainly situations where overlapping quads can reduce your quad count eg the X pentomino. Is this because it makes the problem more complicated, or because there are other problems with overlapping quads?

I have indeed thought about it. While it is true that grouping occluded quads could reduce the face count, in practice it does have some drawbacks, the most noticeable of which is that it could drastically increase the amount of overdraw. If you ever try it, let me know what your results are! For instance, red, green and blue voxels, or multiple voxel textures.

I had an implementation, but it took a lot of time and processing power compared to yours. Say you break a voxel: an efficient way to handle the change would be to remove the vertices and indices associated with that voxel and add vertices and indices associated with the neighbouring voxels.

Your implementations do not consider clockwise or counter-clockwise culling for the faces, which is needed for any practical use. Thank you for your comments! Then when you build the greedy quad, you store the type of the block you are scanning over and only group blocks which are the same type together. If there is enough demand I could put together another demo showing how this works for 3 block types for example. What I am more concerned about is the possibility that there might be a bug in the code.

Can you take a screen shot of a situation where it fails? It is merging faces without regard to the normal direction. This is actually what it is supposed to do, though the effect is a bit unintuitive.

Though I think a Delaunay triangulation is probably not quite the right thing to do. I believe you are really thinking of this problem:.

The most standard way to do polygon triangulation is to perform a monotone subdivision of the volume, then triangulate that. Unfortunately, this would end up being slower than the method in the article; but perhaps the meshes would be small enough that you still gain some advantage.

Unfortunately, polygon triangulation can be quite difficult. For example, the only known O n algorithm for triangulating a simple polygon is due to Chazelle — and it is so complicated that it has never been successfully implemented! This fact is a long standing joke amongst computational geometers. But then again, these polygons we are dealing with are not general simple polygons, and I would not be surprised if there are much simpler ways to do things in the case of Minecraft.

It is certainly something interesting to think about. How could I add some smoothness to the mesh, other than tessellating the mesh after the generation? Hi Alejandro, I am glad you asked this question! I am currently working on a post describing how to deal with smooth surfaces. I will hopefully have it ready and posted shortly! There is some sample code in the demo. Therefore filling everything with one big quad might be a really bad solution even if it reduces the number of vertices.

That is why you would only merge two quads if the lighting values at their vertices are the same. Most likely an issue with three. These models are all so small that differences in rendering time should be inconsequential with the possible exception of the stupid meshers. Robot Technology Trees Vehicles Woman. Special Offer! Off : : : Selected items only.

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