Friday, November 29, 2013

Planet 4 - Part 01 - Making a Mesh

Introduction to this series is here.

Phase 1 is initiated.  I will adhere to a keep-it-simple approach during development.  So setting up the window went very quickly; no config files, no nothing.

The first step was to create the nodes in a spherical arrangement.  I divided the space into unit cubes and looped through them.  Where the innermost and outermost vertices of the cube was on opposite sides of the surface, I placed a node.

Each node represents a surface of about 128m×128m.  Since they're neither square nor uniform the area differs from node to node.  I experimented with different sized planets a bit and decided that radius 60 makes a medium-sized planet.  The circumference is about 48km, which is definitely much further than I have ever traveled in Minecraft.  What's the point of an infinite world, then?

Fig. 1: Radius 60 Sphere - Each node represents a 128m×128m area
Next I move the nodes so that they are no longer on the sphere, but rather arranged in a grid.  This will make noticing and understanding bugs much easier.
Fig. 2: Nodes are now aligned to a grid for debugging visualization
I reduce the size of the planet to radius 5, implement an algorithm for finding their nearest neighbours and link them up.  Ideally, I would have used Delaunay triangulation, but I decided to keep it simple instead; I have never done Delaunay triangulation before.  So I looped through the nodes and checked adjacent grid points for neighbours to link to.  The first time I only checked in the positive directions.  Figure 3 shows the inadequacy of this optimization:
Fig. 3: Added node-to-node links - My first bug is apparent
A simple fix is to just check the negative directions as well.  This does slow it down a bit though.  There is a faster way, but it is complicated, so I stuck to the simple way.
Fig. 4: Fixed the links bug
While my current algorithm does find every node that will contain a part of the sphere's surface, this causes some foreseeable problems, in that the double layer in some regions will stop the erosion algorithm from working correctly.  I take a cross section to see what it looks like (Fig.5) and see two possible solutions.  I could get rid of all diagonal links, or I could remove the extra layer before linking up the nodes.
Fig. 5: Cross section shows double layering
I decided to implement the node removal method, since this will simplify the eventual creation of the zones.  Each node that has three axis-aligned neighbours that are closer to the center is removed.  The cross section looks okay, so I switch back to the full sphere view:
Fig. 6: Removing the extra layer is not working as intended
That's no sphere!  I tried a few different techniques, with no positive results.  Eventually I realize the problem:  When I remove a node I change the circumstances of the surrounding nodes.  Near the axis-aligned planes that pass through the sphere's origin this causes that some nodes will no longer be removed.  The solution is to keep a set of nodes that need to be removed, and only remove them after all have been found.
Fig. 7: It works now
And finally I put the nodes back into their original spherical arrangement.
Fig. 8: Off the grid and back to a sphere

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