Kernal
Joined: 01 Jul 2007 Posts: 16

Posted: Fri Apr 22, 2011 6:29 pm Post subject: Curvature of World Geometry 


As far as I am aware, every MUD has either no formal geometry or exists in a flat geometry. While these are likely to be most convenient in terms of coding ease, it occurs to me that there are better options from a game design perspective.
First a word on "Curvature":
For the sake of simplicity, I'll discuss this in terms of twodimensional planes. Not only is it easier to imagine, but most MUDs are set in effectively 2dimensional worlds anyway, so this has reasonable application. Here's an image to help visualize: http://abyss.uoregon.edu/~js/images/universe_geometry.gif
A flat geometry, or one with zero curvature, is a simple plane like a sheet of paper. This is the geometry we are typically familiar with, but not the only one we can imagine. Consider instead the surface of a sphere; this is also a welldefined surface of constant curvature, but now its curvature is positive. A negative curvature is also possible, but harder to discuss in this context.
My preferred method of thinking about curvature is in terms of these simple toys: http://cgi.ebay.com/ws/eBayISAPI.dll?VISuperSize&item=180644073958
These are plastic equilateral triangles with links on their edges so they can snap together. If we place six of these triangles around every vertex, we'll get a flat plane of zero curvature. If we place only five of these triangles around every vertex, we'll get a surface of positive curvature; if we keep building, we'll eventually construct a large sphere. Finally, if we place seven of these triangles around every vertex, we'll get a surface of negative curvature.
One interesting property of these surfaces is that the circumference of a circle varies. On a flat surface, we know the circumference of a circle to be 2*pi*r, where r is the radius (measured along the surface). On a sphere, some consideration should reveal that the radius of the circle is in fact less than 2*pi*r. Interestingly, on our surface of negative curvature the radius of a circle is greater than 2*pi*r; in fact exponentially so.
What Geometry do MUDs have?
Many MUDs have no geometry at all, although flat is usually implied. These MUDs have distinct rooms connected by their exits in ways that may (or may not) make sense when interpreted to lie in a flat geometry. This interpretation is what mapping programs tend to do.
The remaining MUDs have flag geometries; if you can print a map on a rectangular, cartesian grid then the geometry is probably a flat geometry. These MUDs do in fact have such maps (printed or not), at least for parts of the world.
What Geometry should MUDs have?
Well, flat geometries are the simplest to code, so that is an obvious and reasonable choice. Ignoring this convenience factor, however...
A spherical geometry (positive curvature) is the next obvious choice  this is appropriate for any world set on a physical planet, as the geometry will in fact define a spherical surface. Walking in any direction will lead you back to where you started after the same distance travelled.
A hyperbolic geometry (negative curvature) has no realworld analog, but may be the best suited to a MUD world nonetheless. Recall the "circumference of a circle" analog. With the reasonable assumption that "rooms" (or other representations of constant area) are equally spaced everywhere, this means that as you move away from a singular location there's an exponentially increasing set of locations you could end up, rather than a linearly increasing set of locations. The walking distance (and time) required to access any point increases logarithmically rather than polynomially, which could be a great boon for large worlds.
Summary:
A hyperbolic geometry in a MUD allows for a world with an area that scales exponentially with walking distance, allowing many exploration options with minimal travel time.
Cheers,
Kernal 
