Layman's Guide to 2-Manifolds
(Saturday, July 6, 2002)

Introduction

This article is targeted at anyone with a grasp of basic 3D math and a motivation to gain a better understanding of what 2-manifolds are and how they're beneficial for graphics algorithms.

The number "2" in "2-manifold" refers to the dimensionality of the manifold surface. The term "manifold" means that this 2-dimensional surface obeys certain rules when it's embedded in a 3-dimensional space.

Some Simple 2-Manifolds


Figure 1: Basic 2-Manifolds

Figure 1 displays some familiar objects that have the good fortune to be 2-manifolds. The first thing to notice is that these 2-manifolds have no boundary. This means that there for any point we pick on the surface, we can always find a nearby point within some arbitrarily small distance.

Notice how the torus is the only object in Figure 1 that has a hole in it, yet it's still defined as a 2-manifold. In topology terms, we say that a torus has a genus of 1, which is equivalent to saying that there's a single hole through the object. All the other objects have a genus of 0 since there aren't any holes through them.


2-Manifolds with Boundary

If a 2-manifold has a boundary, then its boundary is a 1-manifold (i.e., a curve) with no boundary. We can think of this as meaning that the curve

Non-2-Manifolds


Figure 3: Some non-2-manifolds

So now that we know about all kinds of 2-manifolds, let's find out what kinds of objects aren't 2-manifolds. In Figure 3 we've got three distinct objects (although it may look like there's more): the union of two cones sharing an apex, the union of two cubes sharing an edge, and the union of two cubes sharing a face1. The shared primitives for each union are highlighted in yellow, and the cubes sharing a face are rendered with partial transparency.

The problem with these surfaces is that they don't resemble the cartesian 2D plane, no matter how we deform them.


A Novel Way of Thinking About 2-Manifolds

One way to think about a boundaryless 2-manifold to imagine yourself driving a tiny car on its surface. The car is constrained so that all four wheels must always touch the surface of the object (i.e., no car-chase jumps or wheelies). If the car is too big to drive over some indentation in the object while satisfying the contact constraints, then we can magically shrink the car to whatever size is necessary to drive into the indentation. The important part of being boundaryless is that no matter where the car drives, it can never fall off the surface of the object. If a 2-manifold has a boundary, then it's possible for the car to "fall of the edge of the world" by crossing this boundary.

Using the car metaphor, we can explain the conditions for an object to be non-2-manifold. Recall the non-2-manifold union of two cones sharing an apex. If we start out driving the car somewhere on the slope of one of the cones and attempt to drive onto the other cone, we'll find it impossible to do using only the shrinking method. As we approach the apex of the cones, we see that we'll have to shrink our car a great deal in order to fit all four wheels onto the apex. In fact, we'll have to shrink the car by an infinite amount, so that the wheels occupy a single point. But then we've lost all sense of direction, as we can't derive a unique orientation of the car unless the four wheels are is separate locations.

Further Reading

Footnotes

  1. I'm on thin rigor ice here when I declare that the union of the cubes sharing a face is a non-2-manifold. Taking the union can be defined many different ways, depending on what we want to accomplish. In this case, we'll just assume that taking the union preserves shared faces that should normally be eliminated. Or more accurately, the union minus the boundary. I define the union operation this way because I'm presenting 2-manifolds with regard to their use in computational geometry, the implementation of which will often allow some funky b-reps such as the union of two face-sharing cubes as I've defined it. Long story short, I just want to be sure that you know that it can happen, and when it does, the resulting object isn't manifold.

See Also:
Layman's Guide to Polyhedra