A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

168 Topology: How Are Objects Connected?

only two possible directions to go. For a 2D manifold, the neigh-

borhood around each point looks like a disk or a plane. Lines,

circles, or any simple curve are examples of 1D manifolds, while

spheres and torii are examples of 2D manifold surfaces. If a man-

ifold of dimension d can be embedded or placed in an Euclidean

space R

n

, then d ≤ n, i.e., the dimension of a manifold is always

less than the dimension of its embedding space.

In geometric modeling, we often encounter non-manifold sur-

faces, especially when we have data corrupted with noise and

outliers, say, as data obtained from a range scanner. In Figure ...