| In this
research we focus on detecting and computing loops on surfaces that are topologically
and geometrically meaningful. We define handle and tunnel loops on surfaces
using homology groups. Intuitively, handle loops can be contracted to a point
in the interior of the shape and cannot be contracted to a point in the exterior
of the shape. Tunnel loops are similarly defined in a complementary way. See
the paper below for precise definitions.
T. K. Dey, K. Li, and J. Sun. On computing handle and tunnel loops. IEEE Proc. NASAGEM 2007, to appear.
We compute handle and tunnel loops for a class of surfaces that are graph retractable (versus surfaces that are "knotted"). The interiors and exteriors of a graph retractable surface retract to inside and outside curve skeletons respectively. They are called core graphs based on which handle and tunnel loops are computed. The curve skeletons are generated by an algorithm described in Defining and computing curve skeletons with medial geodesic function.
We characterize handle and tunnel loops on graph retractable surfaces in terms of their linking with the core graphs. This characterization gives detection and generation algorithms for the loops. Our algorihtm incorporates geometry into the topological algorihm that enable our software to generate ``very good" handle and tunnel loops (as shown below).
We also apply handle and tunnel loops to feature detection and topological simplifications. For example, handle / tunnel features could be obtanined by sweeping the initial handle / tunnel loops in both directions for some distance. To remove insignificant topologies, one can further cut / fill the corresponding handle / tunnel features. Examples are shown below.
The HandleTunnel software based on this result is available but we recommend the new HanTun software
based on a follow-up result.