# Projects

## Nonmanifold Subdivision

Commonly used subdivision schemes require manifold control meshes and produce surfaces which are manifolds. In this project, we design a subdivision algorithm that makes it possible to model nonmanifold surfaces. Our approach is to extend existing subdivision rules to handle a more general class of control meshes. In fact, we allow any triangle mesh as input, as long as for any polygon no two vertices of this polygon coincide. Our scheme extends the well-known Loop subdivision scheme, but the same approach can be applied with few changes to extend any subdivision scheme to nonmanifold control meshes. Additional data assigned to vertices and edges of the mesh allows one to specify how patches are joined together.

We should note that non-manifold surfaces form a very broad class; we restrict our attention to surfaces which have well-defined tangent planes everywhere, possibly several tangent planes at singular features such as creases and nonmanifold curves. For such surfaces, at any point behavior can be described up to the first order by a collection of tangent planes of manifold surface patches meeting at the point. Thus, if we can ensure that a certain part of the surface has prescribed tangent plane at a point, we should be able to model any desired first-order surface behavior.

- Nonmanifold Subdivision
- Lexing Ying, Denis Zorin
*Proceedings of IEEE Visualization 2001.*

**Examples**

Quite often, we want to model the surface where several manifold patches attaching together at one point. The following pictures, we use normal modification at the common point to achieve it. The left pictures are the outlook, while the right pictures are cross-sections.

The following examples demonstrate the case where several manifold patches attaching along a singular curve. Tangent modification is also applied in the last two model to achieve tangent continuity along the singluar curve. Please note the smoothness of the surface in the neighborhood of the endpoints of the singular curve.

Here, we display three different cups, where we specify different patches to have same tangent plane along singular curve. The first row contains the outlooks of three models and the second row contains the cross-sections.

Two cylinders connected by two rectangluar patches. The left picture is the outlook, and the second picture is the cross-section. Notice the smoothness connection at the singular curve in the right picture.

The human heart model. This is a typical biological model with nonmanifold structures. Different muscle layers are abstracted as different manifold patches and these patches are connected along singular curves.

The X29 plane model. The original model is modified to have correct nonmanifold connectivity. It has nonmanifold structure at the cockpit and the places where the wings are connected to the plane body. The left picture is the outlook of the plane, the middle one is the cross-section at the wings. The right one is the cross-section at the cockpit.