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.
