In this project we introduced a 3D shape representation that is based solely on mesh connectivity -- the connectivity shape. Given a connectivity, we defined its natural geometry as a smooth embedding in space with uniform edge lengths and describe efficient techniques to compute it.

In the Figure above we see the well-known polygonal mesh of a cow (a). Ignoring the geometry, we have mapped the cow's connectivity onto the unit sphere (b), where the different densities hint to the features of the cow. In (c) the corresponding connectivity shape is shown. It is a smooth embedding with uniform edge lengths of the connectivity graph of (a) and (b) in three dimensional space.
Imagine all edges of the cow being springs of the same equilibrium length. In the embedding (b) we forced the spring system into a high energy state. In (c) we released all vertices and the spring system relaxed into a low energy state, with more or less uniform edge lengths. This is the connectivity's natural shape. More poetically, the sphere embedding in (b) has the body of a sphere, but the soul of an animal. The natural embedding in (c) reveals the geometric soul of the cow's connectivity.
Furthermore, we show how to generate connectivity shapes that approximate given shapes. This is done by (re)meshing the given shape with uniform edge lengths. For example, the connectivity shape in (d) bears a striking resemblance to the original (a). The only information in this mesh is its connectivity, in the sense that it induces the mesh geometry.
Finally we describe applications of connectivity shapes to modeling, mesh coding, graph drawing, and connectivity creatures . See also our project webpage at http://www.cs.unc.edu/~isenburg/connectivityshapes/ .