Mesh parameterization is a key problem in digital geometry rocessing. By cutting a surface along a set of edges (a seam), one can map an arbitrary topology surface mesh to a single chart. Unfortunately, high distortion occurs when protrusions of the surface (such as fingers of a hand and horses’ legs) are flattened into a plane. This paper presents a novel skeleton-based algorithm for computing a seam on a triangulated surface. The seam produced is a full component Steiner tree in a graph constructed from the original mesh. By generating the seam so that all extremal vertices are leaves of the seam, we can obtain good parametrization with low distortion.

When two topologically identical shapes are blended, various possible transformation paths exist from the source shape to the target shape. Which one is the most plausible? Here we propose that the transformation process should obey a quasi-physical law. This paper combines morphing with deformation theory from continuum mechanics. By using strain energy, which reflects the magnitude of deformation, as an objective function, we convert the problem of path interpolation into an unconstrained optimization problem. To reduce the number of variables in the optimization we adopt shape functions, as used in the finite element method (FEM). A point-to-point correspondence between the source and target shapes is naturally established using these polynomial functions plus a distance map.

We investigate the propagation of errors through geometric transformations, such as re°ections, rotations, similarity transformations, and projections, and also the scalar product of vectors. This means computing tolerance zones of points which undergo such transformations, if these points and the transformations themselves are given by toleranced input data.

Strain fields provide a method of deformation measurement based on physics. Using these as a tool, we can analyze deformation of objects in a measurable way. We have developed a new morphing technique based on strain field interpolation. Shape shaking and squeezing, which often happen when using linear interpolation for morphing, do not arise in our approach. We have also developed a new method to create isomorphic meshes from corresponding objects in two images. Meshes generated by this method have much fewer triangles than other methods, which greatly decreases calculation loads in the morphing process.

We present a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.