מירלה בן-חן, הרצאה סמינריונית לדוקטורט
Geometric modeling deals with representing real objects in a virtual world. A popular geometric representation is the polygonal model. For many applications the need arises to improve the model while preserving its intrinsic geometric property, that it still describes the same "real" object. In this talk, we present a few applications, which although are quite different, use similar geometric concepts and mathematical machinery, the most noticeable being the notion of conformality as a change to the shape which preserves its essence.
First, we consider the problem of planar mesh parameterization. We show how a simple discretization of a classical equation for conformal maps on continuous surfaces can be applied to generate high quality planar mappings in an efficient manner. In addition, we show how the conformal factor, a function on the mesh, which is related to the local scaling the surface should undergo in order to be flattened to the plane, can be used a shape signature, for shape matching and retrieval.
Finally, we address the problem of 3D shape editing and deformation, useful in animation applications. Deformation tasks are extremely time consuming, so the challenge there is "say less, do more", the user should specify as little as possible, and the deformation method should deduce the rest. Here, again, conformal and harmonic maps play an important role, as they allow the user to modify the global shape, while preserving the small details. We describe a number of algorithms which translate to user-friendly methods to intuitively and naturally deform shapes.