Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt results from the book authored by Anderson-Vamanamurthy-Vuorinen [AVV6]. Part 1. Quasiconformal maps an...

The manipulation of surface homeomorphisms is an important aspect in 3D modeling and surface processing. Every homeomorphic surface map can be considered as a quasiconformal map, with its local non-conformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. Efficient and ac...

It has been long observed that area-preserving maps and reversible maps share similar results. This was certainly known to G.D. Birkhoff [5] who showed that these two types of maps have periodic orbits near a general elliptic fixed point. The KAM theory, developed by Kolmogorov-ArnoldMoser for Hamiltonian systems [9], [1] and area preserving maps [15], has also been extended a great deal to rev...

The paper proposes a fully automatic approach for finding a mapping between two nonisometric surfaces. The approach is to blend low-dimensional conformal maps with weights varying smoothly over the surface. The advantage of the approach is that it can handle a wide variety of deformations while keeping the search space of the possible maps polynomial. The paper first details how to find candida...

Computational conformal geometry is an interdisciplinary field, combining modern geometry theories from pure mathematics with computational algorithms from computer science. Computational conformal geometry offers many powerful tools to handle a broad range of geometric problems in engineering fields. This work summarizes our research results in the past years. We have introduced efficient and ...

Introduction A map : M ! N of Riemannian manifolds is harmonic if it extremises the energy functional: Z jdj 2 dvol on every compact subdomain of M. Harmonic maps arise in many diierent contexts in Geometry and Physics (for an overview, see 15,16]) but the setting of concern to us is the following: take M to be 2-dimensional and N to be a Riemannian symmetric space of compact type. In this case...

Continuous conformalmaps are typically approximated numerically using a trianglemeshwhich discretizes the plane. Computing a conformal map subject to user-provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’.We address themore general case of non-triangular elements, and provide a complete analysis of the casewhere the plane is discretized using...