Generating an Approximate Trivariate Spline Representation for Contractible Domains

Abstract

We present an approach to build a bijective parameterization for a contractible domain in $\mathbb{R}^{3}$ represented by a closed triangular mesh. The boundary of the domain is first decomposed and parameterized in order to make it topologically equivalent to the unit cube. The map between the domain and the unit cube is constructed by successively designing its coordinate functions as harmonic maps on the intersection of the previous coordinate functions’ level sets. The computed map is proved to be bijective by the maximum principle. The map is also compatible to the boundary parameterization by using a reparameterization technique. A trivariate spline representation for the domain is finally found by least squares fitting to the inverse map.