Technical Report MSC-2019-13

Title: Generalized Volumetric Foliation From Inverted Viscous Flow
Authors: David Cohen
Supervisors: Mirela Ben-Chen
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Abstract: The theory of foliations emerged as a distinct field in the 1940’s. Since then, it has seen a major progress and a rapid development. This field has it origins in the study of the solution curves of ordinary differential equations, and of vector fields on surfaces. Intuitively speaking, a regular foliation is the decomposition of a manifold into immersed submanifolds, namely leaves, of the same dimension that ”fit together nicely”. In this work we propose a controllable geometric flow that decomposes the interior volume bounded by a triangular mesh into a collection of encapsulating layers, which we denote by a generalized foliation. For star-like genus zero surfaces we show that our formulation leads to a foliation of the volume with leaves that are closed genus zero surfaces, where the inner most leaves are spherical. Our method is based on the three-dimensional Hele-Shaw free-surface injection flow, which is applied to a conformally inverted domain. Every time iteration of the flow leads to a new free surface, which, after inversion, forms a foliation leaf of the input domain. Our approach is simple to implement, and versatile, as different foliations of the same domain can be generated by modifying the injection point of the simulated three-dimensional Hele-Shaw flow in the inverted domain. We demonstrate the applicability of our method on a variety of shapes, including high-genus surfaces and collections of semantically similar shapes.
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