Can Mean-Curvature Flow be Modified to be Non-singular?

Authors:
Michael Kazhdan;Jake Solomon;Mirela Ben-Chen
Affiliations:
Johns Hopkins University;Stanford University;Hebrew University
Venue:
Computer Graphics Forum
Year:
2012

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Abstract

This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the numerical instability in the discretization and show that this modification results in a simpler expression for both the discretized and continuous formulations. We discuss the properties of the modified flow and present empirical evidence that not only does it define a stable surface evolution for genus-zero surfaces, but that the evolution converges to a conformal parameterization of the surface onto the sphere. © 2012 Wiley Periodicals, Inc.