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Computing minimal surfaces with differential forms

Authors:

Stephanie Wang,

Albert ChernAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 40, Issue 4

Article No.: 113, Pages 1 - 14

https://doi.org/10.1145/3450626.3459781

Published: 19 July 2021 Publication History

Abstract

We describe a new algorithm that solves a classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve. Existing numerical methods face challenges due to the non-convexity of the problem. Using a representation of curves and surfaces via differential forms on the ambient space, we reformulate this problem as a convex optimization. This change of variables overcomes many difficulties in previous numerical attempts and allows us to find the global minimum across all possible surface topologies. The new algorithm is based on differential forms on the ambient space and does not require handling meshes. We adopt the Alternating Direction Method of Multiplier (ADMM) to find global minimal surfaces. The resulting algorithm is simple and efficient: it boils down to an alternation between a Fast Fourier Transform (FFT) and a pointwise shrinkage operation. We also show other applications of our solver in geometry processing such as surface reconstruction.

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Code for the paper "Computing minimal surfaces with differential forms" presented in SIGGRAPH 2021 and published in ACM Transactions on Graphics (TOG). The code is also available via GitHub: http://www.replicabilitystamp.org#https-github-com-evastgh-minimal-current (repository.zip)

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Index Terms

Computing minimal surfaces with differential forms
1. Computing methodologies
  1. Computer graphics
    1. Animation
      1. Physical simulation
    2. Shape modeling
2. Mathematics of computing
  1. Continuous mathematics
    1. Topology
      1. Geometric topology
  2. Mathematical analysis
    1. Differential equations
      1. Partial differential equations
    2. Numerical analysis

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 40, Issue 4

August 2021

2170 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3450626

Editor:
Sylvain Paris
Adobe Inc.

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Copyright © 2021 Owner/Author.

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Published: 19 July 2021

Published in TOG Volume 40, Issue 4

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