Computing Minimal Surfaces with Differential Forms (SIGGRAPH 2021)

We compute surfaces of least areas bordered by given boundary curves. Our new geometry representation of curves and surfaces turns this non-convex surface optimization problem into a convex one.

Publication:
Stephanie Wang and Albert Chern. 2021. Computing Minimal Surfaces with Differential Forms. ACM Transactions on Graphics (TOG) Vol. 40, Issue 4, Article 113.

Paper, code, and demo are available on the project page:
https://cseweb.ucsd.edu/~alchern/proj...

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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