Computing minimal surfaces with differential forms (SIGGRAPH 2021)

This is the supplementary video for the paper "Computing minimal surfaces with differential forms," where we develop an algorithm to solve the classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve.

Publication:

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

Preprint:

https://evastgh.github.io/files/mincurr_paper.pdf

Houdini source code:

https://evastgh.github.io/files/mincurr_demo_04282021.hipnc

Houdini demo video:

https://vimeo.com/543382749

Project page:

https://cseweb.ucsd.edu/~alchern/projects/MinimalCurrent/

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.

Music in the video:

"Sappheiros - Fading" is under a Creative Commons license (CC BY 3.0) https://creativecommons.org/licenses/by/3.0/

Music promoted by BreakingCopyright: https://www.youtube.com/watch?v=uhqMKpppnpo