Discrete Minimal Surfaces: Critical Points of the Area Functional from Integrable Systems

Oriented Mixed Area and Discrete Minimal Surfaces

Recently a curvature theory for polyhedral surfaces has been established which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed ...

From Discrete Integrable Systems to Cellular Automata

Smooth surfaces from bilinear patches: Discrete affine minimal surfaces☆

Motivated by applications in freeform architecture, we study surfaces which are composed of smoothly joined bilinear patches. These surfaces turn out to be discrete versions of negatively curved affine minimal surfaces and share many properties with their classical smooth counterparts. We present computational design approaches and study special cases which should be interesting for the archite...

The Plumbing of Minimal Area Surfaces

⋆ Partially supported by the National Science Foundation; Alfred P. Sloan Research Fellow. † Permanent address: Center for Theoretical Physics, MIT, Cambridge, Mass. 02139. Supported in part D.O.E. contract DE-AC02-76ER03069 and NSF grant PHY91-06210. We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than ...

Minimal Discrete Curves and Surfaces

In this thesis, we apply the ideas from combinatorial optimization to find globally optimal solutions arising in discrete and continuous minimization problems. Though we limit ourselves with the 1and 2-dimensional surfaces, our methods can be easily generalized to higher dimensions. We start with a continuous N -dimensional variational problem and show that under very general conditions it can ...